A characterization of wavelet convergence in sobolev spaces

A Characterization of Wavelet Convergence in Sobolev Spaces Mark A. Kon1 Boston University Louise Arakelian Raphael2 Howard University Dedicated to Prof. Robert Carroll on the occasion of his 70th birthday. Abstract We characterize uniform convergence rates in Sobolev and local Sobolev spaces for multiresolution analyses. 1. Introduction and definitions In [KR1] it is shown that convergence rates of wavelet and multiresolution expansions depend on smoothness of the expanded function  . Specifically, if is not larger than a fixed parameter and   / then the error of approximation is 6²2c² c°³ ³, with  dimension and  the number of scales used in the expansion. This result is expected (see [Wa]) and comparable to Fourier approximation orders. In this paper we study a very different phenomenon which occurs for function spaces beyond a certain degree of smoothness. In these cases the rate of convergence “freezes” and fails to improve, no matter what the smoothness of  . Such behaviors have been studied in the context of approximation theory. We show here that the smoothness level at which such freezing occurs depends on the wavelet or scaling function in a well-defined way, and that it more generally depends on the reproducing kernel of the multiresolution analysis (MRA). This completes a characterization of pointwise convergence rates in Sobolev spaces for general MRA's begun in [KR1], to include Sobolev spaces / with large. In addition (Theorem 8), we extend these results to local Sobolev spaces which are related to spaces with uniformly bounded derivatives. See [Ma, Me, Wa] for some results on 3 and 3B convergence rates of -regular multiresolution expansions. 1Research partially supported by the National Science Foundation, Air Force Office of Scientific Research, and U.S. Fulbright Commission 2Research partially supported by the Air Force Office of Scientific Research and the National Security Agnency 1991 AMS Mathematics Subject Classifications: primary 42C15; secondary 40A30 1 In [KR1] it is assumed wavelets  or scaling functions  have sufficient regularity that the regularity of  is the limiting factor in convergence rates. Here we assume  has sufficient regularity, and show that limitations on approximation rates then depend on regularity properties of  or . We indicate more generally how the interaction of the regularities of  and of  limits convergence. This complements results (see [D2, Ma, Me]) which relate convergence rates for functions  to exact characterizations of function spaces. These results rely on sufficient regularity for  , and so do not give information when wavelets have lower regularity relative to  . Our conditions on behavior of wavelets near the origin are more precise versions of Strang-Fix type conditions ´SF]. They can be translated into moment conditions on wavelets in the case that the moment powers are integers; see also [KR2]. For detailed definitions and theory of an MRA we refer to [D2]. An MRA is defined as an increasing sequence of subspaces ¸= ¹ of 3 ²l ³ ( ‚ ³ such that  ²%³  = iff  ²%³  =b , the intersection of the spaces is ¸¹, the closure of their union is all of 3 , and = is invariant under translations of integers. It is also generally assumed (though we do not require it here) that there exists a function ²%³ (the scaling function) whose integer translates form an orthogonal basis for = . Let > be the orthogonal complement of = in =b , i.e., > = =+1 m = , so that =+1 = = l > . From existence of  it follows (see, e.g., [D2], [Me], [Wo]) that there is a set of  (%) – basic wavelets ¸ (%)¹$ (with $ a finite index set) such that  °  ² % c ³ (  t,   td ) form an orthonormal basis for > for fixed , and form an orthonormal basis for 32 (ld ) as Á Á  vary. Our results will hold for any wavelet set ¸ ¹ related to > whose translations and dilations form an orthonormal basis for 32(ld), regardless of how they are constructed (see [D2], Ch. 10; [Me]; [HW]). It follows from the above definitions that there exist numbers ¸ ¹t such that the scaling equation B  ²%³ ~   ²% c ³ (1À1) ~cB holds. We define  ²³ –   c  B (1.2) ~cB V²³ ~  ²°³ V²°³, where V denotes to be the symbol of the MRAÀ Note it satisfies  Fourier transform. Our convention for the Fourier transform is V²³ – < ²³²³ – ²³c°  l ²%³ch% % where  h % ~ %À Definitions 1.1: We define 7 and 8 to be the 3 orthogonal projections onto = and > , respectively, with kernels (when they exist) 7 (%,&) and 8 (%,&). We define 7 ~ 7 . 2 Given   32 , (i) the multiresolution approximation of  is the sequence ¸7  ¹ ; (ii) the wavelet expansion of  is   ²%³ —  ,   (1.3a)   the 32 expansion coefficients of  , and — denoting convergence in 3 ; with  (iii) the scaling expansion of  is   ²%³ —  ,   ²%³ b     (1.3b) ‚  are 32 expansion coefficients, and  ²%³ ~ ²% c ³À where the  ,   We say such sums converge in any given sense (e.g., pointwise, in 3 , etc.) if the sums are calculated in such a way that at any stage in the summation there is a uniform bound on the range (largest minus smallest) of  values for which we have only a partial sum over , . Definitions 1.2: A multiresolution analysis (MRA) or family of wavelets  yields pointwise order of approximation (or pointwise order of convergence) €  in H r if for any f  H r , the th order approximation 7  satisfies +7  c  +B ~ 6²c ³Á ²1À4³ as  tends to infinity, if  c ° €  (if  c °   the left side of (1À4) is in fact infinite for some  ). It yields best pointwise order of approximation (or convergence) €  in /  if s is the largest positive number such that (1À4) holds for all f  H  À If the supremum of the numbers for which (1À4) holds is not attained, then we denote the best pointwise order of convergence by c À The MRA yields optimal pointwise order of approximation (or convergence) if is the best pointwise order of approximation for sufficiently smooth  , i.e. for   /  for sufficiently large . Thus this order of convergence is the best possible order in any Sobolev space. We say ~ B if the best order of approximation in /  becomes arbitrarily large for large . By convention best order of approximation  means that the supremum in (1À4) fails to go to 0; thus ‚  by our definitions. We remark that our use of the term best approximation order differs from the term best approximation as used, e.g., by Singer [Si]. In addition the word best is used for technical reasons associated with the formulations of our statements. Specifically, in this paper an expansion has order of approximation if the optimal exponent in (1À4) is or better, while it has best order of approximation if the optimal exponent is and no larger than . Definitions 1.3: The Sobolev space / s is defined by 3 / s – F   32 (l ): ”  ” s – n O ^ ()O 1 + || !s  < B G. The homogeneous Sobolev space is: /s – F   32 (l ): ”  ” Á – n O^ ()O ||2  < B G. Note the spaces contain the same functions (by virtue of the fact that / is restricted to 3 )À Only the norms differ, and the second space is incomplete as defined (its completion contains non-3 functions which grow at B³À Definitions 1.4: A function  (%) on ld is radial if  depends on |%| only. A real valued radial function is radial decreasing if O (%)O  O (&)O whenever |%| ‚ |&|. A function  ²%³ is in the radially bounded class [RB] (c.f. [GK1,2]) if it is absolutely bounded by a positive 31 radial decreasing function (%), i.e., (% ³ ~ ²% ³ when O% O ~ O% O Á with (%1 )  (%2 ) whenever |%1 | ‚ |%2 |, and (%)  31 (ld ) (note we assume is defined and finite at the origin, so that  must be bounded). Less general forms of the following two theorems were announced in [KKR2, KR2]; Theorems 1 through 4 were proved in [KR1] (see http://math.bu.edu/people/mkon/). These theorems say that under mild assumptions on the MRA (i.e., the scaling function or wavelets have a radially decreasing 3 majorant), for   / ²l ³, the rate of convergence to  of the error P c 7  PB has sharp order c² c°³ À We emphasize that the conditions in Theorems 1 through 4 are equivalent. Theorem 1 [KR1]: Given a multiresolution analysis with either (i) a scaling function   [RB], (ii) basic wavelets   [RB] or (iii) a kernel 7 ²%Á &³ for the basic projection P satisfying |P²%Á &³|  H²% c &³ with H  [RB], then the following conditions (a to b) are equivalent for € °, with  the dimension: (a) The multiresolution approximation yields pointwise order of approximation s c ° in H s. (aZ ) The multiresolution approximation yields best pointwise order of approximation s c d/2 in H s . (aZZ ) The multiresolution approximation yields best pointwise order of approximation r c ° in H  for all °  r  À (b) The projection 0 c Pn : Hhs ¦ LB is bounded, where 0 is the identityÀ Theorem 2 is related to the vanishing moments property of the wavelets  À Theorem 2 [KR1]: Under the assumptions of Theorem 1, if there exists a family ¸ ¹ of basic wavelets corresponding to ¸Pn ¹ with  (x)  [RB], then the following conditions are equivalent to those of Theorem 1: 4 (c) For every such family of basic wavelets and each ,   /c , the dual of / À (cZ ) For every such family of basic wavelets and for each  , ^ | ()|2 ||c2s d < B ²1À5³ ||<  for some (or for all)  > 0. (cZZ ) For some such family of basic wavelets, (1À5a) holds. Definition 1.5: We define the space < / to be the Fourier transforms of functions in / , with the analogous definition for < / À Theorems 3 and 4 are related to the so-called Strang-Fix conditions on the scaling function and the low pass filter  . Theorem 3 [KR1]: Under the assumptions of Theorem 1, if there exists a scaling function  corresponding to ¸Pn ¹, the following conditions are equivalent to those of Theorems 1 and 2: V|  < / c . (d) For every such scaling function, 1 c (2) /2 |  (d Z ) For every scaling function   [RB] corresponding to ¸Pn ¹ ^ ()|) ||c2s d < B ( 1 c (2)°2 | ²1À5³ ||<  for some (or all)  > 0. (d ) For some scaling function  corresponding to ¸Pn ¹, (1À5) holds. (d ZZZ ) For every scaling function   [RB] corresponding to ¸Pn ¹ ZZ ^ ( + 2M)|2 ||c2  < B  | ²1À5³ ||<  M£0 We define - = ¸0,1¹d to be the set of all -vectors with entries from the pair ¸0,1¹. Theorem 4 [KR1]: If m ²³ is a symbol of a multiresolution expansion corresponding to a sequence of projections 7 as in Theorem 1, the following conditions are equivalent to those in Theorems 1-3 for € ° ¢ (e) For every symbol m () corresponding to ¸Pn ¹, ||<  (1 c |0 ()|2 ) ||c2  < B for some (or all)  > 0 (eZ ) For some symbol m0 () corresponding to ¸Pn ¹, (1À5 ) holds. (ZZ ) Every (or some) symbol m () corresponding to ¸Pn ¹ satisfies 5 ²1À5³ |c|<  |m ²³|2 | c |c d < B ²1À5³ for some (or all)  >  and for every   - Z , where we define - Z – - ±¸¹, ²1À6³ where - ~ ¸Á ¹ and  denotes the zero vector in ²1.6³. For the remainder of the paper, we assume the following: Assumptions: We assume in all of the following theorems that one of the following holds: ²³ The projection 7 onto = satisfies O7 ²%Á &³O  /²% c &³ for some /  [RB]À ²1 À 7 ³ If a scaling function  exists, ²³   [RB]. If a wavelet family  exists, ²³  ²%³²ln² b O%O³  [RB] for all . Remark: It is shown in [KKR1] that ²³ ¬ ²³ and ²³ ¬ ²³À This follows from the representations of 7 ²%Á &³ in terms of sums involving  or  when they exist. Note that the condition on  in ²³ is somewhat stricter than that required for Theorems 1 through 4 above. It is required for existence of a kernel 7 ²%Á &³ for the projection 7 satisfying O7 ²%Á &³O  /²% c &³, with /² h ³ a radial decreasing 3 function. This class of wavelets includes all r-regular wavelets (see [Me]) for any r ‚ 0. The assumptions are also needed for appropriate 3 and a.e. convergence properties of wavelet expansions [KKR1]. Theorems 1 - 4 apply only to expansions of functions in Sobolev spaces / for which , ~ 0 c 7 ¢ / ¦ 3B is bounded (see (b) of Theorem 1). They say nothing about the case of unbounded , . We show here that for larger (for which , is unbounded), approximation rates are essentially the same as for the largest for which , is bounded. Details of the approximation rates, however, depend somewhat delicately on the wavelets or scaling function. Before giving an overview of our new results in Theorem 5, we refer the reader to formulas ²1À5a, b, d³ as motivation for the following definition. Definitions 1.6: We define for Á  ‚ 0 6 0 ( ) – 1‚||‚c 2 ( ) – sup  3 ( ) – ^ ()|7 ||-2s d 6 c ²³° |  1‚||‚c 1‚||‚c V ²³O ||-2s d O (1 c |0 ()|2 ) ||-2s dÀ In this paper an often-used consequence of Theorems 1 through 4 is the existence of a least upper bound  (best Sobolev parameter), depending only on the MRA, for which ²³ of Theorem 1 holds. This motivates the following definition. Definition 1.7: The best Sobolev parameter  of an MRA is  = sup ¸ €  | (0 c 7 ): / ¦ 3B is bounded¹. By convention  ~  if the set in the supremum is empty. Some bounds on  follow from Theorem 1 above: Proposition 1.8: If the best Sobolev parameter  £ , then  € °, and the set ' – ¸ €  | (0 c 7 ): / ¦ 3B is bounded¹ satisfies ' = ²°, µ or ' = ²°, ). ²1À8³ Proof: Assume  £ Á so that  €  (recall  ~  means there is no positive order of convergence). Under any one of the assumptions (1.7), the kernel 7 ²%Á &³ of the projection onto = is bounded by 2²% c &³, with 2  [RB]. Thus 2 is bounded and in 3 , and hence 2²% c &³  3 ´&µ and is bounded in 3 ´&µ, uniformly in %. Thus for   32 , 7   3B À ° But for every nonnegative integer , there exist unbounded   / . For such  , ° , ~ ²0 c 7 ³ ~  c 7  is not in 3B , and so , ¢ / ¦ 3B is unbounded. Similarly, , ¢ / ¦ 3B is unbounded for    °. Thus if  £ , i.e., if , is bounded for some € , then , must be bounded for an € °. Therefore , the supremum of for which , is bounded, must satisfy  € °. To prove statement (1.8) we need only show ' is connected. This on the other hand follows by the equivalence of ²Z ³, ²ZZ ³, and ²³ of Theorem 1, showing ' ~ ²°Á ³ or ' ~ ²°Á µÀ From Theorems 1 and 4 we then have immediately Proposition 1.9: If the best Sobolev parameter  £ , then 7  ~ sup ¸ €  | 0 ²³  B¹ ~ sup ¸ €  | 2 ²³  B¹ ~ sup ¸ €  | 3 ²³  B¹. In Theorems 1-4,  is important in that all statements hold only if  . For approximation rates in / , we prove the following theorem. This summarizes convergence rates in all / in terms of properties of the projections 7 or of integrals involving the wavelets or scaling functions. Theorem 5: Given a multiresolution approximation ¸P ¹, ²³ If  ~ , there is no positive order of approximation for the MRA ¸7 ¹ in any / ,  lÀ If ²³ does not hold then  € ° and: (i) For    °, the best pointwise order of approximation in / is ; (ii) If °  s  , the best pointwise order of approximation in H is  ~ s c d/2; (iii) If s = , the best pointwise order of approximation in /  c  / if 0 (0)  B  ~H c ( c /2) if 0 ²0³ ~ B (iv) If € , the best pointwise order of approximation in /  c ° if 0+1/2 ( ) = 6(1° ) ( ¦ ³ ~H  c ( c /2) otherwise (v) In (iii) and (iv) above, 0 ²³ can be replaced by 2 ²³ or by 3 ²³. is is Another way to say (iv) is that if € , then there exists   / ²l ³ such that sup ² c°³ P c 7 PB ~ BÀ This says the convergence rate cannot be improved for  functions belonging even to very smooth Sobolev spaces. Moreover we note that the value  b ° used above in (iv) is not crucial for its statement. Equivalent conditions to those in (#) exist in the form 0 b° ²³ ~ 6² c ³ for any (or all)  € . In terms of the Sobolev order of the expanded function  and the best Sobolev parameter  of the MRA, the following diagram gives rates for an MRA expansion in any Sobolev space (or local Sobolev space; see below). The rates on the boundary region ~  in (iii) above are not indicated in the diagram. 8 Figure: Approximation rate diagram; see Theorem 3²³ for rates on the boundary ~ . The ²c³ in ² c °³²c³ indicates that the superscript c is present only in some cases. We will show that this diagram applies to expansions of functions  in / and in uniform local spaces /" , when the decay rate ! of the scaling function satisfies ! c  ‚ (see Theorem 8³. In addition, on compact subsets the rates in the diagram apply to functions in local Sobolev spaces / , when the wavelet has compact support. See Definitions 1.10 below for definitions of the spaces /" and / . We now establish several equivalent conditions for failure of convergence in all Sobolev spaces. Corollary 6: The following ((a) through (e)) are equivalent for the MRA ¸7 ¹: (a) 0 c 7 ¢ / ¦ 3B is unbounded for all ‚ . (b) This operator is unbounded for  ²°Á ° b ³ for some  € . If there exists a family of wavelets ¸ ¹ ¢  V ²³O OOc  ~ B for some . ( ) For every € °Á O If there exists a scaling function  ¢ V²³³OOc  ~ B. () For every € °Á ² c ²³°  () In every Sobolev space / of nonnegative order, the MRA fails to have any positive order of convergence, i.e., the optimal order of convergence is À We now state our results for optimal pointwise orders of convergence in Sobolev spacesÀ Recall  denotes the best Sobolev parameter of ¸7 ¹, and that optimal order of approximation denotes the highest order of approximation in sufficiently smooth Sobolev spaces. 9 Corollary 7: If the best Sobolev parameter  £ , then the wavelet collection  [or scaling function ] yield optimal pointwise order of approximation  c ° if 0b° ²³ ~ 6²°³ [where 0 can be replaced by 2 or L], and ² c °³c otherwise. This optimal order is attained for all functions  with smoothness greater than , i.e., for   / with €  Corollary 7 gives “best possible” pointwise convergence rates, i.e., convergence rates for the smoothest possible functions. In fact this optimal rate in fact is largely independent of how smoothness is defined, i.e., which particular scale of spaces we are working with. Such a statement is possible because when the smoothness parameter is sufficiently large, the most used scales of “smoothness spaces” satisfy inclusion relations. For example for Z Z large the space / is contained in the sup-norm Sobolev space 3B and in other 3B -type Sobolev spaces. Therefore the optimal rates of convergence given here are upper bounds for convergence rates in all 3B spaces, no matter how smooth. What is most important is that such inclusions work in both directions for the uniformly local spaces in Definitions 1.10 below. For example, for sufficiently large , the uniformly local 3 -Sobolev space 3," is contained in spaces in the scale ¸3Á" ¹ for any fixed values of  and  (including B³. In addition, for sufficiently large the space 3," contains Sobolev spaces 3 . This includes 3B and its related smoothness spaces of functions with bounded derivatives. This observation can then be used as follows. If ²%³ has decay rate ! (Def. 1.10) with ! c  €  c ° (which holds for many wavelets of interest), then by Theorem 8 below the optimal convergence rate in all of the scales of uniformly local spaces 3Á" (including  ~ B), is either  c ° or ² c °³c , i.e., the same as in Corollary 7. Now the extension of Corollary 7 to the spaces 3," (Corollary 9) can be broadened, by the above argument, to more general scales of smoothness spaces, including smoothness spaces based on sup norms. (The caveat, however, is the scaling function  must have sufficiently rapid decay.) With this motivation, we now give the results for uniformly local Sobolev spaces. Our results will also extend to local Sobolev spaces / with some caveats. Definitions 1À10: The decay rate of a function  is sup¸! ¢ O²%³O  2O%Oc! for some 2 € ¹À We will assume here our decay rates ! are positive unless otherwise specified. The local Sobolev space / is ¸ ¢   / D  *B ¹, where *B is compactly supported * B functions. The uniform local Sobolev space /" is ¸ ¢ P P"  B¹Á where the uniformly local norm P h P" is defined by ²here )% is the unit ball centered at %): P P" – sup P P %l Above, the local norm is defined by 10 Á)% . P P Á)% – P i P À inf  i e ²1À9³ ~ ,  / i )% Similarly, the space 3 – ¸  3 ¢ ² c "³ °   3 ¹ has a local version 3Á" defined analogously to the above with the norm P i P in ²1À9³ replaced by the norm of the Sobolev space 3 À Thus /" consists of functions locally in / with local / norms uniformly bounded. The following results for /" are effectively local versions of our rates of convergence results, modulo the spatial uniformity assumptions on functions in /" . Such uniformity assumptions also hold, e.g., for 3B Sobolev spaces. We require our working spaces /" to have uniformly bounded local 3 Sobolev norms rather than 3B Sobolev norms, since the latter would make our work more difficult. As shown above, however, most other scales of smoothness spaces based on uniform (3B type) bounds satisfy inclusion relations with the uniform Sobolev spaces /" , extending optimal convergence rate results to these spaces. Additionally, our results of course become entirely local (valid for local Sobolev spaces) if wavelets involved have compact support. Recall from the definitions that approximation order 0 in a space ? means the error ,  fails to have any positive rate of decay for some   ?À Theorem 8 (Localization): The multiresolution or wavelet expansion corresponding to a scaling function   [RB] has a best pointwise approximation order of at least min ²Á ! c ³ in /" , with  the rate of best approximation in / and ! €  the decay rate of . Corollary 9: If the best Sobolev parameter   ! c ° (where ! is the decay rate of ), then (a) The optimal approximation order in the scale of spaces /" is exactly  c ° if 0b° ²³ ~ 6²°³ [where 0 can be replaced by 2 or L], and ² c °³c otherwise. (b) The same exact optimal approximation order holds in the scale of uniform local spaces 3Á" for fixed     B, and in particular also in the scale LBÁ" and thus 3B . Indeed, note that when is an even integer (i.e. the operator ²c"³ ° is local) the spaces 3B and 3BÁ" are identical, since the first space always is contained in the second, and if  ¤ 3B , then there is a sequence of unit balls ) for which sup ²c"³ °  is %) B unbounded, so that  ¤ 3BÁ" À Thus for each , the scale ¸3Á" ¹ is eventually contained in B B 3 for  sufficiently large, and similarly ¸3 ¹ is eventually in 3BÁ" , so the two scales have identical optimal orders of convergence. This type of inclusion also works for other scales of 3B Sobolev spaces, yielding identical optimal orders of approximation. 11 Proposition 10: If  is compactly supported, the best pointwise approximation rate for the expansion of any   H on any compact 2 ‰ l is the same as the rate for the global space / À Examples: To illustrate these results we give applications to some well-known wavelet approximations. 1. Haar wavelets We calculate the exact approximation order for Haar wavelets. The scaling function  is the characteristic function of the unit interval, whose Fourier transform is V ² ³ ~   l V²³O ~ In this case O  c% % ~  l b 6²OO ³ c° sin ²°³ ²°³l so by Proposition 1À9,  ~ °Á and 0 ²³ ~ BÀ In addition 0b° ²³ ~ ~ ~ €OO€ €OO€ €OO€ V²³O³OOc c  ² c ²³° O 8 c  sin ° c 9 O O    ² ° b 6² ³³ c  ~ 6² c ³. Thus by Theorem 5, in / Haar expansions have best order of convergence ~J Á ° c°Á ° ° c Á ~° Á €° , with the same orders in the uniform local Sobolev spaces /" by Theorem 8. By the same theorem, since  is compactly supported, these orders of convergence to  ²%³ hold uniformly for % in a compact set, for any  ²%³ locally in / . Finally by Corollary 7, the optimal approximation order for such expansions (i.e., for arbitrarily smooth functions) is 1. By Corollary 9 this optimal order also holds, for example, in the scale 3B of 3B Sobolev spaces. 2. Meyer wavelets We now consider standard Meyer wavelet expansions. The Fourier transform of the scaling function is [D2, page 137] 12 V ² ³ ~  J ²³c° Á ²³c° cos ´   ²  OOc³µÁ OO °  °OO °  otherwise , V  *B . In this case where  is an appropriately chosen smooth function for which   ~ Bc , so we have order of convergence c ° in each Sobolev space / , € °Á and convergence order  for  °À Note this implies that for functions in the intersection q / of all Sobolev spaces, we have convergence faster than any finite order . The same holds in the uniform local spaces /" by Theorem 8. Thus the optimal order of convergence in both these cases is B, iÀe., convergence rates have no intrinsic limitations based on the wavelet for very smooth  . 3. Battle Lemarié wavelets Consider now Battle-Lemarié wavelets, which effectively yield spline expansions of a given order. For splines of order 1 the B-spline is ²%³ ~ H  c O%OÁ    O%O   . otherwise The Fourier transform is V²³ ~ ²³c° 8 sin ° 9 .  °  Here  is not a scaling function, since it does not have orthonormal translates. The ~ orthogonalization trick ([D2], section 5.4) yields a scaling function  with orthogonal translates, whose Fourier transform is Ṽ²³ ~ l²³c°   sin ° À  ´ b  cos °µ° The corresponding wavelet has Fourier transform  b  sin ° c° °  l V  sin ° @ ²³ ~ ²³ A  b  cos ° °  sin °  B  ´ b  cos °µ° C ~ 6² ³À From this it follows from Proposition 1À9 that  ~ °À Further, 2 ²³ ~ B, while 2b° ²³ ~ 2 ²³ ~ OO€ V ²³O OOc  ~ 6²°³À O By Theorem 5, Battle-Lemarié expansions (and of course order one spline expansions, since the scaling spaces = are the same) have order of convergence 13 ~J Á 1° c°Á 1° 5° c Á ~ ° Á € ° in / À In the uniform local spaces /" the same approximation rates hold by Theorem 8. Analogous results hold for the higher order versions of these spline wavelets, and the corresponding spline expansions. 4. Daubechies wavelets For standard Daubechies wavelets of order 2, we consider the symbol  ²³ (see ²1.2³; note the definition of the coefficients  in equation ²1À1)): 1 [(1 + l3) + (3 + l3)c + (3 - l3)c + (1 c l3)ec ] 8  ~ ´ b ' b '  b '  µ 0 () = Here ' ~ c , and  ~  b lÁ  ~  b lÁ  ~  c lÁ  ~  c l. Note  ²³ ~ , while O ²³O  ~ >² b  b   b  ³ b ² b  b ³ cos  b ² b ³ cos  b  cos ?À  Therefore     O ²³O e~ ~ . Since  b  ~ ,  b  b  ~  , and  ~ cÁ     O ² ³O ~ c ² b  b ³ b    ~ c  cos  cos  " cos  c  cos #, > ?      and so  O ²³O f  ~ À In addition, ~     O ²³O f ~  but ~     O ² ³O ~ >36 cos  c  cos ?,     so  O ²³O f  ~ £ À Therefore O ²³O ~  b 6²OO ³ ( ¦ ³, so Theorem 4 implies  ~ °. Thus by Theorem 5, in / the best order of approximation for these Daubechies wavelets is 14 ~J Á 1° c°Á 1° 5° c Á ~5° Á €5° . Similar analyses can of course be done for higher order Daubechies expansions. As before, by Theorem 8, the global space / can be replaced by the uniform local space /" . We see the optimal order of convergence for Daubechies wavelets of order 2 is 2. For the compactly supported Daubechies order 2 wavelets, these are entirely local results. Thus for any   /oc , the above exact approximation rates hold uniformly on any compact 2 ‰ l À Remark: Our results imply that for one dimensional [RB] scaling functions with * B Fourier transforms (e.g., for compactly supported ones), optimal orders of convergence are always integers. The reason is clear from Theorem 5, since  for such  is always a halfV is always infinitely integer (see Def. 1.6 and Prop. 1.9), given the Fourier transform  c VO ~ m  VV is also infinitely differentiable at 0, differentiable at the origin. Specifically O V²³O ~  b 6²OO ³ with  an integer and  ~ ° b ° (note  ~  here)À and so O l V²³O always has a maximum of However  must be even since O l at  ~ À In such cases the Strang-Fix conditions, which indicate integer convergence orders and are related to moment and polynomial representation conditions, are entirely equivalent to those above. However, for non-compactly supported scaling functions supported cases the two theories can diverge, in particular our results allow for non-integer optimal convergence rates (see [KR1]). The proofs for the new results 5-9 above are given in section 7. These hold for multiresolution, scaling, and wavelet expansions when they are defined. 2. Preliminaries for proofs Let 7 and 8 be the kernels of the 3 projections onto the spaces = and > , respectively. We inverse Fourier transform and obtain V  ²%Á c³ ~ <&c 7 (%Á &); 7 V ²%Á c³ ~ <&c 8 (%Á &) 8 ²2 À 1 ³ with V  ²%Á c³ ~ ²³c° 7 7 ²%Á &³& & ²2 À 2 ³ V ²%Á c³ defined similarly. The transforms converge everywhere and are continuous and 8 in  if O7 ²%Á &³O  /²% c &³ with /  [RB] (see Def. 1À4). As usual, we have defined 7 ~ 7 here. The same conclusions hold if ²%³ ln (2 + |%|)  [RB] [KKR1]. The error , – 0 c 7 is bounded in 3 À In Fourier space its kernel is [KR1] 15 ~ (,  )(%) = , <  , — where ,n has the kernel (in the variable ) ~ ,  ²%Á ³ ~ (2)c/2 % c 7V (%,c), ²2 À 3 ³ We denote , ~ , À Recall the scaling property 7 (%Á &) = 2 70 (2 %, 2 &), ²2 À 4 ³ — — , ²%Á ³ ~ , ² %Á c ³À ²2 À 5 ³ which implies — Also under our assumptions on the scaling function , the Fourier kernel , ²%Á ³ of the remainder operator , is c — V² ³Á , ²%Á ³ ~ ²³c° % c A²%Á –³ ²2 À 6 ³ where A²%Á ³ –  (% c ) c =  (% b )  (2À7) is the Zak transform of . For later reference, it follows from [KR1], equation (3.12) and its sequel, along with properties of the Zak transform, the Poisson summation formula, and the scaling function  , that the Zak transform can be written ^ (2 c )2% A²%Á ³ ~ c%  (2)/2  t ~ ²2³° c%  0 ( c /2) - (2.8) ^ ( c °2) 2% ,   2t b where - ~ ¸Á ¹ . In addition, as calculated in ´KR1], we have from (2.6) — ,²%Á ³ ~ ~ ^ () (2)c/2 % c A (%,–)  ^ (2 b ) 2%  ^ ( ) À (2)c/2 % :1 c  (2)  ; t The second factor can be written in the form 16 ²2À9³ ^ (2 b )2%  ^ () 1 c (2)  t ²2À10³ ^ ()|2 c  (2)  ^ ()  ^ (2 b )2% . ~ 81 c (2) | 9 £0 For completeness we state a proposition relating approximation orders and operator norms, and two propositions relating operator norms and kernels. Proposition 2.1 [KR1]: Assume a Banach space (, a normed linear space ), and a sequence of bounded operators Q ¢ ( ¦ ). Then the sequence 8 has order of approximation ²³, i.e., +²0 c 8 ³ +)  * ²³ for all   (, if and only if the operator norm +0 c 8 +  * Z ²³, where * Z Á * are constants (the latter depending on  ). — Proposition 2.2: An operator 9: / s ¦ 3B with kernel 9 ²%Á ³ defined by 9 ²%³ – — V ²³ 9 ²%Á ³ ²2À11³ has operator norm P9P/ ¦3B — ~ sup O9 ²%Á ³O ² b OO ³c À % Proposition 2.3 [KR1]: For  l, the operator R: / ¦ 3B defined by equation — ²2À11³ is bounded if and only if the kernel 9 ²%Á ³ satisfies — O9 ²%Á ³O OOc   *  B . Replacing the operator 9 by , ~ 0 c 7 we get: Corollary 2.4: For € ° the MRA ¸Pn ¹ has best pointwise order of approximation — c ° in H s if and only if , (%Á )  < /c in the variable , uniformly in x, i.e., iff — |, ²%Á )| O|c2  is essentially bounded in x. Proof: This follows from equivalence of ²Z ³ and ²³ of Theorem 1 and Proposition 2.3. 3. Growth rates of functions The following results on growth of functions are required in our proofs of sharpness of the best Sobolev parameter , and our main result, Theorem 5. The proofs are available for reference in an appendix to this paper on the Internet at http://math.bu.edu/people/mkon/, with the same title as this paper. 17 Definitions 3.1: A function  ²%³ on an open set E is locally bounded if it is bounded on compact sets. We denote by ) the unit ball of l À Lemma 3.2: Given a locally bounded positive function (²%³ on ) c ¸¹ and  £ Á (a) We have (²%³ ~ 6²O%Oc ³ if and only if O(²%³ c (²%°³O ~ 6²O%Oc ³ ²3 À 1 ³ where if    we assume (²³ ~ lim (²%³ ~ À %¦ () If    and (²³ ~ lim (²%³ exists, but we do not assume (²³ ~ , then in statement %¦ (a), (²%³ ~ 6²O%Oc ³ is replaced by O(²%³ c (²³O ~ 6²O%Oc ³ . ( ) In (a), 6² h ³ may be replaced by ² h ³ (as % ¦ ). Definitions 3.3: Two functions ( h ) and ( h ) are equivalent, ( h ) —  ( h ), if there exist positive constants c1 and c2 such that for every  in their domain, c1 ( )  ( )  c2 ( ). For * € Á we define B* ~ B* ²l ³ to be the class of positive radial functions ²O%O³ on l satisfying ²³ — ²³ for  €  and     , i.e., such that for all  € , °*  ²³  *À ²³ We also define the norm PPB – % |%|c ²O%O³ Henceforth we assume all statements involving the order parameter  hold for     . Theorem 3.4: The following statements are equivalent for any fixed  €  and a positive function  ²%³ on the unit ball ) of l , with      (all integrals are restricted to the unit ball): (a) The integral O%O€  ²%³% ~ 6² c ³À () For some (or all)   , O%O€  ²%³ O%O % ~ 6² c ³ ´and if  € ,  ²%³O%O %  BµÀ (') For some (or all)   l, 18 °O%O  ²%³ O%O % ~ 6² c ³À ('') For some (or all)  € , and some (or all)  with  b  c   ,  ²%³O%O ² b O%O³ % ~ 6²+ c ³ ´and if  €  and  b  c  € Á then lim  ²%³O%O ² b O%O³ % exists and is finiteµÀ ¦ ( ) For any function ²O%O³  BC such that |%|c ²O%O³ %  B , it follows that ²O%O³ O%O  ²%³ %  BÁ for some (or all) * € À Statements in brackets ´ h µ may be included or excluded without changing the equivalences. In addition, 6² h ³ may be replaced by ² h ³ simultaneously in all of the above statements excluding (c), and the equivalences of (a)-(ZZ ) (i.e. all statements excluding ()) continue to hold. Remark: For completeness (though this will not be used in the paper) we remark that the conditions in the above Theorem are also equivalent to the following conditions, listed below: (b''') For some (or all)  € , O%O<  ²%³ O%O % ~ 6² c ³À (b'''') For some (or all) Á   l , c/2<O%O<  ²%³ O%O ² b O%O³ % ~ 6² bc ³À (d) °O%O  ²%³ % ~ 6² c ³À (e) °O%O |%|  ²%³ % ~ 6²³À We now state a corollary which gives uniformity for Theorem 3.4. Corollary 3.5: Let ¸q ¹qQ be a family of positive functions from ) to lÀ The following statements (with all inequalities uniform in  ) are equivalent for fixed  €  (note all integrals below are restricted to ) ) and     : ²³ The integral O%O€  ²%³%  2  c . ²³ For some (or all)   , 19 O%O€ q ²%³ O%O %  2  c ´and for  € ,  ²%³O%O %  2 for some 2 independent of µÀ ²Z ³ For some (or all)   l , °O%O q ²%³ O%O %  2  c . ²ZZ ³ For some (or all) choices of  with  €  and  with  b  c   , q ²%³O%O ² b O%O³ %  2  bc ´gand if  €  and  satisfy  b  c  € Á then  ²%³O%O ² b O%O³ %  2 for some 2 independent of µ. ²³ For any function ²O%O³  B* such that PPB – |%|c ²O%O³%  B , it follows that ²O%O³ O%O q ²%³%  2 PPB Á for some (or all) * € À ² Z ³ For any function ²O%O³  B* such that PPB  B , it follows that ²O%O³ O%O q ²%³ %  2²³Á for some (or all) * € , where 2²³ depends on  but not on  . The above constants 2 are all equivalent, i.e., there is a constant  such that   2  2 Á 2 ,à Á 2  2 , for any fixed choice of Á  Á and  . The bracketed statements in ²³ and ²ZZ ³ can be included or excluded without changing the equivalences. Remark: For completeness (though this will not be used in this paper), we remark that the conditions of Corollary 3.5 are also equivalent to the following: (b''') For some (or all)  € , O%O< q ²%³ O%O %  2  c . (b'''') For some (or all) Á   l , c/2<O%O< q ²%³ O%O ² b O%O³ %  2  bc À (d) °O%O q ²%³ %  2  c . (e) °O%O |%| q ²%³ %  2 . with the constants 2 through 2 equivalent to 2 through 2 . The next Corollary relates divergence rates of two integrals as  ­  ¢ 20 Corollary 3.6: Let ¸i ¹8 denote a family of positive functions i ¢ ) ¦ lÀ The following statements (with all inequalities uniform in  ) are equivalent for given  Á  Á  lÁ with  c , and Á c  positive. (a) ) i ²%³² b O%O³ %  2  c , for some 2 €  and all     À (b) ) i ²%³² b O%O³ b %  2  c b , for some 2 €  and all     . Furthermore, if the above assumptions hold except that c  is negative, then (a) implies that the left side of (b) is bounded uniformly in À Proof of Corollary 3.6: We first prove () ¯ () under the initial assumptions. Note that for fixed , statement ²ZZ ³ of Corollary 3.5 is equivalent to itself if "for some" is replaced by "for all". By the symmetry of ²³ and ²³, it suffices to prove ²³ ¬ ²³. Thus assume ²³ holds. We define constants  € Á  Á  which satisfy  ~  and  b  c  ~ c . Note that since ²3À2³  c it follows  € . Defining the function  by i ²%³ –  ²%³O%O , we see that ²ZZ ³ of Corollary 3.5 is satisfied for our choice of Á Á  . Let   , and now replace  by  b  , and replace by c  . With these new values of  and , we keep  and  unchanged, so that (3.2) is still satisfied. By the equivalence of the "for some" and "for all" versions of statement ²ZZ ³ in Corollary 3.5, it follows that for this new value of  , ²ZZ ³ still holds. However, ²ZZ ³ of Corollary 3.5 with the new value of  is the same as ²³ of this Corollary, proving ²³ as desired. Now consider the case where c  is negative. We will show that () implies that the left side of () is uniformly bounded. We maintain all of the original assumptions of this Corollary, with the only change that now c  is assumed negative instead of positive. Again define  € Á  Á  so that (3.2) holds. Then with these values of Á Á  Á ²³ above is again equivalent to the unbracketed part of ²ZZ ³ of Corollary 3.5. Since the unbracketed part of ²ZZ ³ for one value of  implies the bracketed part for all values of  such that  b  c  € , it follows that the bracketed part of Corollary 3.5 holds for the new value of  . Thus ) q ²%³O%O ² b O%O³ b %  2 for some 2 € . This completes the proof. 4. Convergence rates and the best Sobolev parameter The main result of this section, Theorem 4.3, shows that for any wavelet expansion the best pointwise rate of convergence in / is independent of for € , where  is the best Sobolev parameter. We recall that , ~ 0 c 7 Á and 8 ~ , c ,b ; see ²2À1³ and ²2.3³ for definitions — V of 8 and ,  . V 21 The next proposition will be used later to establish that the function 2 ²³ can replace 0 ²³ (see Def. 1.6) in the statements of our theorems. Proposition 4.1: If € °,  £  c , and  € , then for      4 ²³ – j ||€ — |,n ²%,³| ||cs dj ~ 6² c ³ B if and only if 5 ²³ – j ||€ Vn ²%,³| ||cs dj |Q ~ 6² c ³, B where the norms are in xÀ Proof: Assume 4 ²³ ~ 6² c ³À Using scaling properties of the kernels ²2À5³, 5 ²³ ~ ” ~P ||>c ||>c V (%Á )| ||c2s  ” B |8 — — O, (%,°³ c , (%Á )O OOc  ” B  2 6P ||€ c b = 2² — O, (%Á °³O OOc  ” B + ” — O, (%Á )O OOc  ” B 7 OO€ 4 (°) + 4 ( )³, proving 5 ²³ ~ 6² c ³À Conversely assume 5 ²³ ~ 6² c ³À Then we have 5 ²³ ~ ” ||> ‚ :” — —  O, (%Á °) c , (%Á )O ||-2s ” B — °  || O, (%Á °)O PB c P ||> ~ 6°c ” °c ~ ²  -2s ||>c° OO€ —  ||c2s O, (%Á )O P° B c P ° 4 ²°³ — °  || |, (%Á )| ” B ; -2s ° 5 c 4 ²³ OO€  — °  ||c2s O, (%Á )O ” B 7 , where all 3B norms are in %. To show this implies 4 ²³ ~ 6² c ³, define :²³ ~  c° 4 ²³° À Then 5 ²³ ~ 6°c :²°³²°³°c c °c :²³7 ~ c ²:²°³ c :²³³ À  Since 5 ²³ ~ 6² c ³, O:²°³ c :²³O ~ 6² c°b 22 c° ³ ²4 À 1 ³ By ²³ of Lemma 3.2, if  €  c Á we have :²³ ~ 6² c°b c° ³, and so 4 ²³ ~ 6² c ³ as desired. If on the other hand    c , in order to apply Lemma 3.2 (a) we show :²³ – lim ¦ :²³ ~ . To this end, we first bound 4 ²³ as follows. Define c — @ ²³ ~ P, ²%Á ³PB ~ P²³c° % c A²%Á –³V² ³PB Á where the above norms are in %. It is not difficult to show (see [KR1] and the remarks for - ²³ before equation (2.3.16³ there) that @ ²³ ­ . ¦ Note also that °|| — P, ²%,³PBÁ% ||-2s d ~ °||  2  c  2  ~ ² c c @ ²³ ||-2s d °||  @ ²³ d  sup @ ²³ OO ³À Thus 4 ²³  ||>c — P,n ²%,³PBÁ% ||c d ~ ² c ³, using the equivalence of (b) and (b') (in the case where 6 is replaced by  ) in Theorem 3.4, since by our assumptions  c   À Thus lim :²³ ~ lim ¦ ¦ c° 4 ²³° ~ ²³ ² ¦ ³Á so that :²³ ~ À Now applying Lemma 3.2 (a), we have by ²4À1³ in the case      c  that :²³ ~ 6² c°b c° ³, so 4 ²³ ~ 6² c ³ in this case as well, completing the proof. Theorem 4.2: For if and only if ‚ , the MRA ¸Pn ¹ has pointwise order of convergence   l in H — |, (%Á )|2 ( + ||)c   2  ²c ³+ for     , uniformly in %. Proof: Assume first ²4À2³ holds. Then for   / (letting  = 2c ) 23 ²4 À 2 ³ |,  (%)|2 = e 2 — ,  (%Á ) ^ ()   e — |,  (%,)|2 (1 + ||)c   = 2 c |^ ()|2 (1 + ||)   — |, (%°Á )|2 ( + ||)c   2  cc = 2 2c , |^ ()|2 (1 + ||)  bb independently of %. In the third line we used the scaling property ²2À5³ for kernels. Conversely assume the MRA has approximation order  in H . Then for f  / we have ” ,  ” B  2 2c = 2  . Thus by Proposition 2.1 P, P/ ¦3B ~ 6²c ³, implying by Proposition 2.2 (and the equivalence of the factors ² b ||³c and ² b || ³c ³ that 2c ~ 2  — ‚ ess sup |, n (%Á )|2 ² b ||³c d % — ~  cb ess sup |, (%Á )|2 ² b O|³c2 Á % implying — ess sup |, (%Á )|2 ² b O|³c2   2 ²c ³b % as desired. Theorem 4.3: If the best Sobolev parameter  £ , then the best pointwise order of convergence of the MRA ¸7 ¹ in / is independent of for € À Proof: Assume we have approximation order  in / . Then uniformly a.e. in %, — referring to the definition of , (%Á ) in (2.3) and Theorem 4.2, |,̃ (%Á )|2 ( + ||)c2   2 c2 +2+ ²4À3) . — Assume initially that €  b °À We apply Corollary 3.6 with %i ²³ ~ |, (%Á )|2 and  ~ c , ~  c  c , and  ~ . We show the hypotheses and (a) of the corollary are satisfied as follows. First, b  ~ c  c   . Second, %i ²³² b OO³  ~ ˜ O,²%Á ³O ² b OO³c   2  c b ~ 2  c . ²4À4³ Note that since €  b °, it follows that ²0 c 7 ³ ¢ / ¦ 3B is unbounded. The integral on the left side of (4.4) diverges as  ¦ , since by the equivalence of ²Z ³ and ²³ 24 in Theorem 1 and Corollary 2.4, |,̃²%Á )| O|c2s  is unbounded in % for € . Thus € À Now, for an  €  (to be determined later), we claim that (using  ~ ³, uniformly a.e. in %, |,̃ (%Á )|2 ( + ||)c2² c(°+)³   2 c2² c(°+)³+2+ ~ 2 c b . ²4À5) We first define  more precisely. We will require that  be sufficiently small that c ²° b ³ € À (4.6) Further, we require that  be chosen so that the exponent  c b  in (4.5) is nonzero. Before continuing we will show that the part of the integral in (4.5) which is outside the unit ball remains uniformly bounded a.e. in % and À To this end, note that the exponent in the integral satisfies c² c (°+)³  c  cÁ since  € °. Thus the integral over the outside of the ball remains uniformly bounded, since ,̃²%Á ³ is uniformly bounded (this follows from the definition of , and from the fact that 7 ²%Á &³ is in all cases bounded by an 3 convolution kernel /²% c &³ with / radially bounded; see [KR1]). Now we will show that under the above assumptions in fact  c b   . Indeed, Corollary 3.6 implies that if  c b  € , then the left side of (4.5) is uniformly bounded a.e. in % as  varies in ²Á ³ (this includes the portion of the integral outside the ball by the above remark). However, by (4.6), the left side of (4.5) diverges as  ¦  since (as above) by Theorem 1 and Corollary 2.4, |,̃²%Á )| O|c2s  is unbounded in % for € . Hence  c b   . Thus by Corollary 3.6, (4.5) holds, and by Theorem 4.2 we conclude that we have order of convergence  in / c²°b³ À Thus since c ° € c ²° b ³, we also have order of convergence  in / c° (when €  b °³. Thus if €  + 1/2, and we have order of approximation  in H , then we also have order of approximation  in / c1/2 . This means that if : is the set of orders of convergence in / , then : c° Œ : for €  b °À We know also that any order  of convergence in / also applies to / for Z > , and so : as a set is nondecreasing with . Thus : c° ‹ : À Combining the above inclusions, : c° ~ : , i.e., as a function of the set : is periodic with period 1/2. Combining this with the fact that : is nondecreasing, we conclude the set : must be constant as a function of for > . Thus the valid orders of convergence are the same in / for >  . We remark that above any positive constant could have been used in place of ° in the term  b °. Z Proposition 4.4: If E: / ¦ 3B is bounded, then for all ‚ , the MRA ¸Pn ¹ has best order of convergence  c /2 in / . Proof: Our assumption on the boundedness of , implies by Proposition 1.8 that  € °À 25 By Theorem 1 if      c °, the MRA has approximation order  in /  , and hence in / ‰ /  for € . Thus the best approximation order is at least  c /2 in / , and so : = ´Á  c /2], where : denotes the set of approximation orders in / À Note we have used the equivalence of () and (') in Theorem 1, which implies that if we have approximation order  - d/2 in /  this order is the largest possible in /  . By Theorem 4.3 : is independent of €  . We claim for € , : = ´Á  c /2]. First note since : is increasing with and : = ´,  c  /2], we have : Œ ´Á  c  /2] for > . We show : ‹ ´,  c /2] as follows. If it were true that  €  c ° and   : then Theorem 4.3 would imply that   :b° , since  b ° € À This would give by the equivalence of ²Z ³ and ²³ in Theorem 1 that 0 c 7 ¢ / b° ¦ 3B is bounded. By definition of  (Definitions 1.7), this would imply  ‚  b °, giving the desired contradiction. Thus it is impossible that   : if  >  c /2 and € . Therefore : = ´Á  c /2] for >  as claimed, completing the proof. 5. Conditions for convergence rates With Corollary 2.4 as motivation, we define (for     1³ 1 ( ) = sup % ||‚c — |, (%,)|2 ||c2 d , where the sup as usual is a.e. The following theorem is the analog of Theorem 5, using the 1 instead of the 0 integrals as criteria for approximation orders. Theorem 5.1: Given an MRA ¸Pn ¹ with  £ : () If °  s   the best order of approximation of ¸Pn ¹ in H s is - d/2À if 1 (0)B () If s ~ , the best order of approximation in / is  ~ F (c° À c°2) if 1 (0)~B c () If s € , the best order of approximation in H s is r = F  cd°2 if 1+1/2 ( ) = 6(° ) ²¦³ ( c°2) otherwise c À Proof: Since  £  we have  € ° by Proposition 1.8. Statement () follows from the definition of  and from the equivalence in Theorem 1 of (ZZ ) and (). If 1 ²³  B, statement (ii) follows from Corollary 2.4. On the other hand if 1 ²³ ~ B, then by Corollary 2.4, approximation order  c ° fails. However, by Proposition 1.8 the set ' satisfies ' ~ ²°Á ³À Thus for any Z with °  Z  , the operator , ¢ / ¦ 3B is bounded, and so the MRA has approximation order Z c ° in / and hence also in /  À Therefore, in /  we have all orders of approximation less than  c °, which means the order of approximation is ² c °³c À It remains to prove (iii). Since by Theorem 4.3 we have best order of approximation independent of s for s > , we need only consider a specific value of , say =  + /2, and Z Z 26 find the best approximation order in / s . By Theorem 4.2 this is the supremum of values of  for which — |, (%,)|2 ( + ||)c2 c d  2  c2+bc . ²5À1³ for     . For any  where ²5À1³ holds, we have order of convergence  in / +/2 . We first show (letting  =  c °) — |, (%Á )|2 ( + |)-2- d  2  c2+2( cd/2)bc ~ 2° ²5À2³ if 1+/2 ( ) = 6( c ), ²5À3³ ²recall all order statements in  hold for      only). Then we show that ²5À2³ is true for  replaced by c (i.e. it holds for all Z < ) if ²5À3³ fails. We also need to show these choices of  and c are best possible (largest possible) in ²5À2³. To prove the first statement, i.e., equation ²5À2³, assume ²5À3³ holds, i.e., sup % ||‚c — |, (%Á )|2 ||c2 c   21  c . — With the goal of applying Corollary 3.5, let % ²³ ~ |, (%Á )|2 OOcc and  ~ . Let  ~  b  and  ~ c c , so that  b  c  ~ c   and  c  ~  € . Applying the equivalence of ²³ and ²ZZ ³ in Corollary 3.5, we conclude sup % — |, (%Á )|2 ² b O|³c2 c   21  c . This proves ²5À2³ and shows that if 1+1/2 () = 6( c ) we have approximation order  c /2 in / +1/2 by Theorem 4.2. Note that this order is in fact best by Theorem 1. Indeed, since this order is the same in all / for € , it holds for ~  b  for  € À However by Theorem 1 the approximation order in / in this case cannot be better than  b  c °, and so the (constant) order of approximation in / for €  cannot be better than this for all  € , and hence cannot be better than  c °. Thus for ~  b ° (and so for all €  ) the best approximation order in / is  c ° as desired. Now we consider when 1+1/2 ( ) = 6 (° ) fails to hold, and show we have best approximation order ( c /2)c in /  b° . By (ii) above the best approximation order is at least ( c /2)c and we must show it cannot be better. However since ²5À3³ fails by our assumption, it is easy to show by the same arguments as above that ²5À2³ fails. By Theorem 4.2 therefore, we fail to have order of convergence ( c  /2) in / +1/2 , so the best approximation order must be ( c  /2)c . Thus by Theorem 4.3 this is the best approximation order in / for all ‚ . 6. Preliminaries for the proof of Theorem 5 27 We present some technical lemmas required in the proofs of the main results, parts (iv) and (v) of Theorem 5. Recall that the class B* and the norm P h PB are given in Def. 3.3, and that the integral 0b° ²³ – 1‚||‚c ^ ()|7 ||c2c1 . 6 c (2)/2 | (6.1) Lemma 6.1: Let  € ° and * € . Assume 0b° ²³ ~ 6²°³ (for     ). Then for ²OO³  B* with PPB  B, we have (defining ²³ ~ ²OO³OOc ) ²³ ||< (1 c O0 (/2)|2 ) ²³  < B ²³ OO |0 ( b /2)|2 ²³  < B for   - ~ ¸Á ¹ À Proof: Using the assumption 0b° ²³ ~ 6²°³ and the equivalence of () and ( ) in V()|7OOc c and  ~ , we have Corollary 3.5, letting  ²³ ~ 61 c (2)°2 | OO V()|7²³  B. 61 c (2)°2 | In addition, OO V()| 7²³  ~ 61 c (2) | OO V()|7 61 + (2)°2 | V()|7²³, 61 c (2)°2 | so || Thus since V()| 7 ²³  < B. 61 c (2) | V(2)|2 c 17 c |0 ()|2 6(2) | V()|2 c 17, |0 ()|2 c 1 = 6(2) | ²6À2) we have ||< (1 c O0 (°2)|2 ) ²³ d < BÁ (6.3) proving ²³À Additionally, it is known that (e.g., [KR1, equation (2.3.9) and Lemma 2.3.1]),  |0 ( + )|2 = 1 - ^ ² + M³|2 = ²³cd À  | Mtd If  £ 0, by ²6À4³Á 28 (6.4) |0 ( b /2)|2 = 1 c |0 (Z b °2)|2  1 c |0 (°2)|2 ,  Z £ so by ²6À3³ OO |0 ( b /2)|2 ²³  < BÁ (6.5) yielding ²³ and completing the proof. The Fourier kernel ,̃²%Á ³ is given in ²2.6³. We also have: Lemma 6.2: Let  € ° and * € . Assume 0b° ²³ ~ 6²°³ (    )À Then the ˜ kernel ,²%Á ³ satisfies OO ˜ ²%Á ³PB c P, ˜ ²%Á °³PB O ²³   B O P, for any ²OO³  B* with PPB  B. Here ²³ ~ ²³OOc , and the P h PB norm is taken with respect to %. Proof: We have from (2.8): ^ () A (%Á –)  ^ (°2)A (2%,–°2) b  (°2) ^ (°2) ( b °2)A (2%,–°2 c ), ~ |0 (°2)|2  0 0 £0 and so ^ () c A (2%Á –°2)  ^ (/2) A (%Á –)  ²6À6³ ^ (°2)A (2%,–°2) b  (°2) ^ (°2) ( b /2)A (2%,c/2 c )À ~ (|0 (°2)|2 c 1) 0 0  £0 We now use (6.6) and ²³ and ²³ of Lemma 6.1 (with  and  as in the Lemma), noting ˜ ²%Á °³PB ~ P, ˜ ²%Á °³PB , to obtain P, 29 8 ~8 OO ||< ˜ ²%Á ³PB c P, ˜ ²%Á °³PB O ²³ 9 O P, ° ^ () c ²³c° ” c ” c% A (2%Á –°2) ^ (/2) c ²³c° ” 2 c% A (%Á –) B B f ² ³  9 f”  8 = : ^ () c A (2%,–/2)  ^ (°2) ” 2 ²³ d ” A (%,–)  9 B ||< ||< 1/2 ^ (°2) A (2%,–/2) ” (|0 (°2)|2 c 1)  ^ (°2)   ( b /2) A (2%,–/2 c ) ” 2 ²³  + 0 (°2)  0 B ; 1/2  £0 8  + r s ||< ||< ^ (°2) ” A (2%Á –°2) ” O 2 ²³  O (|0 (°2)|2 c 1)  B 9 1/2 ^ (°2)  O ( b °2)O ” A (2%Á –°2 c ) ” O 2 ²³ O m0 (°2)  0 B £0  BÁ V²³, and  ²³ are uniformly bounded in % and À where we have used that A (%Á ),  Lemma 6.3: Assume that  € °. There exists a number * €  such that for any  ²OO³  B* ²l ³, the following holds (defining  ²³ ~  ²³OOc ): For any positive @ ²³ with @ ²³ ­  , if ||<1 | @ () c @ (/2)|2  ²³   B, then OO |@ ¦ 2 ²³| ²³  B.  ²OO³ if OO   Proof: Define ²OO³ ~ H  , with   B* , and * to be determined. Let  otherwise ²³ ~ ²OO³OOc , so that from the assumption of the Lemma |@ () c @ (/2)| 2 ²³ d  B. ld Now choose * €  so c * °  c° Á which is possible since  € °À Then l O@ ²³ c @ ²°³O ²³  ~ l O@ ²³²³° c @ ²°³²°³° ²°³O  where (recall   B* ; see Definitions 3.3) 30 ²6 À 7 ³ u v 1/2 1/2 ° ²°³ – ²³ ° °²°³ c ~ ²OO³°  c * °  c° . ° ²O°O³ We have defined ° ~  above. Let @ () be a positive sequence in 32 which converges pointwise to @ () such that @ ²³  @ ²³À We may assume convergence occurs such that, defining ° .²³ ~ @ ²³²³ and . ²³ ~ @ ²³²³° Á |. () c ²°³. (°)|  |.() c ²°³.(°)|. ²6 À 8 ³ For example, since @ ²³ ­ , we could choose @ ²³ ~ ²@ ²³ c °³b , with + ¦ denoting the greater of the argument and 0. Then O@ ²³ c @ ² ³O  O@ ²³ c @ ² ³O for all Á   l , and O. ²³ c ²°³. ²°³O ~ |@ ²³²³° c @ ²°³² ³° |  |@ ²³²³° c @ ²°³²³° | ~ |.²³ c ²°³.²°³O as desired. Now ” . () c ²°³. (°)P2 ‚ ” . ()P2 c ” ²°³. (°)P2 ‚ ² c °c * ° ³ ” . ()P2 ²6 À 9 ³ By our choice of * we have  c °c * ° € . By ²6À7³ l |.²³ c ²°³.²°³O  ~ l |@ () c @ (°)|2 ²³ d  B. d Thus by dominated convergence and ²6À8³, the left side of ²6À9³ converges, so the right side is bounded in . Thus the sequence P. ()P2 is bounded, and since . () converges to .() pointwise from below, P.()P is finite, proving the lemma. Recall for an inner product space = (with inner product h ³, a family of vectors ¸A  ¹ ‰ = forms a frame if there exist constants ( €  and )  B such that for all   = , (P P  O h A  O  )P P À  Lemma 6.4: In a finite dimensional space, the optimal frame bound is a continuous  function of the frame. Specifically, if the vectors ¸A  ²³¹m ~ form a frame in l for each  , and if the A  ²³ vary continuously in , then O h A  ²³O  sup ~  l O  O  O h A  ²³O  Á inf  l are continuous functions of . 31 ~ O O (6.10) Proof: We write  O h A  ²³O  sup ~ O O  l O h A  ²³O .  ~ sup  l Á OO~ ~ (6.11) Because suprema of equicontinuous function families are continuous, it is only necessary that we check that the family ¸ O h A  ²³O ¹ S  is equicontinuous, where ~ : ~ ¸ ¢ O O ~ ¹. But for this it suffices to show that for each , ¸O h A  ²³O ¹ S forms an equicontinuous family of functions, which is clear by the Schwartz inequality. Lemma 6.5: Let f() and A ²%Á ³ be vector functions on l , let  € , and assume ²³ The vectors ¸A ²%,³¹ span l , i.e., there is no nonvanishing vector  such that A ²%Á ³ h  ~  a.e. ´%µ. ²³ A ²%Á ³ is continuous in . Then if  ²³ is a vector function in 3 ²l ³ such that OO€ O ²³ h A ²%Á ³O  ~ 6² c ³ a.e. ´%µ, (6.12) it follows that OO€ O ²³O  ~ 6² c ³À Proof: Assume (6.12). Let ¸% ¹~ be such that ¸A²% Á ³¹ is a basis for l , and such that the equality in (6.12) holds for % ~ % D. Then by the previous Lemma, for OO sufficiently small, say OO   Á O ²³O  2 O ²³ h A ²% Á ³O Á  since ¸A (% Á 0)¹ forms a basis and thus a frame. Now write (for small  ) OO€ O ²³O  ~   €|O€ O ²³O  b OO€ O ²³O  2 O ²³ h A ²% Á ³O  b  €|O€  c  6² ³ b 2 ~ 6² c ³À OO€ O ²³O  7. Proof of Theorem 5 The next theorem establishes the equivalence of the functions 0 and 3 . 32 Theorem 7.1: For  € Á     , ! £  b , || ^ ()|) ||c!  ~ 6² c ³ ( 1 c (2)°2 | ²7À1³ (1 c |0 ()|2 ) ||-t  ~ 6² c ³ ²7À2³ if and only if || Proof: Let the left side of ²7.1³ be ; ²³ and that of ²7.2) be < ²³À Recall V²³ ~  ²°³ V²°³À Let :²³ – V ²³O ³OOc! À Note by factoring  ² c ²³ O OO the integrand that :²³ ~ 6² c ³ iff ; ²³ ~ 6² c ³. Assume ²7.1³. Then (6.2) gives ²7.2³ (after factoring the differences of squares on the right of ²6À2³). Conversely assume ²7.2³ holds, i.e., that < ²³ ~ 6² c ³À Then note V(2)|2 7 c 61 c (2) | V()|2 7 = (2) (1 c |0 ()|2 ) | V()|2 . 61 c (2) | ²7À3³ The factors in the integrand of ²7.2³ are positive since they symbol  ²³ assumes its V²³O ~  for maximum of 1 at  ~ . Thus, without loss redefining  c ²³° O OO € °, 6² c ³ ~ ‚f OO OO ^ ()|2 ) c ( 1 c (2) | ^ (/2)|2 ) | ||c!  |( 1 c (2) | ^ ()|2 )OOc!  c c! ( 1 c (2) | c! ~ O:²³ c  °OO° ^ )|2 )||c!  (1 c (2) |( f :²°³OÀ Defining 7 ²³ ~  !c :²³ we have 6² c ³ ~ O c! ²7 ²³ c 7 ²°³³OÁ so O7 ²³ c 7 ²°³O ~ 6² cb!c ³. ²7 À 4 ³ Hence if ! c  c   , by Lemma 3.2, 7 ²³ ~ 6² !cc ³, so ; ²³ ~ 6² c ³Á as desired. On the other hand, if ! c  c  € Á then we can again apply Lemma 3.2 (a) if we can ^ ()|) ­ Á show 7 ²³ – lim7 ²³ ~ À To this end, note that since (1 - (2)/2 | ¦ ¦ °|| ^ ()|) ||c!   ²°³c! O)²³O sup ² c ²³° O V²³O³ ~ ² c! ³Á ( 1 c (2)/2 | OO 33 where O)²³O denotes the volume in  dimensions of the ball of radius . Thus by the second paragraph of the statement of Theorem 3.4 (note since ! c  c  €  we have ! € ) relating to replacement of 6 by , ; ²³ ~ ||1 ^ ()|) ||c!  ~ ² c! ³À ( 1 c (2)°2 | So by factoring the integrand below, :²³ ~ ||1 ^ ()| ) ||c!  ~ ² c! ³, ( 1 c (2) | and thus 7 ²³ ~  !c :²³ ­ . Thus we can apply Lemma 3.2 to (7.4), to again obtain ¦ 7 ²³ ~ 6² !cc ³Á and ; ²³ ~ 6² c ³Á completing the proof. Note for later reference that  (%)) = 2c /2 c2c  ^ (2c ). < ( — Recall also formula (2.6) for , ²%Á ³. We now give the complete proof of Theorem 5, the main result of this paper. Proof of Theorem 5: (o) If  ~  it suffices to show that we fail to have any positive order of convergence in / for € °. Recall  ~  means 0 c 7 ¢ / ¦ 3B is unbounded for all ‚ À To begin we claim that in this case the convergence rate of the MRA for € ° is independent of À The rest of the proof is similar to that of Theorem 4.3. Assume we have approximation order  €  in / . Then uniformly (as usual a.e.) in %, we have by Theorem 4.2 — |, (%Á )|2 ( b ||)c2 d  2 c2 b2b ~ 2 c . ²7À5) Assume for the moment that € ° b °À We apply Corollary 3.6 with — 2 i % ²³ ~ |, (%Á )| ,  ~ c , ~  c  c  , and  ~  b , with  €  to be determined belowÀ Note that the supremum of the left side of ²7À5³ diverges as  ¦  by Corollary 2.4, the equivalence of ²Z ³ and ²³ in Theorem 1 and the fact that € ° € , so that c  À Similarly to the proof of Theorem 4.3, it follows from Corollary 3.6 that for some  €  we have uniformly a.e. in %: — |, (%Á )|2 ( + ||)c2² c²°b³³   2 c2² c²°b³³+2+ ~ 2 c b . ²7À6) More precisely, we choose  so small that c ²° b ³ € °, and also so that the exponent  c b  £ 0. Then as in the previous proof, we conclude that in fact 34  c b   , since the left side of ²7À6) diverges as  ¦ . The divergence of the left side of ²7À6) follows by Corollary 2.4, the equivalence of ²Z ³ and ²³ in Theorem 1, and the fact that c ²° b ³ € ° € À Corollary 3.6 then gives ²7À6). From this we conclude we have pointwise approximation order  in / c²°b³ À With appropriate choice of  € , it follows (as in the proof of Theorem 4.3) that the best pointwise approximation order  (as a function of ) is periodic with period ° and nondecreasing. Thus the set of orders is constant for € °, as desired. We have established the rate of convergence  in / is independent of . By Theorem 1, it is also less than c ° in / for € °À Since can be arbitrarily close to °, we conclude   c ° must be smaller than any positive number. Hence there is no positive rate of convergence  in any Sobolev space / with € °. Since for Z  ° Z and € ° we have / Š / , it also follows that there are no positive convergence rates for Z  °. Thus we have proved (o). ²i³ For    °, there exist unbounded   / . However, for any   / , 7   3B . This follows since by our assumptions, the scaling function satisfies O²%³O  ²O%O³ with bounded, decreasing, and in 3 ²l ³. Thus O7  O ~ O ²% c ³O  :O O ; °    : ²% c ³ ; °  2P7  P3 P P .  Above, the sum involving can be bounded by the norm on the right because it can be bounded by an integral. Thus for unbounded   / we have , ~  c 7  is unbounded. Therefore , ¢ / ¦ 3B is never bounded, and hence P, P/ ¦3B fails to have any decay rate. This yields approximation order 0 in these spaces (by our definition of approximation order ). (ii) This follows from equivalence of (ZZ ) and () in Theorem 1. (iii) This follows from the equivalence of () and (Z ) of Theorems 1 and 3. Specifically if 0 ²³  B then by these theorems the best pointwise approximation order in /  is  c °. But if 0 ²³ ~ B, then Theorems 1 and 3 imply the best approximation order in /  cannot be  c °. However for ~  c  Theorem 1 implies the best approximation order in / is c ° ~  c ° c . Thus for any €  c  the best approximation order is at least  c ° c À Since this holds for all  € , for €  the MRA has approximation order  c ° c  for all  € , and hence has best order ² c °³c as desired. ²iv³ Since  £ , we have  € °. By Theorem 4.3 the best approximation order in / is independent of for > . Thus to determine this order for any >  we only consider =  + 1/2. By part ²³ of this Theorem, since best approximation order in / 2 cannot be worse than in / 1 for 2 >  , the best order in / is at least ( c d/2)c . We now show that if 35 0b° ²³ – ‚||‚ ^ ()|7 ||c2c1  ~ 6(1° ) 6 c (2)°2 | ²7 À 7 ³ then we have approximation order  c /2 in / s for € . To do this we will verify 1 b1/2 ( ) – sup % ||‚ ˜ (%Á )|2 ||c2 c1  ~ 6²°³, |, (7.8) and use Theorem 5.1. To verify (7.8) we will use equivalence of () and ( Z ) in Corollary 3.5. Specifically we let % ²³ ~ |,̃ (%Á )|2 OOcc and  ~ . It then suffices to show for some fixed * €  that for any  ²||³  B* such that ||c  ²OO³   B, ˜ (%Á )|2 ||c2   2  B  ²||³ sup |, % ||1 To prove ²7À9³ for some * (we will choose * later), let ²OO³ ~ H ²7À9)  ²OO³ if OO   .  otherwise Defining ˜ @ () – P,²%Á ³PB , we have by Lemma 6.2 that if ²³ – ²OO³OOc , | @ () c @ (°2)|2 ²³   B. ||<1 Note that @ ²³ ­  (see remarks before equation (2.3.16³ in [KR1]). Now choose * as ¦ in the statement of Lemma 6.3. Then, using Lemma 6.3, we have ˜ (%Á )|2 ||c2  ~ ²||³ sup |, % OO OO @ 2 ²³ ²³   B, so ²7À9³ has been established, proving we have approximation order  c ° in / for € . To prove the second case of ²#³, suppose now that 0b° ²³ – 1‚||‚ ^ ()|7 ||c2c1 d £ 6(1° ) 6 c (2)°2 | for   À By the equivalence of ²³ and ²³ in Theorem 3.4 for any * €  there then exists ²OO³  B* with ||c ²OO³   B and OO ^ ()|7 ||c2 ²OO³  ~ BÀ 6 c (2)/2 | 36 We wish to show that in this case the order of convergence in / for €  is ² c °³c À To this end it suffices to show, by Theorem 5.1Á that 1b° ²³ £ 6²°³ for   . Again let ²³ ~ ²OO³OOc . We have ^ ()|) ²³  = BÀ (1 c (2)°2 | ²7À10³ ||<1 Letting * denote the unit cube in l , we have by (2.9) and (2.10): ²7À11³ ˜  ²%Á ³f ²³  f,  * ~ * % ||< % ~ ||< ‚ f(2) c /2 ||< r s f(2) c /2 ^ 2 ^ ^ 2ix  /2 81 c (2) |()| 9 c (2) ()(2 b ) f ²³    £0 u ^ 2 ^  /2 ^ ²³  81 c (2) |()| 9f b  e(2) ()(2 b )e v £0   ^ () ^ (2 b )e ²³   e(2)°2  ||< £0 The second equality follows from the Parseval identity for Fourier series, since the % integration (once % and  integrations are interchanged) is the square of the 3 -norm of a Fourier series in %. By ²6À4³ and ²7À10³, factoring the difference of squares below, ²³ V² b M³O )²³  ~ ²O OO M£ OO ^ ()|2 )²³  ~ B, (1 c (2) | V²³ £  and  V is continuous) so comparing with ²7À11³, (since  * % ˜  ²%Á ³e ²³  ~ BÀ e,  OO The above is an 3 norm over the unit cube * in %, so the 3B norm is also infiniteÀ Thus the error operator , with Fourier kernel ^ () ,˜ ²%Á ³ ~ (2)c/2 % c A (%Á –)  satisfies j OO O,˜ ²%Á ³O ²³ j 37 ~ B. B ²7À12³ But by % ²³ ~ the equivalence — O, ²%Á ³O OOc c , of parts () and ( Z ) of Corollary 3.5, letting it follows 1b° – j O, ²%Á ³O OOcc j — OO £ 6²°³À B Now by Theorem 5.1, it follows that the best pointwise order of approximation in / is ² c °³c , as desired. This completes the proof of (#). ²#³ We wish to show the above statements hold with 0b° ²³ replaced by 2 b° ²³ or 3b° ²³À Note by Theorems 1 through 4 the proof of ²³ does not change if we replace 0 by 2 or 3. Now consider the proof of ²#³ in these cases, first with replacement of 0b° ²³ with 3b° ²³. We wish to show 0b° ²³ – 1‚||‚ ^ ()|7 ||c2 c1  ~ 6(1/ ), 6 c (2)d/2 | if and only if 3b° ²³ – 1‚||‚ (1 c |0 ()|2 ) ||c c  ~ 6²°³. We apply Theorem 7.1 with ! ~  b , and  ~ , so  b  ~  b . We have  € , so that ! ~  b  £  b . It follows that 0b° ²³ may be replaced by 3 b° ²³, as desired. To show we can replace 0b° ²³ by 2 b° ²³, we show first that if 0b° ²³ ~ 6²°³, ²7À13³ then the same holds for 2b° ²³ – sup  1‚||‚c O ²³O ||c2 c . Note that if ²7.13) holds then by part ²#³ of this theorem and by Theorem 5.1, 1b° – ” ||€ — |,  ²%Á ³| ||c c d ” B ~ 6²°³ ²7À14³ which by Proposition 4.1 implies that ” ||€ V ²%Á ³| ||c c d ” B ~ 6²°³À |Q c c V ²c³c À We then have We write 8 ~ 8 and , ~ , À Note that < ² ²& c ³³ ~  that the kernel [KKR1] 38 V 8²%Á ³ ~ <& ²8²%Á &³³ c ~ <& ² ²% c ³ ²& c ³³  , c ~  ²% c ³<& ² ²& c ³³  , c V ²c³c ~  ²% c ³  , c V ²c³A  ²%Á ³Á ~   where A  ²%Á ³ ~  ²% b ³ is the Zak transform of  . But for a set . of % of  positive measure, A  ²%Á ³ is nonzero at  ~  (see [KR1] after (2.2.9)), and it is always continuous in , since its Fourier coefficients  ²% b ³ are in M . We have, letting c cM c V ²c³Á  V ²c³Á à Á  V ²c³³, A ²%Á ³ ~ ²A  Á A  Á à Á A M ³ and ²³ – OOc c° ² e²³ h A ²%Á ³e  ~  OO€ ~ OO€ ||>c c  V ²c³A  ²%Á ³e OOc c  e  ²7À15³  V  ²x,³| ||c c d |Q ~ 6²°³À The set ¸A ²%Á ³¹%. spans l in that there is no nonvanishing vector  (see [KR1], before (2.2.9)) such that A ²%Á ³ h  ~  for almost all %  .À Thus we may apply Lemma 6.5 to conclude that OO€ O²³O  ~ 6²°³ Thus 2b° ²³ ~ sup  €OO€ O ²³O OOc c  ~ 6²°³Á as desired. Conversely, assume 2b° ²³ ~ 6²°³; we wish to show that then 0b° ²³ ~ 6²°³. But if (7.16) holds, then OO€ O8V ²%Á c³O OOc c  ~ OO€ (7.16) c  V ²c³A  ²%Á ³e OOc c  ~ 6²°³Á e   39 ( ¦ ), by the boundedness of A  ²%Á ³À Then by Proposition 4.1, ²7À14³ holds, and so by Theorem 5.1 the best order of approximation in / is  ~  c °. Thus by ²#³ of this theorem, which has already been proved, it follows that 0b° ²³ ~ 6²°³, as desired. This completes the proof of (#). Proof of Corollary 6: The implication ²³ ¬ ²³ is clear, while ²³ ¬ ²³ and ²³ ¬ ²³ follow from Theorems 1 through 3. Also ²³ implies that  ~ , which by Theorem 5 implies ²³. Thus ²³ implies ²³, ²³, ²³, and ²³. On the other hand, by Proposition 1.8, ²³ implies  ~  and thus ²³. By Theorems 1 through 3, ²³ and ²³ imply 0 c 7 ¢ / ¦ 3B is unbounded for € °, which implies () and hence also () (recall , ~ 0 c 7 ¢ / ¦ 3B is unbounded for    °³. In addition, by Theorem 1, () implies () and hence (). 8À Proof of Theorem 8 Proof of Theorem 8: Note that the statement of the theorem yields positive convergence rates only if ! €  , which we assume throughout. Our assumption easily implies that uniformly in % and &, O7 ²%Á &³O  2O% c &Oc! . ²8À1³ Indeed, since 7 ²% b Á & b ³ ~ 7 ²%Á &³ it suffices to check this for % in the unit cube 9. For %  9, since O²% c ³O  ² b OO³c! , we have c O7 ²%Á &³O ~ O²% c ³²& c ³O  2 ² b OO³c! ² b O& c O³c!   ~ 2 r  sO&cOO&O° u b O&cO€O&O°v ² b OO³c! ² b O& c O³c! r  2 O&Oc!  ² b O& c O³c! b s O&cOO&O°  2 O&Oc! Á u ² b OO³c! . v O&cO€O&O° O&Oc!  recalling that ! € À Then ²8À1³ follows from the boundedness of 7 ²%Á &³À In this case it suffices without loss to find a uniform local rate of convergence in the unit ball ) ‰ l , since the same rate will hold in any other unit ball. We consider   /" with P P/ ~ . For such an  we write  ~  b  , where  is supported in ),  is supported outside ), and P P/  * with * independent of  . Then " 40 P,  PB  P,  P B b P,  PB À The first term satisfies P,  PB  2 h c , with 2 independent of  . The second term satisfies (for %  )) |,  | ~ f  7 ² %Á  &³ ²&³&f ~ f  7 ² % c ´ %µÁ  & c ´ %µ³ ²&³&f 2 ~2  2  2 — )  O7 ² % c ´ %µÁ  & c ´ %µ³O& — b )c´ %µ O7 ² % c ´ %µÁ &³O & — b )c´ %µ —  ) O&Oc! & O&Oc! &  2  c²!c³  — above ´ h µ denotes the greatest integer function applied componentwise to vectors, and ) is the complement of )À We have used the fact that   3B , with norm 2 bounded by the B norm P P/" , since /" ‹ 3 , given € °À Proof of Corollary 9: (a) The assumptions of the Corollary imply  c °  ! c . For € , therefore, the best order of convergence in /" is the same as in / , i.e.,  ~ ² c °³²c³ , by Theorem 8, where ² c ³ indicates the possibility of c or its absence in the superscript. Since this statement is independent of € , we conclude the optimal order of convergence is ² c °³²c³ À (b) For any   Á   B and  € , the inclusion 3Á" ‹ 3Á" holds for  sufficiently large. Indeed in this case P P3 ‚ 2P P3 is clear from the standard  Á"  Á" Sobolev inclusion relations. In particular, 3 Á" ‹ /" for  sufficiently large, so in the scale ¸3 ¹  ‚ , the optimal order of convergence is at least that in the scale ¸/" ¹ € and thus the scale ¸/ ¹ € , as seen above. The reverse inclusion (for sufficiently large  ³ shows that it cannot be greater, and so is the same as in the scale ¸/ ¹À Proof of Proposition 10: This proposition follows immediately from Theorem 5, since on any compact 2 a function in / is a restriction of a function in / , and convergence 41 properties for a function  on 2 depend only on the properties of  in 2 , since the wavelet is compactly supported. We conclude by remarking that most of the present results and their variations hold in general spaces of functions to which global versions of these theorems apply. Acknowledgments: The first author thanks the U.S. Fulbright Commission and the University of Warsaw for its support. The second author is grateful to Anneli and Peter Lax for encouragement and useful comments on this work, and Louis Nirenberg for an invitation to spend a year at the Courant Institute. References [D1] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. [D2] Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, 1992. [GK1] Gurarie, David, and Mark A. Kon, Radial bounds for perturbations of elliptic operators, J. Functional Analysis 56 ²1984³, 99-123. [GK2] Gurarie, David, and Mark A. Kon, Resolvents and regularity properties of elliptic operators, in Operator Theory: Advances and Applications , C. Apostol, Ed., Birkhauser Verlag, 1983. [HW] Hernandez, E. and G. Weiss, First Course in Wavelets, CRC Press, 1996. [KKR1] Kelly S., M. Kon, and L. Raphael, Local convergence of wavelet expansions, J. Functional Analysis 126 (1994), 102-138. [KKR2] Kelly, S., M. Kon, and L. Raphael, Pointwise convergence of wavelet expansions, Bull. Amer. Math. Soc. 30 (1994), 87-94. [KR1] Kon, M. and L. Raphael, Convergence rates of multiresolution and wavelet expansions, to appear in Wavelet Transforms and Time-Frequency Signal Analysis, CBMS Conference Proceedings, L. Debnath, Ed., Chapter 2. [KR2] Kon, M and L. Raphael, Characterizing convergence rates for multiresolution approximations, in Signal and Image Representation in Combined Spaces, J. Zeevi and Ronald Coifman, Eds., 1998, 415-437. [Ma] Mallat, S., Multiresolution approximation and wavelets, Trans. Am. Math. Soc. 315 (1989), 69-88. 42 [Me] Meyer, Yves, Ondelettes, Hermann, Paris, 1990. [Si] Singer, I., The Theory of Best Approximation in Functional Analysis, CBMS Conferences in Applied Mathematics, vol. 13, SIAM, 1974. [SF] Strang, G. and Fix, G., A Fourier analysis of the finite element variational method, in Constructive Aspects of Functional Analysis, Edizioni Cremonese, Rome, 1973. [Wa] Walter, G., Approximation of the delta function by wavelets, J. Approximation Theory 71 (1992), 392-343. [Wo] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge, 1997. Mark A. Kon Department of Mathematics Boston University Boston, MA 02215 [email protected] Louise A. Raphael Department of Mathematics Howard University Washington, DC 20059 [email protected] 43 44 Note: This appendix is not included in the published version of this paper. 9. Appendix for “A Characterization of Wavelet Convergence in Sobolev Spaces” by Kon and Raphael This appendix contains the technical proofs of statements in Section 3 of this paper. The following is a standard fact whose proof (using the closed graph theorem) we omit. Below ) again denotes the unit ball of l . Lemma 9À1: On a normed linear space (NLS) ? , two norms in which ? is complete are equivalent if whenever  ¦  in the first norm and ¸ ¹ converges in the second norm, then  ¦  in the second normÀ Recall a seminorm P h P on a vector space ? satisfies P P ~ OOP P for   ? and   ], and satisfies the triangle inequality, but not necessarily positive definiteness. We say seminorm P h P on a vector space ? is complete if for every Cauchy sequence ¸ ¹ there exists   ? such that  ¦  , i.e., P c  P ¦ . Note that seminorm convergence  ¦  does not in general determine  uniquely. Lemma 9À2: A nonnegative seminorm P h P on a vector space ? is complete if and only if whenever a sequence ¸ ¹ is absolutely convergent (i.e., P P  B), it follows  converges (i.e., there exists   ? such that i c  i ¦ ).    Lemma 9À3: Let ¸P h P ¹ ; be a family of seminorms on a vector space ? . Assume that for any ¸ ¹ ‰ ? , if for each  ,  ­ ¦ B   ? in seminorm  and sup P P  B, then  there is an   ? such that  ­ ¦ B  in all the seminorms P h P . Then if ? is complete in each of these seminorms, it is also complete in the seminorm P P – sup P P , assuming  the latter is always finite. Proof of Lemma 3.2: (a) The forward implication is clear, and we prove the reverse. First assume  € . Then if O%O  °, we have (²%³  f <(²2 %³ c (²b %³=f b (²5b %³ 5 ~ 5  *  Ob %Oc b (²5b %³ ~ c  ² * °² c c ³³O%Oc b (²5b %³Á 45 ²9 À 1 ³ where 5 ~ 5 ²%³ is the largest integer such that O5b %O is less than 1. This yields (²%³  * O%Oc , ²9À2³ with * ~ 2c * °² c c ³ b sup (²%³, where we note that by the definition of 5 , °O%O 5b °  O %O  À Also note (² %³O%Oc ‚ (²5b %³ since O%O  . On the other hand, if °  O%O  , then (9.2) also clearly holds, yielding the desired bound for all %  ) À On the other hand if    5b O(²%³O ~ f (²2c %³ c (²cc %³f  *  Oc %Oc ~ * O%Oc °² c  ³, B B ~ ~ ²9 À 3 ³ yielding * ~ * °² c  ³À (b) In this case O(²%³ c (²³O ~ lim f 6(²2c %³ c (²cc %³7 b (²c5c %³ c (²³f 5¦B 5 ~ ~ f (²2c %³ c (²cc %³f B ~  * O%Oc °² c  ³, where the last inequality follows as in ²9À3³. (c) Here we again need only prove the reverse implication. Our assertion is equivalent to showing that if O(²%³ c (²%°³O  + ²%³O%Oc where + ²%³ is positive with + ²%³ ­ , then (²%³  + ²%³O%Oc , where + has the same properties as + . First consider the case  € À Then under our assumptions %¦ (²%³  f <(²2 %³ c (²b %³=f b (²5b %³ 5 ~ 5  *  + ²b %³Ob %Oc b (²5b %³À ~ where 5 is chosen as above. We now redefine 5 ²%³ to remain an integer for all %, but so that as % ¦  we have 2 %°  5b %  2 %° À Then by part (a), since we know at least that (²%³ ~ 6²O%Oc ³, (²5b %³  2 O5b %Oc  2 O%Oc° ~ ²O%Oc ³À Consider the ratio (recall 5 ~ 5 ²%³) 46 b b c f + ² %³O %O f 5 ~ c²5 b³ c O%Oc c cc b b c f + ² %³O %O f 5 ~ ~  Ob %Oc 5 À ~  * sup + (2b %³ 5 ­ À %¦ Thus (²%³ ~ ²O%Oc ³, as desired, if  € À Now assume   , and that (²³ ~ À In that case, O(²%³O ~ f (²2c %³ c (²cc %³f, B ~  *  + ²c %³Oc %Oc B ~ ~ sup + ²c %³ * O%Oc °² c  ³ B ~ ²O%Oc ³, as desired. Below is a more general version of Theorem 3.4, with proof included. Theorem 3.4: The following statements are equivalent for  €  and a positive function  ²%³ on the unit ball ) of l with      (where all integrals are restricted to the unit ball): (a) The integral O%O€ %  ²%³ ~ 6² c ³À (b) °O%O %  ²%³ ~ 6² c ³À (c) °O%O % |%|  ²%³ ~ 6²³À (d) For some (or all)   , O%O€ %  ²%³ O%O ~ 6² c ³ ´and for  € , % ²%³O%O  BµÀ (d') For some (or all)   l, c/2<O%O< %  ²%³ O%O ~ 6² c ³À (d'') For some (or all)  € , O%O< %  ²%³ O%O ~ 6² c ³À (d''') For some (or all)  €  , and some (or all)  with  b  c   , % ²%³O%O ² b O%O³ ~ 6²+ c ³ ´and for  €  and  b  c  € Á then lim % ²%³O%O ² b O%O³ exists and is finiteµÀ ¦ (d'''') For some (or all) Á   l , c/2<O%O< %  ²%³ O%O ² b O%O³ ~ 6² bc ³À (e) For any function ²O%O³  BC such that % |%|c ²O%O³  B , it follows that % ²O%O³ O%O  ²%³  BÁ for some (or all) * € À Statements in brackets ´ h µ may be included or excluded without changing the equivalences. 47 In addition, 6² h ³ may be replaced by ² h ³ simultaneously in all of the above statements excluding (c), and the equivalences of (a)-(ZZ ) (i.e. all statements excluding ()) continue to hold. Proof of Theorem 3.4: We first prove the statements above are equivalent without the bracketed inclusions. Whenever we apply Lemma 3.2, we implicitly extend  ²%³ to have value 0 for O%O € . That (a) ¯ (b) follows from Lemma 3.2, choosing (²³ ~ O%O€ %  ²%³À To show (b) ¬ (c) assume (b) holds. Then °O%O %    ²%³ ~ 6²³À However, °O%O % ²°³  ²%³  °O%O % O%O  ²%³  °O%O %    ²%³ ²9 À 4 ³ so (c) follows. This argument can be reversed to yield (c) ¬ (b). The equivalence of ( ) and (Z ) is proved in essentially the same way as that of () and ( ). To show ( ) ¬ () assume ( ) holds and let * € À Assume %O%Oc ²|%|³  B for some ²|%|³  B* . Letting  ~ sup ²³, and  the same quantity with sup replaced c cb by inf ,  c O%Ocb % O%O  ²%³   c O%Ocb  % ²O%O³O%O  ²%³ c O%Ocb % O%O  ²%³À Since ²O%O³  B* , we have for some 2 € Á °2  ²³°²³  2 for 1    À Thus if O+O is the surface area of the unit sphere in  dimensions,   2  2 h cb c  ²³  % O%Oc ²O%O³ O+O c O%Ocb 2  % O%Oc ²O%O³À O+O c O%Ocb ~2 Thus 48 % ²O%O³O%O  ²%³ ~  B c cb ~  O%O B   c O%Ocb ~  8sup ‚ % O%O  ²%³ % O%O  ²%³ 9  B ~ 8 sup ‚ % ²O%O³O%O  ²%³ c O%Ocb c O%Ocb 2 O+O ~ % O%O  ²%³ 9 h 2 O+O c O%Ocb % O%Oc ²O%O³ % O%Oc ²O%O³  BÁ implying (³. To show () ¬ ( ), assume () for some * € . We wish to show   ²%³ ~ 6²³, or equivalently  – % | % | % | % |  ²%³ ~ 6²³ as the  °O%O c O%Ocb integer  becomes largeÀ For this it suffices to show that for any positive summable sequence ¸ ¹, the sequence H c O%Ocb % |%|  ²%³I ²9 À 5 ³ is also summable. Further it suffices to show the collection of sequences for which this holds includes the summable sequences ¸ ¹ satisfying °*  b °  *À ²9À6³ Indeed if  £ 6²³ (i.e., is unbounded), let ¸ ¹ be a subsequence satisfying  ‚ c À We could then choose a family of sequences ¸ ¹ , defined by  ~ c * cOc O , and then define  ~  À Since for each  the sequence  (as a function of  ) satisfies ²9À6³, it  follows that  does as well. Furthermore, clearly  is summable. And finally we would have   ‚   ‚ : ; 0 ‚   0 ~ c   ~ B.       Thus it would be false that ²9À5³ is summable for all ¸ ¹ satisfying ²9À6³. This shows that it suffices to show ²9À5³ is summable for all summable ¸ ¹ satisfying ²9À6³. Given an arbitrary summable ¸ ¹ satisfying ²9À6³, there exists a function ²O%O³ ¢ l ¦ l such that  ~ c O%Ocb % O%Oc ²O%O³. Indeed let  be chosen so c ²³ is constant on each dyadic interval c    cc and equals   °O+O, where + is the surface of the unit  -ball. In this case 49 % O%Oc ²O%O³ ~ O+O c O%Ocb c cb  c   °O+O ~  . Then %O%Oc ²O%O³ ~    BÀ B  =1 For any constant     2 we have c ²³ ~   °O+O for c    cb , while ²³c ²³ ~   °O+O or 2c c °O+OÀ Then  c  c ²³c ²³   Á ~ ~ J c  ²³ ²³ or  c ²³ while c  c    J   * or   max²*°Á ³ implying that for        c ²³°²³  max²*°Á ³Á * and so   ²³°²³  max²*Á ³. * Further, letting  ~  above, we have ²2³c ²2³ ~ 2c c °O+O, so ²2³c ²2³ c c ~ Á c ²³   and so  ²2³c ²2³ *   Á c  ²³ *  and  ²2³   *Á * ²³ so ²³  B* À 50 Since we are assuming (),  % |%|  ²%³ ~  B  =1 B c O%Ocb  =1 B c O%Ocb  =1 c O%Ocb  % c O%Oc ²O%O³O+O |%|  ²%³ % O%OO%Oc ²O%O³O+O |%|  ²%³  B, proving ²9À5³ is summable for an arbitrary summable sequence ¸ ¹ satisfying ²9À6³. Thus c O%Ocb % |%|  ²%³ ~ 6²³, implying ²³, and showing () ¬ ( ). We have thus showed equivalence of (), (), ( ), (Z ), and (). For   , Lemma 3.2 implies () and (Z ) are equivalent. If ( Z ) holds for some   l, then by the same arguments as earlier (showing equivalence of () and ( )) it holds for all   l, proving the equivalence of ( ) and (Z ) if in (Z )  ‚  as well. To show equivalence of (d'') and ( Z ), note the implication (ZZ ) ¬ ²Z ³ is clear. The reverse follows once we observe that if (Z ) holds, then (recall  ²%³ may be assumed 0 outside the unit ball) lim ¦B O%O€c %  ²%³ O%O ~ lim ¦B  c  Z ~c c  ' O%O 'b %  ²%³ O%O  lim  * h ² c³  BÁ ¦B Z  Z ~c so % ²%³O%O  B. Therefore if (²³ – O%O %  ²%³ O%O , we have (²³ ~ Á and so by Lemma 3.2 (²³ ~ 6² c ³. That (ZZZZ ) is equivalent to (Z ) is clear since if °  O%O   , then Á O%OÁ and  b O%O are all of the same order. To show (ZZZ ) is equivalent to (ZZZZ ) it is first clear (ZZZ ) implies (ZZZZ ). Now assume ZZZZ ( ). Then if  €  and  b  c   , it follows that   , so that %  ²%³O%O ² b O%O³ ~   %  ²%³O%O b O%O  b c ~ 6² O%O %  ²%³O%O ² b O%O³ b O%O€ O%O€ %  ²%³ O%O ² b O%O³ %  ²%³ O%Ob ³ b 6² bc ³ as desired, where we have used ( ) and (ZZ ). Thus the ()-() are equivalent. To prove that we may also include the bracketed statements in ( ) and (ZZZ ), it suffices to prove the statement in brackets in () follows from the unbracketed statement there, and similarly for (ZZZ ). First assume the initial part of () holds, i.e., that for some (or 51 all)   , O%O€ %  ²%³ O%O ~ 6² c ³. Then by what has already been proved (i.e., the equivalences of the unbracketed statements), if  € , O%O % ²%³ O%O ~ 6² c ³Á so %  ²%³ O%O  BÀ Similarly, if the initial part of ( ZZZ ) holds and  b  c  € Á then we have by the dominated convergence theorem %  ²%³O%O ² b O%O³ ­ ¦ %  ²%³O%Ob  BÁ where the right side is finite by the fact that the bracketed part of () holds as shown above. Note the left side is always finite if  € , again by the bracketed part of ( ). This completes the proof of the equivalence of statements ()-(). To complete the proof we now assume statements (a)-(d'''' ) have 6² h ³ replaced by ² h ³À The equivalence of (a) and (b) then follows directly from Lemma 3.2( ). The equivalence of parts (b), (c), (d'), and (d'''') follows from the fact that, multiplied by appropriate powers of  , the left sides of the expressions in all of these parts have the same order ²i.e., are equivalent as functions of ). The equivalence of (d) and (d') is proved in the same way as that of (a) and (b). The equivalence of (d') and (d'') again follows from Lemma 3.2( ). To show that (d''') and (d'''') are equivalent, it is first immediate that (d''') ¬ (d''''). To prove the reverse implication, assume that (d'''') holds. Then the proof of (d''') follows identically to the proof of (d'''') ¬ (d''') in the previous case above. This completes the proof. Proof of Lemma 9.2: Assume ? is complete. If P P  BÁ let 5 ~   À Then 5  ~ ¸5 ¹ forms a Cauchy sequence, and so since ? is complete there is a  such that j  c j ­ ¦ BÀ 5 Conversely assume that whenever P P  B it follows  converges. Then if ~   ¸ ¹ is a Cauchy sequence let  be a subsequence satisfying P c  P  c for all  €  . Then  ~  b  ² Z c  Z c ³À By our assumption since the infinite sum  Z  of the norms in the previous expression is finite, P c  P ­  for some   ?. Thus ¦B by the triangle inequality P c  P ­ ¦ B  and ? is complete. Proof of Lemma 9.3: Assume these hypotheses and let P P  BÀ Then for each  P P  B, and since P h P is complete,  converges in P h P À By our assumption   5 P  c  P ¦  for all  for some fixed  .  5¦B ~ Note if  ¦  in the  norm, then P P ¦ PP À Indeed P P  PP b P c PÁ and PP  P P b P c  P. Thus if  ~  in P h P (recall such sums are not unique), then P P  P P , since i  i 5  ~  ­  5 ¦B P P . 52 Now j  c  j ~ supj  c  j 5 5  ~ ~ B ~ supj   j  ~5b B    sup  P P   ~5b B  sup P P ~5b B  ~  P P ­ ~5b 5 ¦ B Á proving completeness of P h PÀ Proof of Corollary 3.5: Define the norms associated with the above statements as follows (subscript refers to statement): P P – sup   O%O€ % | ²%³|  P P – sup   °O%O % | ²%³| ´Áµ ´Áµ P P – sup °O%O % |%| | ²%³| P P – sup  c ´Áµ P PZ – sup  c ´Áµ P PZZZ ´Áµ % | ²%³| O%O % | ²%³| O%O  P PZZ – sup  c c/2<O%O< – sup  cc % | ²%³|O%O ² b O%O³  ´Áµ P PZZZZ – sup  cc ´Áµ  P P  – sup PP B B* O%O€ °O%O % | ²%³|O%O ´Áµ % | ²%³| O%O O%O<   ² b O%O³  % ²O%O³ O%O O ²%³O. These are norms since the triangle inequality can be verified for all of them, and they are all positive definite. Further, defining as each norm's domain the space of functions  ¢ ) ¦ ] on which it is finite, each of these norms has the same domain ( by Theorem 3.4À We claim each of these norms is complete on (À The proof of completeness is similar for all the norms. To show P h P is complete for example, note P P is equivalent to the norm P Pi – sup c  tb cc O%Oc % | ²%³|Á where tb denotes the nonnegative integers. Defining the seminorm c  P Pi Á –  cc O%Oc % | ²%³|, we see this seminorm is complete, being equivalent  to an 3 seminorm. In addition, if  is a sequence which converges in each seminorm i Á  to  , and supP Pi Á  B, then clearly there is a   ( such that  ¦  in the norm  53 P h Pi Á Á  being equal to  in the interval cb  O%O  c À Thus by Lemma 9.3 the norm P h Pi is complete, and hence so is P h P À To show P h P is complete on the domain (, define for each positive  with PPB ~  the norm P PÁ ~ % ²O%O³ O%O O ²%³OÀ Then ( is complete in this norm, since it is a weighted 3 norm. If we have  ¦  in each of the norms P h PÁ , then since each  is positive, it follows that the functions  must all be the same, i.e.,  –  for some fixed  . Thus by Lemma 9.3 P h P is complete in (À The proofs of completeness for the other norms follow similarly. By Lemma 9.1 in order to prove equivalence of the norms on their common domain ( it now suffices to show that if  ¦ 0 in one of these norms and converges in a second, then  ¦ 0 in the second norm. Since convergence in all the norms implies 3 convergence on compacts not containing 0, it is easy to see that the same limit must be obtained in all the norms if it exists. Thus the norms are all equivalent, proving the equivalence of (a) - (e). To prove equivalence of (e) and (e'), note that for fixed * , defining the space A* ~ ¸  B* ¢ PPB  B¹Á each  defines a linear functional - on A* , defined by - ²³ ~ % ²O%O³ O%O q ²%³À By the uniform boundedness principle, the family ¸- ¹ is uniformly bounded on A* if and only if it is uniformly bounded for each   A* À However, uniform boundedness of ¸- ¹ on 2* is equivalent to (e), while uniform boundedness for each   A* is equivalent to ²Z ³, proving equivalence of (e) and (e'). That the bracketed statement in (d) follows from the unbracketed statement follows from the fact that we have already showed that (d) implies (d''), which completes the proof. In (d'''), if  b  c  € Á then it follows from (d'''') that c/2<O%O< %q ²%³O%O ² b O%O³  2  bc À (9.6) Summing the left side for  ~ ° , with  ~ Á Á Á à , we get  %q ²%³O%O ² b O%O³ À On the other hand, the right side of (9.6) adds up to a finite number, giving the bracketed part of (d'''). This completes the proof.  54