The Schwartz Space: A Background to White Noise Analysis

THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS JEREMY J. BECNEL AND AMBAR N. SENGUPTA Abstract. An account of the Schwartz space of rapidly decreasing functions as a nuclear space is presented, along with a description of the Gaussian measure on the dual space. 1. Introduction In this expository paper we present an account of • the Schwartz space S(Rd ) of rapidly decreasing functions as a nuclear space, in a largely self-contained way, concluding with a construction of the standard Gaussian measure on the dual space S ′ (Rd ), directed primarily to those who plan to delve further into white noise analysis. We work out the properties of the useful operator (1.1) T =− x2 1 d2 + + 2 dx 4 2 on S(R), in terms of creation and annihilation operators, and describe in brief the origins of these notions in quantum mechanics. The operator T arose in quantum mechanics as the Hamiltonian for a harmonic oscillator and, in that context as well as in white noise analysis, the operator N = T − 1 is called the number operator. Our exposition of the properties of T and of S(R) follows Barry Simon’s [4], but we provide more detail (and our notational conventions are slightly different). For the Gaussian measure, we describe a direct construction using the Kolmogorov theorem, instead of the more general (and more difficult) Minlos theorem. 2. Objectives In this section we summarize the basic notions, notation, and results that we discuss in more detail in later sections. Here, and later in this paper, we will work mainly with the case of functions of one variable and then describe the generalization to the multi-dimensional case. We will use the letter W to denote the set of all non-negative integers: (2.1) W = {0, 1, 2, 3, ...} Date: August 2004. Research supported by US NSF grant DMS-0201683. 1 2 JEREMY J. BECNEL AND AMBAR N. SENGUPTA 2.1. The Schwartz Space. The Schwartz space S(R) is the linear space of all functions f : R → C which have derivatives of all orders and which satisfy the condition def pa,b (f ) = sup |xa f b (x)| < ∞ x∈R for all a, b ∈ W = {0, 1, 2, ...}. The finiteness condition for all a ≥ 1 and b ∈ W , implies that xa f b (x) actually goes to 0 as |x| → ∞, for all a, b ∈ W , and so functions of this type are said to be rapidly decreasing. 2.2. The Schwartz Topology. The functions pa,b are semi-norms on the vector space S(R), in the sense that pa,b (f + g) ≤ pa,b (f ) + pa,b (g) and pa,b (zf ) = zpa,b (f ) for all f, g ∈ S(R), and z ∈ C. For this semi-norm, an open ball of radius r centered at some f ∈ S(R) is given by (2.2) Bpa,b (f ; r) = {g ∈ S(R) : pa,b (g − f ) < r} Thus each pa,b specifies a topology τa,b on S(R). A set is open according to τa,b if it is a union of open balls. The topologies τa,b put all together, generate the standard Schwartz topology τ on S(R). This is the smallest topology containing all the sets of τa,b for all a, b ∈ W . There is a different approach to the topology on S(R) that is very useful for analysis, and much of our effort in this paper will go into demonstrating the equivalence of the two ways of understanding the topology on S(R). 2.3. The operator T . The operator x2 1 d2 + + 2 dx 4 2 plays a very useful role in working with the Schwartz space. As we shall see, there is an orthonormal basis {φn }n∈W of L2 (R, dx), where T =− (2.3) W = {0, 1, 2, ...}, consisting of eigenfunctions φn of T : (2.4) T φn = (n + 1)φn The functions φn , called the Hermite functions are actually in the Schwartz space S(R). Let B be the bounded linear operator on L2 (R) given on each f ∈ L2 (R) by X (2.5) Bf = (n + 1)−1 hf, φn iφn n∈W It is readily checked that the right side does converge and, in fact, X X (2.6) ||Bf ||2L2 = |hf, φn i|2L2 = ||f ||2L2 (n + 1)−2 |hf, φn i|2L2 ≤ n∈W n∈W Note that B and T are inverses of each other on the linear span of the vectors φ n : (2.7) T Bf = f and BT f = f for all f ∈ L, THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 3 where L = linear span of the vectors φn , for n ∈ W (2.8) 2 2.4. The L approach. For any p ≥ 0, the image of B p consists of all f ∈ L2 (R) for which X (n + 1)2p |hf, φn i|2 < ∞. n∈W Let Sp (R) = B p L2 (R) (2.9)
This is a subspace of L2 (R), and on Sp (R) there is an inner-product h·, ·ip given by X def (2.10) hf, gip = (n + 1)2p hf, φn ihφn , gi = hB −p f, B −p giL2 n∈W which makes it a Hilbert space, with L, and hence also S(R), a dense subspace. We will see later that functions in Sp (R) are p-times differentiable. We will prove that the intersection ∩p∈W Sp (R) is exactly equal to S(R). In fact, (2.11) S(R) = ∩p∈W Sp (R) ⊂ · · · S2 (R) ⊂ S1 (R) ⊂ S0 (R) = L2 (R) We will also prove that the topology on S(R) generated by the norms || · ||p coincides with the standard topology. Furthermore, the elements (n + 1)−p φn ∈ Sp (R) form an orthonormal basis of Sp (R), and X X n2p < ∞, ||(n + 1)−(p+1) φn ||2p = n2(p+1) n∈W n≥1 showing that the inclusion map Sp+1 (R) → Sp (R) is Hilbert-Schmidt. The topological vector space S(R) has topology generated by a complete metric, and has a countable dense subset given by all finite linear combinations of the vectors φn with rational coefficients. 2.5. Coordinatization as a Sequence Space. All of the results described above follow readily from the identification of S(R) with a space of sequences. Let {φn }n∈W be the orthonormal basis of L2 (R) mentioned above, where W = {0, 1, 2, ...} Then we have the set CW . An element a ∈ CW is a map W → C : n 7→ an . So we shall often write such an element a as (an )n∈W . We have then the coordinatizing map (2.12) I : L2 (R) → CW : f 7→ (hf, φn i)n∈W For each p ∈ W let Ep be the subset of CW consisting of all (an )n∈W such that X (n + 1)2p |an |2 < ∞ n∈W On Ep define the inner-product h·, ·ip by X (2.13) ha, bip = (n + 1)2p an bn n∈W 4 JEREMY J. BECNEL AND AMBAR N. SENGUPTA This makes Ep a Hilbert space, essentially the Hilbert space L2 (W, µp ) where µp is the measure on W given by µp ({n}) = (n + 1)2p for all n ∈ W . The definition (2.9) for Sp (R) shows that it is the set of all f ∈ L2 (R) for which I(f ) belongs to Ep . We will prove in Theorem 11.1 that I maps S(R) exactly onto def E = ∩p∈W Ep (2.14) This will establish essentially all of the facts mentioned above concerning the spaces Sp (R). Note the chain of inclusions: (2.15) E = ∩p∈W Ep ⊂ · · · ⊂ E2 ⊂ E1 ⊂ E0 = L2 (W, µ0 ) 2.6. The multi-dimensional setting. In the multidimensional setting, the Schwartz space S(Rd ) consists of all infinitely differentiable functions f on Rd for which sup xk11 ...xkdd x∈Rd ∂ m1 +···+mk f (x) < ∞, md 1 ∂xm 1 ...∂xd for all (k1 , ..., kd ) ∈ W d and m = (m1 , ..., md ) ∈ W d . For this setting, it is best to use some standard notation: (2.16) (2.17) |k| = k1 + · · · + kd for k = (k1 , ..., kd ) ∈ W d xk = xk1 ...xkd Dm = (2.18) ∂ |m| md 1 ∂xm 1 ...∂xd For the multi-dimensional case, we use the indexing set W d whose elements are d–tuples j = (j1 , ..., jd ), with j1 , ..., jd ∈ W , and counting measure µ0 on W d . The d d sequence space is replaced by CW ; a typical element a ∈ CW , is a map a : W d → C : j = (j1 , ..., jd ) 7→ aj = aj1 ...jd (2.19) The orthonormal basis (φn )n∈W of L2 (R) yields an orthonormal basis of L2 (Rd ) consisting of the vectors (2.20) φj = φj1 ⊗ · · · ⊗ φjd : (x1 , ..., xd ) 7→ φj1 (x1 )...φjd (xd ) The coordinatizing map I is replaced by the map Id : L2 (Rd ) → CW (2.21) where Id (f )j = hf, φj iL2 (Rd ) (2.22) Replace the operator T by (2.23) Then def Td = T ⊗d = d     x21 x2d ∂2 ∂2 + 1 ··· − 2 + +1 − 2 + ∂xd 4 ∂x1 4 Td φj = (j1 + 1)...(jd + 1)φj for all j ∈ W d . In place of B, we now have the bounded operator Bd on L2 (Rd ) given by X [(j1 + 1)...(jd + 1)]−1 hf, φj iφj (2.24) Bd f = j∈W d THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 5 Again, Td and Bd are inverses of each other on the linear subspace Ld of L2 (Rd ) spanned by the vectors φj . d d The space Ep is now the subset of CW consisting of all a ∈ CW for which X [(j1 + 1)...(jd + 1)]2p |aj |2 < ∞ j∈W d and this is a Hilbert space with inner-product X [(j1 + 1)...(jd + 1)]2p aj bj ha, bib = j∈W d Again we have the chain of spaces def E = ∩p∈W Ep ⊂ · · · E2 ⊂ E1 ⊂ E0 = L2 (W d , µ0 ), with the inclusion Ep+1 → Ep being Hilbert-Schmidt. To go back to functions on Rd , define Sp (Rd ) to be the range of Bd . Thus Sp (Rd ) is the set of all f ∈ L2 (Rd ) for which X [(j1 + 1)...(jd + 1)]2p |hf, φj i|2 < ∞ j∈W d The inner-product h·, ·ip comes back to an inner-product, also denoted h·, ·ip , on Sp (Rd ) and is given by (2.25) hf, gip = hBd−p f, Bd−p giL2 (Rd ) The intersection ∩p∈W Sp (Rd ) equals S(Rd ). Moreover, the topology on S(Rd ) is the smallest one generated by the inner-products obtained from h·, ·ip , with p running over W . 3. Topological Vector Spaces The Schwartz space is a topological vector space, i.e. it is a vector space equipped with a Hausdorff topology with respect to which the vector space operations (addition, and multiplication by scalar) are continuous. In this section we shall go through a few of the basic notions and results for topological vector spaces. Let V be a real vector space. A vector topology τ on V is a topology such that addition V × V → V : (x, y) 7→ x + y and scalar multiplication R × V → V : (t, x) 7→ tx are continuous. If V is a complex vector space we require that C × V → V : (α, x) 7→ αx be continuous. It is useful to observe that when V is equipped with a vector topology, the translation maps tx : V → V : y 7→ y + x are continuous, for every x ∈ V , and are hence also homeomorphisms since t −1 x = t−x . A topological vector space is a vector space equipped with a Hausdorff vector topology. A local base of a vector topology τ is a family of open sets {Uα }α∈I containing 0 such that if W is any open set containing 0 then W contains some Uα . If U is any open set and x any point in U then U − x is an open neighborhood of 0 and hence contains some Uα , and so U itself contains a neighborhood x + Uα of x: (3.1) If U is open and x ∈ U then x + Uα ⊂ U , for some α ∈ I 6 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Doing this for each point x of U , we see that each open set is the union of translates of the local base sets Uα . 3.1. Local Convexity and the Minkowski Functional. A vector topology τ on V is locally convex if for any neighborhood W of 0 there is a convex open set B with 0 ∈ B ⊂ W . Thus, local convexity means that there is a local base of the topology τ consisting of convex sets. The principal consequence of having a convex local base is the Hahn-Banach theorem which guarantees that continuous linear functionals on subspaces of V extend to continuous linear functionals on all of V . In particular, if V 6= {0} is locally convex then there exist non-zero continuous linear functionals on V . Let B be a convex open neighborhood of 0. Continuity of R × V → V : (s, x) 7→ sx at s = 0 shows that for each x the multiple sx lies in B if s is small enough, and so t−1 x lies in B if t is large enough. The smallest value of t for which t−1 x is just outside B is clearly a measure of how large x is relative to B. The Minkowski functional µB is the function on V given by µB (x) = inf{t > 0 : t−1 x ∈ B} Note that 0 ≤ µB (x) < ∞. The definition of µB shows that µB (kx) = kµB (x) for any k ≥ 0. Convexity of B can be used to show that µB (x + y) ≤ µB (x) + µB (y) If B is symmetric, i.e. B = −B, then µB (kx) = |k|µB (x) for all real k. If V is a complex vector space and B is balanced in the sense that αB = B for all complex numbers α with |α| = 1, then µB (kx) = |k|µB (x) for all complex k. Note that in general it could be possible that µB (x) is 0 without x being 0; this would happen if B contains the entire ray {tx : t ≥ 0}. 3.2. Semi-norms. A typical vector topology on V is specified by a semi-norm on V , i.e. a function µ : V → R such that (3.2) µ(x + y) ≤ µ(x) + µ(y), µ(tx) = |t|µ(x) for all x, y ∈ V and t ∈ R (complex t if V is a complex vector space). Note that then, using t = 0, we have µ(0) = 0 and, using −x for y, we have µ(x) ≥ 0. For such a semi-norm, an open ball around x is the set (3.3) Bµ (x; r) = {y ∈ V : µ(y − x) < r}, and the topology τµ consists of all sets which can be expressed as unions of open balls. These balls are convex and so the topology τµ is locally convex. If µ is actually a norm, i.e. µ(x) is 0 only if x is 0, then τµ is Hausdorff. A consequence of the triangle inequality (3.2) is that a semi-norm µ is uniformly continuous with respect to the topology it generates. To see this, consider any x, y ∈ V . From µ(x) = µ(x − y + y) ≤ µ(x − y) + µ(y) we have µ(x) − µ(y) ≤ µ(x − y) Interchanging x and y, we conclude that (3.4) |µ(x) − µ(y)| ≤ µ(x − y), THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 7 which implies that µ, as a function on V , is continuous with respect to the topology τµ it generates. Now suppose µ is continuous with respect to a vector topology τ . Then the open balls {y ∈ V : µ(y − x) < r} are open in the topology τ and so τµ ⊂ τ . 3.3. Topologies generated by families of topologies. Let {τα }α∈I be a collection of topologies on a space. It is natural and useful to consider the the least upper bound topology τ , i.e. the smallest topology containing all sets of ∪ α∈I τα . In our setting, we work with each τα a vector topology on a vector space V . Theorem 3.1. The least upper bound topology τ of a collection {τα }α∈I of vector topologies is again a vector topology. If {Wα,i }i∈Iα is a local base for τα then a local base for τ is obtained by taking all finite intersections of the form Wα1 ,i1 ∩ · · · ∩ Wαn ,in . Proof. Let B be the collection of all sets which are of the form Wα1 ,i1 ∩ · · · ∩ Wαn ,in . Let τ ′ be the collection of all sets which are unions of translates of sets in B (including the empty union). Our first objective is to show that τ ′ is a topology on V . It is clear that τ ′ is closed under unions and contains the empty set. We have to show that the intersection of two sets in τ ′ is in τ ′ . To this end, it will suffice to prove the following: (3.5) If C1 and C2 are sets in B, and x is a point in the intersection of the translates a + C1 and b + C2 , then x + C ⊂ (a + C1 ) ∩ (b + C2 ) for some C in B. Clearly, it suffices to consider finitely many topologies τα . Thus, consider vector topologies τ1 , ..., τn on V . Let Bn be the collection of all sets of the form B1 ∩ · · · ∩ Bn with Bi in a local base for τi , for each i ∈ {1, ..., n}. We can check that if D, D ′ ∈ Bn then there is an E ∈ Bn with E ⊂ D ∩ D ′ . Working with Bi drawn from a given local base for τi , let z be a point in the intersection B1 ∩ · · · ∩ Bn . Then there exist sets Bi′ , with each Bi′ being in the local base for τi , such that z + Bi′ ⊂ Bi (this follows from our earlier observation (3.1)). Consequently, z + ∩ni=1 Bi′ ⊂ ∩ni=1 Bi Now consider sets C1 an C2 , both in Bn . Consider a, b ∈ V and suppose x ∈ (a+C1 )∩(b+C2 ). Then since x−a ∈ C1 there is a set C1′ ∈ Bn with x−a+C1′ ⊂ C1 ; similarly, there is a C2′ ∈ Bn with x − b + C2′ ⊂ C2 . So x + C1′ ⊂ a + C1 and x + C2′ ⊂ b + C2 . So x + C ⊂ (a + C1 ) ∩ (b + C2 ), where C ∈ Bn satisfies C ⊂ C1 ∩ C2 . This establishes (3.5), and shows that the intersection of two sets in τ ′ is in τ ′ . Thus τ ′ is a topology. The definition of τ ′ makes it clear that τ ′ contains each τα . Furthermore, if any topology σ contains each τα then all the sets of τ ′ are also open relative to σ. Thus τ ′ = τ, the topology generated by the topologies τα . Observe that we have shown that if W ∈ τ contains 0 then W ⊃ B for some B ∈ B. 8 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Next we have to show that τ is a vector topology. The definition of τ shows that τ is translation invariant, i.e. translations are homeomorphisms. So, for addition, it will suffice to show that addition V × V × V : (x, y) 7→ x + y is continuous at (0, 0). Let W ∈ τ contain 0. Then there is a B ∈ B with 0 ∈ B ⊂ W . Suppose B = B1 ∩ · · · ∩ Bn , where each Bi is in the given local base for τi . Since τi is a vector topology, there are open sets Di , Di′ ∈ τi , both containing 0, with Di + Di′ ⊂ Bi Then choose Ci , Ci′ in the local base for τi with Ci ⊂ Di and Ci′ ⊂ Di′ . Then Ci + Ci′ ⊂ Bi Now let C = C1 ∩· · ·∩Cn , and C ′ = C1′ ∩· · ·∩C ′ n. Then C, C ′ ∈ B and C +C ′ ⊂ B. Thus, addition is continuous at (0, 0). Now consider the multiplication map R×V → V : (t, x) 7→ tx. Let (s, y), (t, x) ∈ R × V . Then sy − tx = (s − t)x + t(y − x) + (s − t)(y − x) Suppose F ∈ τ contains tx. Then F ⊃ tx + W ′ , for some W ′ ∈ B. Using continuity of the addition map V × V × V → V : (a, b, c) 7→ a + b + c at (0, 0, 0), we can choose W1 , W2 , W3 ∈ B with W1 + W2 + W3 ⊂ W ′ . Then we can choose W ∈ B, such that W ⊂ W1 ∩ W2 ∩ W3 Then W ∈ B and W + W + W ⊂ W′ Suppose W = B1 ∩ · · · ∩ Bn , where each Bi is in the given local base for the vector topology τi . Then for s close enough to t, we have (s − t)x ∈ Bi for each i, and hence (s−t)x ∈ W . Similarly, if y is τ –close enough to x then t(y −x) ∈ W . Lastly, if s − t is close enough to 0 and y is close enough to x then (s − t)(y − x) ∈ W . So sy − tx ∈ W ′ , and so sy ∈ F , when s is close enough to t and y is τ –close enough to x.  The above result makes it clear that if each τα has a convex local base then so is τ . Note also that if at least one τα is Hausdorff then so is τ . A family of topologies {τα }α∈I is directed if for any α, β ∈ I there is a γ ∈ I such that τα ∪ τ β ⊂ τ γ In this case every open neighborhood of 0 in the generated topology contains an open neighborhood in one of the topologies τγ . THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 9 3.4. Topologies generated by families of semi-norms. We are concerned mainly with the topology τ generated by a family of semi-norms {µα }α∈I ; this is the smallest topology containing all sets of ∪α∈I τµα . An open set in this topology is a union of translates of finite intersections of balls of the form Bµi (0; ri ). Thus any open neighborhood of f contains a set of the form Bµ1 (f ; r1 ) ∩ · · · ∩ Bµn (f ; rn ) This topology is Hausdorff if for any non-zero x ∈ V there is some norm µα for which µα (x) is not zero. The description of the neighborhoods in the topology τ shows that a sequence fn converges to f with respect to τ if and only if µα (fn − f ) → 0, as n → ∞, for all α ∈ I. We will need to examine when two families of semi-norms give rise to the same topology: Theorem 3.2. Let τ be the topology on V generated by a family of semi-norms M = {µi }i∈I , and τ ′ the topology generated by a family of semi-norms M′ = {µ′j }j∈J . Suppose each µi is bounded above by a linear combination of the µ′j . Then τ ⊂ τ ′ . Proof. Let µ ∈ M. Then there exist µ′1 , ..., µ′n ∈ M′ , and real numbers c1 , ..., cn > 0, such that µ ≤ c1 µ′1 + · · · + cn µ′n Now consider any x, y ∈ V . Then |µ(x) − µ(y)| ≤ µ(x − y) ≤ n X i=1 |ci |µ′i (x − y) So µ is continuous with respect to the topology generated by µ′1 , ..., µ′n . Thus, τµ ⊂ τ ′ . Since this is true for all µ ∈ M, we have τ ⊂ τ ′ .  3.5. Completeness. A sequence (x)n∈N in a topological vector space V is Cauchy if for any neighborhood U of 0 in V , the difference xn − xm lies in U when n and m are large enough. The topological vector space V is complete if every Cauchy sequence converges. Theorem 3.3. Let {τα }α∈I be a directed family of Hausdorff vector topologies on V , and τ the generated topology. If each τα is complete then so is τ . Proof. Let (xn )n≥1 be a sequence in V , which is Cauchy with respect to τ . Then clearly it is Cauchy with respect to each τα . Let xα = limn→∞ xn , relative to τα . If τα ⊂ τγ then the sequence (xn )n≥1 also converges to xγ relative to the topology τα , and so xγ = xα . Consider α, β ∈ I, and choose γ ∈ I such that τα ∪ τβ ⊂ τγ . This shows that xα = xγ = xβ , i.e. all the limits are equal to each other. Let x denote the common value of this limit. We have to show that xn → x in the topology τ . Let W ∈ τ contain x. Since the family {τα }α∈I which generates τ is directed, it follows that there is a β ∈ I and a Bβ ∈ τβ with x ∈ Bβ ⊂ W . Since (xn )n≥1 converges to x with respect to τβ , it follows xn ∈ Bβ for large n. So xn → x with respect to τ .  10 JEREMY J. BECNEL AND AMBAR N. SENGUPTA 3.6. Metrizability. Suppose the topology τ on the topological vector space V is generated by a countable family of semi-norms µ1 , µ2 , .... For any x, y ∈ V define X (3.6) d(x, y) = 2−n dn (x, y) n≥1 where dn (x, y) = min{1, µn (x − y)} Then d is a metric, it is translation invariant, and generates the topology τ . 4. The Schwartz Space S(R) Our objective in this section is to show that the Schwartz space is complete, in the sense that every Cauchy sequence converges. Recall that S(R) is the set of all C ∞ functions f on R for which def def pa,b (f ) = ||f ||a,b = sup |xa Db f (x| < ∞ (4.1) x∈R for all a, b ∈ W = {0, 1, 2, ...}. The functions pa,b are semi-norms, with || · ||0,0 , being just the sup-norm, is actually a norm. Thus the family of semi-norms given above specify a Hausdorff vector topology on S(R). We will call this the Schwartz topology on S(R). Theorem 4.1. The topology on S(R) generated by the family of semi-norms || · || a,b for all a, b ∈ {0, 1, 2, ...}, is complete. Proof. . Let (fn )n≥1 be a Cauchy sequence on S(R). Then this sequence is Cauchy in each of the semi-norms || · ||a,b , and so each sequence of functions xa Db fn (x) is uniformly convergent. Let gb (x) = lim Db fn (x) (4.2) n→∞ Let f = g0 . Using a Taylor theorem argument it follows that gb is Db f . For instance, for b = 1, observe first that
Z 1 Z 1
dfn (1 − t)x + ty) fn′ (1 − t)x + ty (y − x) dt, dt = fn (x) + fn (y) = fn (x) + dt 0 0 and so, letting n → ∞, we have f (y) = f (x) + Z 0 ′ 1
g1 (1 − t)x + ty (y − x) dt, which implies that f (x) exists and equals g1 (x). In this way, we have (4.3) xa Db fn (x) → xa Db f (x) pointwise. Note that our Cauchy hypothesis implies that the sequence of functions x a Db fn (x) is Cauchy in sup-norm, and so the convergence xa Db fn (x) → xa Db f (x) is uniform. In particular, the sup-norm of xa Db f (x) is finite, since it is the limit of a uniformly convergent sequence of bounded functions. Thus f ∈ S(R). Finally, we have to check that fn converges to f in the topology of S(R). We have noted above that xa Db fn (x) → xa Db f (x) uniformly. Thus fn → f relative THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 11 to the semi-norm || · ||a,b . Since this holds for every a, b ∈ {0, 1, 2, 3, ...}, we have fn → f in the topology of S(R).  Now lets take a quick look at the Schwartz space S(Rd ). First some notation. A multi-index a is an element of {0, 1, 2, ...}d , i.e. it is a mapping a : {1, ..., d} → {0, 1, 2, ...} : j 7→ aj If a is a multi-index, we write |a| to mean the sum a1 + · · · + ad , xa to mean the product xa1 1 ...xadd , and Db to mean the differential operator Dxa11 ...Dxadd . The space S(Rd ) consists of all C ∞ functions f on Rd such that each function xa Db f (x) is bounded. On S(Rd ) we have the semi-norms ||f ||a,b = sup |xa Db f (x)| x∈Rd for each pair of multi-indices a and b. The Schwartz topology on S(Rd ) is the smallest topology making each semi-norm || · ||a,b continuous. This makes S(Rd ) a topological vector space. The argument for the proof of the preceding theorem goes through with minor alterations and shows that Theorem 4.2. The topology on S(Rd ) generated by the family of semi-norms ||·||a,b for all a, b ∈ {0, 1, 2, ...}d , is complete. 5. Hermite Polynomials, Creation and Annihilation Operators We shall summarize the definition and basic properties of Hermite polynomials (our approach is essentially that of Hermite’s original [1]). A central role is played by the Gaussian kernel 2 1 (5.1) p(x) = √ e−x /2 2π Properties of translates of p are obtained from y2 p(x − y) (5.2) exy− 2 = p(x) Expanding the right side in a Taylor series we have (5.3) exy− y2 2 = p(x−y) p(x) = P∞ 1 n n=0 n! Hn (x)y , where the Taylor coefficients, denoted Hn (x), are (5.4) Hn (x) = 1 p(x) d − dx
n p(x) This is the n–th Hermite polynomial and is indeed an n–th degree polynomial in which xn has coefficient 1, facts which may be checked by induction. Observe the following Z Z 2 2 p(x − y) p(x − z) − y +z 2 p(x)dx = e e−x(y+z) p(x) dx p(x) p(x) R R y 2 +z 2 2 = e− = eyz + (y+z)2 2 12 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Going over to the Taylor series and assuming integrals and series can be interchanged at will, we have ∞ X X yn zn yn zm = hHn , Hm iL2 (p(x)dx) n!m! n! n=0 n,m≥0 Thus hHn , Hm iL2 (p(x)dx) = n!δnm (5.5) Thus an orthonormal set of functions is given by (5.6) hn (x) = √1 Hn (x) n! Since these are orthogonal polynomials, the n–th one being exactly of degree n, their span contains all polynomials. It can be shown that the span is in fact dense in L2 (p(x)dx). Thus the polynomials above constitute an orthonormal basis of L2 (p(x)dx). Next, consider the derivative of Hn : So Hn′ (x) = (−1)n p(x)−1 p(n+1) (x) − (−1)n p(x)−1 p′ (x)p(x)−1 p(n+1) (x) = −Hn+1 (x) + xHn (x) (5.7) The operator
√ d − dx + x hn (x) = n + 1 hn+1 (x)   d − +x dx
is the creation operator in L2 R; p(x)dx . More officially, we can take the creation operator toP have domain consisting of all functions f which can be expanded in L2 (p(x)dx) as n≥0 an hn , with each an a P complex number, and satisfying the condition n≥0 (n + 1)|an |2 < ∞; the action of √ P the operator on f yields the function n≥0 n + 1 an hn+1 . This makes the creation operator unitarily equivalent to a multiplication operator (in the sense discussed below in subsection 12.5) and hence a closed operator (see 12.1 for definition). For the type of smooth functions f we will mostly work with, the effect of the operator
d on f will in fact be given by application of − dx + x to f . Next, differentiating the fundamental generating relation (5.3) with respect to x we have X 1 2 H ′ (x)y n = yexy−y /2 n! n n≥0 X 1 y n+1 Hn (x) = n! n≥0 X 1 = Hn−1 (x)y n (n − 1)! n≥1 From this we see that (5.8) Hn′ (x) = nHn−1 (x) THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 13 where H−1 = 0. Thus: d dx hn (x) (5.9) = √ n hn−1 (x) The operator d dx
is the annihilation operator in L2 R; p(x)dx . As with the creation operator, we may define it in a more specific way, as a closed operator. 6. Hermite Functions, Creation and Annihilation Operators In the preceding section we
studied Hermite polynomials in the setting of the Gaussian space L2 R; p(x)dx . Let us translate the concepts are results back to
the usual space L2 R; dx . To this end, consider the unitary isomorphism: √ (6.1) U : L2 (R, p(x)dx) → L2 (R, dx) : f 7→ pf Then the orthonormal basis polynomials hn go over to the functions φn given by (6.2) 2 φn (x) = (−1)n √1n! (2π)−1/4 ex 2 /2 /4 dn e−x dxn The family {φm }n≥0 forms an orthonormal basis for L2 (R, dx). We will now determine the annihilation and creation operators on L2 (R, dx). If f ∈ L2 (R, dx) is differentiable and has derivative f ′ also in L2 (R, dx), we have:   i p d d h p(x)−1/2 f (x) U U −1 f (x) = p(x) dx dx
= f ′ (x) + p(x)1/2 −1/2 p(x)−3/2 p′ (x)f (x) 1 = f ′ (x) + xf (x) 2 So, on L2 (R, dx), the annihilator operator is (6.3) A= d dx + 12 x which will satisfy (6.4) Aφn = √ n φn−1 where φ−1 = 0. For the moment, we proceed by taking the domain of A to be the Schwartz space S(R). Next,     1 d −1 f (x) = −f ′ (x) + xf (x) − xf (x) +x U U − dx 2   d 1 = − + x f (x) dx 2 14 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Thus the creation operator is d + 12 x C = A∗ = − dx (6.5) The reason we have written A∗ is that, as is readily checked, we have the adjoint relation     1 d + x g (6.6) hAf, gi = f, − dx 2 with the inner-product being the usual one on L2 (R, dx). Again, for the moment, we take the domain of C to be the Schwartz space S(R) (though, technically, in that case we should not write C as A∗ , since the latter, if viewed as the L2 –adjoint operator, has a larger domain). For this we have √ (6.7) Cφn = n + 1 φn+1 Observe also that (6.8) AC = d2 1 2 1 x − 2+ I 4 dx 2 and CA = 1 2 d2 1 x − 2− I 4 dx 2 which imply: [A, C] = AC − CA = I, (6.9) Next observe that (6.10) CAφn = the identity √ √ n nφn = nφn and so CA is called the number operator N : (6.11) 2 d N = A∗ A = CA = − dx 2 + x2 4 − 1 2 the number operator As noted above in (6.10), the number operator N has the eigenfunctions φn : (6.12) N φn = nφn Integration by parts shows that hf, g ′ i = −hf ′ , gi for every f, g ∈ S(R), and so also hf, g ′′ i = hf ′′ , gi It follows that the operator N satisfies hN f, gi = hf, N gi (6.13) for every f, g ∈ S(R). Now consider the case of Rd . Then for each j ∈ {1, ..., d}, there are creation, annihilation, and number operators: (6.14) Aj = ∂ ∂xj + 21 xj , ∂ + 12 xj , Cj = − ∂x j Nj = Cj Aj THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 15 These map S(Rd ) into itself and satisfy the commutation relations (6.15) [Aj , Ck ] = δjk I, [Nj , Ak ] = −δjk Aj , [Nj , Ck ] = δjk Cj Now let us be more specific about the precise definition of the creation and annihilation operators. Given its effect on the orthonormal basis {φm }m∈W d , the operator Ck has the form: √ φm 7→ mk + 1φm′ , where m′i = mi for all i ∈ {1, ..., d} except when i = k, in which case m′k = mk + 1. The domain of Ck is the set D(Ck ) given by   X D(Ck ) = f ∈ L2 (R) (mk + 1)|am |2 < ∞ where am = hf, φm i m∈W d for The operator Ck is then officially defined by specifying its action on a typical element of its domain:   X X √ am mk + 1φm′ , am φm = (6.16) Ck m∈W d m∈W d ′ where m is as before. The operator Ck is essentially the composite of a multiplication operator and a bounded linear map taking φm → φm′ where m′ is as defined above. (See subsection 12.5 for precise formulation of a multiplication operator.) Noting this it can be readily checked that Ck is a closed operator using the following argument: Let T be a bounded linear operator and Mh a multiplication operator (any closed operator will do); we show that the composite Mh T is a closed operator. Suppose xn → x. Since T is a bounded linear operator, T xn → T x. Now suppose also that Mh (T xn ) → y. Since Mh is closed, it follows then that T x ∈ D(Mh ) and y = Mh T x. The operators Ak and Nk are defined analogously. Proposition 6.1. Let L0 be the vector subspace of L2 (Rd ) spanned by the basis vectors {φm }m∈W d . Then for k ∈ {1, 2, . . . , d}, Ck |L0 and Ak |L0 have closures given by Ck and Ak , respectively (see subsection 12.4 for the notion of closure). Proof. We need to show that the graph of Ck , Gr(Ck ), is equal to the closure of the graph of Ck |L0 , Gr(Ck |L0 ) (refer to subsection 12.1 for the notion of graph). It is clear that Gr(Ck |L0 ) ⊆ Gr(Ck ). Using this and the fact that Ck is a closed operator we have that Gr(Ck |L0 ) ⊆ Gr(Ck ) = Gr(Ck ) P Going in the other direction, take (f, Ck f ) ∈ Gr(Ck ). Now f = m∈W d am φm where am = hf, φm i. Let fN be given by X am φm where WNd = { m ∈ W d | 0 ≤ m1 ≤ N, . . . , 0 ≤ md ≤ N }. fN = d m∈WN Observe fN ∈ L0 . Moreover lim fN = f and lim Ck fN = lim N →∞ N →∞ N →∞ X (mk + 1)am φm = Ck f d m∈WN in L2 (Rd ). Thus (f, Ck f ) ∈ Gr(Ck |L0 ) and we have that Gr(Ck ) ⊆ Gr(Ck |L0 ). The proof for Ak follows similarly.  16 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Linking this new definition for Ck with our earlier formulas (6.14) we have: Proposition 6.2. If f ∈ S(Rd ) then Ck f = − ∂f xk + f, ∂xk 2 Ak f = and ∂f xk + f ∂xk 2 ∂f + x2k f . Since f ∈ S(Rd ), we have that g ∈ L2 (Rd ). So we Proof. Let g = − ∂x Pk can write g as g = j∈W d aj φj where aj = hg, φj i. Let us examine these aj ’s more closely. Observe E D ∂f xk aj = hg, φj i = − + f, φj ∂xk 2 E D ∂φ x k j + φj = f, ∂x 2 p k = hf, jk φj ′ i where ji′ = ji for all i ∈ {1, ..., d} except when i = k, in which case jk′ = jk − 1. Bringing this information back to our expression for g we see that g= X p jk hf, φj ′ iφj j∈W d = X √ m∈W d mk + 1hf, φm iφm′ = Ck f where m′ is as defined above by (6.16) The second equality is obtained by letting m = j ′ and noting that φj ′ = 0 when jk′ is −1. The proof follows similarly for Ak .  7. Properties of Sp (R) functions Our aim here is to demonstrate that functions in Sp (R) are p-times differentiable. The main tool we will use is the Fourier transform: Z e−ipx f (x) dx (7.1) fˆ(p) = Ff (p) = (2π)−1/2 R 1 This is meaningful whenever f is in L (R), but we will work mainly with f in S(R). We will use the following facts: • F maps S(R) onto itself and satisfies the Plancherel identity: Z Z (7.2) |f (x)|2 dx = |fˆ(p)|2 dp R R • for any f ∈ S(R), (7.3) f (x) = (2π) −1/2 Z eipx fˆ(p) dp R • if f ∈ S(R) then (7.4) pfˆ(p) = −iF(f ′ )(p) THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 17 Consequently, we have ||f ||sup ≤ (2π) −1/2 Z ZR |fˆ(p)| dp (1 + p2 )1/2 |fˆ(p)|(1 + p2 )−1/2 dp Z 1/2 = (2π)−1/2 (1 + p2 )|fˆ(p)|2 dp π 1/2 by Cauchy-Schwartz R i h ≤ 2−1/2 ||fˆ||L2 + ||pfˆ||L2 (R,dp) = (2π)−1/2 R (7.5) ≤ 2−1/2 [||f ||L2 + ||f ′ ||L2 ] by Plancherel and (7.4) 2 Recall that for f in Sp (R) ⊂ L (R) we have that X f= an φn n≥0 where an = hf, φn i for every n ≥ 0. Let fN = N X an φn n=0 and observe the following: ′ Lemma 7.1. If f ∈ Sp (R) for p ≥ 1, then {fN } is Cauchy in L2 (R). Proof. Note that  A−C fN = 2 ′ ′ So ||fN − fM ||L2 ≤ 12 ||AfN − AfM ||L2 + 21 ||CfN − CfM ||L2 . Now for M < N we have N X √ an nφn−1 ||2L2 ||AfN − AfM ||2L2 = ||  ′ fN n=M +1 N X = n=M +1 N X ≤ n=M +1 Likewise, ||CfN − CfM ||2L2 = || = ≤ PN N X n=M +1 N X n=M +1 N X n=M +1 |an |2 n |an |2 (n + 1)2 √ an n + 1φn+1 ||2L2 |an |2 (n + 1) |an |2 (n + 1)2 Since f ∈ Sp (R), we know M +1 |an |2 (n + 1)2 tends to 0 as M goes to infinity. ′ Thus {fN } is Cauchy in L2 (R).  18 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Lemma 7.2. If f ∈ Sp (R) for p ≥ 1, then {fN } converges uniformly to f , i.e. ||f − fN ||sup → 0 as N → ∞. Proof. It is enough to show that ||fM − fN ||sup → 0 as M, N → ∞. Note that ′ ′ ||fM − fN ||sup ≤ 2−1/2 [||fN − fM ||L2 + ||fN − fM ||L2 ] by (7.5). Since f ∈ L2 (R) we have ||fN − fM ||L2 → 0 as M, N → ∞ and by Lemma ′ ′ 7.1 we have that ||fN − fM ||L2 → 0 as M, N → ∞. Therefore {fN } converges uniformly to f .  The Lemma above immediately gives us that f ∈ Sp (R) is continuous, being the uniform limit of the continuous functions fN . Next we address the differentiablity of functions in Sp (R). We show that functions in Sp (R) are differentiable and their derivatives lie in Sp−1 (R). ′ Theorem 7.3. If f ∈ Sp (R) for p ≥ 1, then f is differentiable and f ′ = limn→∞ fN 2 ′ in L (R). Moveover f is in Sp−1 (R). ′ Proof. By Lemma 7.1, the sequence of derivatives fN is Cauchy in L2 (R). Let ′ 2 g = limN →∞ fN in L (R). Observe that Z 1
′ (x + t(y − x) (y − x) dt fN (7.6) fN (y) = fN (x) + 0 R1 p ′ Now 0 |fN (x + t(y − x))(y − x) − g(x + t(y − x))(y − x)| dt ≤ ′ by the Cauchy-Schwartz inequality. Since kfN − gkL2 → 0 as N Z 1 0 ′ fN (x + t(y − x))(y − x) dt → Z 0 1 ′ |y − x|||fN − g||L2 → ∞, we have g(x + t(y − x))(y − x) dt Since fN converges to f uniformly by Lemma 7.2, taking the limit as N → ∞ in (7.6) we obtain Z 1
g (x + t(y − x) (y − x) dt f (y) = f (x) + 0 Therefore f ′ = g ∈ L2 (R). Now we have the L2 limits:   N √ 1X √ A−C ′ ′ fN = lim ( n + 1an+1 − nan−1 )φn f = lim fN = lim N →∞ 2 N →∞ N →∞ 2 n=0 Observe that √ √ 1X (n + 1)2(p−1) | n + 1an+1 − nan−1 |2 2 n≥0 X X ≤ (n + 1)2(p−1) (n + 1)|an+1 |2 + (n + 1)2(p−1) n|an−1 |2 n≥0 n≥1 since |a − b|2 ≤ 2(a2 + b2 ) for a, b ∈ R X X ≤ n(2p−1) |an |2 + (n + 2)2(p−1) (n + 1)|an |2 n≥0 n≥0 THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 19 This sum is finite since f ∈ Sp (R) and since, for n large enough, n(2p−1) ≤ (n + 1)2p and (n + 2)2(p−1) (n + 1) ≤ (n + 1)2p . Thus we have f ′ is in Sp−1 (R).  Corollary 7.4. If f ∈ Sp (R), then f (k) exists for k ∈ {0, 1, 2, . . . , p} and f (k) ∈ Sp−k (R). Proof. Recursively apply Theorem 7.3 to f and it’s derivatives.  8. Inner-products on S(R) from N For f ∈ L2 (R), define (8.1) ||f ||t =  X  n≥0 (n + 1)t |hf, φn i|2 1/2   for every t > 0. More generally, define X (8.2) hf, git = (n + 1)t hf, φn iL2 hφn , giL2 , n≥0 for all f, g in the subspace of L2 (R) consisting of functions F for which ||F ||t < ∞. Theorem 8.1. Let f ∈ S(R). Then every t > 0 we have ||f ||t < ∞. Moreover, for every integer m ≥ 0, we also have X (8.3) N mf = nm hf, φn iφn , n≥0 2 2 d x 1 where on the left N m is the differential operator − dx 2 + 4 − 2 applied n times, 2 and on the right the series is taken in the sense of L (R, dx). Furthermore, (8.4) ||f ||2m = hf, (N + 1)m f i This result will be strengthened and a converse proved later. Proof. Let m ≥ 0 be an integer. Since f ∈ S(R), it is readily seen that N f is also in S(R), and thus, inductively, so is N m f . Then we have X hf, N m f i = hf, φn ihφn , N m f i n≥0 = X n≥0 = X n≥0 = X n≥0 hf, φn ihN m φn , f i by (6.13) hf, φn i hnm φn , f i nm |hf, φn i|2 Thus we have proven the relation (8.5) hf, N m f i = An exactly similar argument shows (8.6) hf, (N + 1)m f i = X n≥0 X n≥0 nm |hf, φn i|2 (n + 1)m |hf, φn i|2 = ||f ||2m 20 JEREMY J. BECNEL AND AMBAR N. SENGUPTA So if t > 0, choosing any integer m ≥ t we have ||f ||2t ≤ ||f ||2m = hf, (N + 1)m f i < ∞ Observe that the series X (8.7) n≥0 nm hf, φn iφn 2 is convergent in L (R, dx) since X n2m |hf, φn i|2 = hN 2m f, f i < ∞. n≥0 2 So for any g ∈ L (R, dx) we have, by an argument similar to the calculations done above: X hN m f, gi = nm hf, φn ihφn , gi n≥0 = X n≥0 = * hnm hf, φn iφn , gi X n≥0 This proves the statement about N m f . m n hf, φn iφn , g +  We have similar observations concerning C m f and Am f . First observe that since C and A are operators involving d/dx and x, they map S(R) into itself. Also, hAf, gi = hf, Cgi, for all f, g ∈ S(R), as already noted. Using this we have for f ∈ S(R), we have Therefore, (8.8) m hφn+m , C m f i = hA p φn+m , f i (n + m)(n + m − 1) · · · (n + 1) hφn , f i. = m C f= X  (n + m)! 1/2 hf, φn iφn+m 1/2 hf, φn iφn−m n≥0 n! Similarly, (8.9) Am f = X n≥0 n! (n − m)! More generally, if B1 , ..., Bk are such that each Bi is either A or C then X (8.10) B1 ...Bk f = θn,k hf, φn iφn+r , n≥0 where the integer r is the excess number of C’s over the A’s in the sequence B1 , ..., Bk , and θn,k is a real number determined by n and k. We do have the upper bound (8.11) 2 θn,k ≤ (n + k)k ≤ [(n + 1)k]k = (n + 1)k k k THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 21 Note also that (8.12) ||B1 ...Bk f ||2 = h(B1 ...Bk )∗ B1 ...Bk f, f i = X n≥0 2 θn,k |hf, φn i|2 Actually, it seems that (B1 ...Bk )∗ B1 ...Bk is a polynomial in N of degree k, and so 2 θn,k would be a polynomial in n of degree k. Lets look at the case of Rd . The functions φn generate an orthonormal basis by tensor products. In more detail, if a ∈ W d is a multi-index, define φa ∈ L2 (Rd ) by φa (x) = φa1 (x1 )...φad (xd ) Now, for each t > 0, and f ∈ L2 (Rd ) define  1/2  X  def , (8.13) ||f ||t = [(a1 + 1)...(ad + 1)]t |hf, φa i|2   d a∈W and then define (8.14) hf, git = X a∈W d [(a1 + 1)...(ad + 1)]t hf, φa iL2 (Rd ) hφa , giL2 (Rd ) , for all f, g in the subspace of L2 (Rd ) consisting of functions F for which ||F ||t < ∞. Let Td be the operator on S(Rd ) given by Td = (Nd + 1)...(N1 + 1) Then, for every non-negative integer m, we have ||f ||2m = hf, Tdm f i The other results of this section also extend in a natural way to Rd . 9. L2 –type norms on S(R) For integers a, b ≥ 0, and f ∈ S(R), define (9.1) Recall the operators A= ||f ||a,b,2 = ||xa Db f (x)||L2 (R,dx) 1 d + x, dx 2 C=− d 1 + x, dx 2 N = CA and the norms ||f ||m = hf, (N + 1)m f i Our objective for this section is summarized as: Theorem 9.1. The system of semi-norms given by ||f ||a,b,2 and the system given by the norms ||f ||m generate the same topology on S(R). Proof. Let a, b be non-negative integers. Then ||f ||a,b,2 = ||(A + C)a 2−b (A − C)b f ||L2 ≤ a linear combination of terms of the form ||B1 ...Bk f ||L2 , where each Bi is either A or C, and k = a + b. Writing cn = hf, φn i, we have X ||B1 ...Bk f ||2L2 = || cn θn,k φn+r || n≥0 22 JEREMY J. BECNEL AND AMBAR N. SENGUPTA = X n≥0 where 2 |cn |2 θn,k , r = #{j : Bj = C} − #{j : Bj = A}, and, as noted earlier in (8.11), So 2 θn,k ≤ (n + k)k ≤ [(n + 1)k]k = (n + 1)k k k ||B1 ...Bk f ||2L2 ≤ X n≥0 |cn |2 (n + 1)k k k = k k ||f ||2k Thus ||f ||a,b,2 is bounded above by a multiple of the norm ||f ||a+b . It follows, that the topology generated by the semi-norms || · ||a,b,2 is contained in the topology generated by the norms || · ||k . Now we show the converse inclusion. From ||f ||2k = hf, (N + 1)k f i ≤ ||f ||2L2 + ||(N + 1)k f ||2L2 and the expression of N as a differential operator we see that ||f ||2k is bounded above by a linear combination of ||f ||2a,b,2 for appropriate a and b. It follows then that the topology generated by the norms || · ||k is contained in the topology generated by the semi-norms || · ||a,b,2 .  Now consider Rd . Let a, b ∈ W d be multi-indices, where W = {0, 1, 2, ...}. Then for f ∈ S(Rd ) define ||f ||a,b,2 = Z Rd |xa Db f (x)|2 dx 1/2 These specify semi-norms and they generate the same topology as the one generated by the norms || · ||m , with m ∈ W . The argument is a straightforward modification of the one used above. 10. Equivalence of the three topologies We will demonstrate that the topology generated by the family of norms || · ||k , or, equivalently, by the semi-norms || · ||a,b,2 , is the same as the Schwartz topology on S(R). Recall from (7.5) we have that for f ∈ S(R) ||f ||sup ≤ 2−1/2 [||f ||L2 + ||f ′ ||L2 ] Putting in xa Db f (x) in place of f (x) we then have (10.1) ||f ||a,b ≤ a linear combination of ||f ||a,b,2 , ||f ||a−1,b,2 , and ||f ||a,b+1,2 Next we bound the semi-norms ||f ||a,b,2 by the semi-norms ||f ||a,b . To this end, observe first Z ||f ||2L2 = (1 + x2 )−1 (1 + x2 )|f (x)|2 dx R ≤ ||(1 + x2 )|f (x)|2 ||sup π
≤ π ||f ||2sup + ||xf (x)||2sup 2 ≤ π (||f ||sup + ||xf (x)||sup ) THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 23 So for any integers a, b ≥ 0, we have ||f ||a,b,2 = ||xa Db f ||L2 ≤ π 1/2 (||f ||a,b + ||f ||a+1,b ) (10.2) Thus, the topology generated by the semi-norms ||·||a,b,2 coincides with the Schwartz topology. Now lets look at the situation for Rd . The same result holds in this case and the arguments are similar. The appropriate Sobolev inequalities require using (1+|p| 2 )d instead of 1 + p2 . For f ∈ S(Rd ), we have the Fourier transform given by Z (10.3) F(f )(p) = fˆ(p) = (2π)−d/2 e−ihp,xi f (x) dx Rd 2 Again, this preserves the L norm, and transforms derivatives into multiplications:   ∂f ˆ pj f (p) = −iF (p). ∂xj Repeated application of this shows that (10.4) |p|2 fˆ(p) = −F (∆f ) (p), where ∆= n X ∂2 ∂x2j j=1 is the Laplacian. Iterating this gives, for each r ∈ {0, 1, 2, ...} and f ∈ S(Rd ), |p|2r fˆ(p) = (−1)r F (∆r f ) (p), (10.5) which in turn implies, by the Plancherel formula (7.2), the identity: Z Z 2 |∆r f (x)|2 dx |p|2r |fˆ(p) dp = (10.6) Rd Rd Then we have, for any m > d/4, Z ||f ||sup ≤ (2π)−d/2 |fˆ(p)| dp d ZR = (2π)−d/2 (1 + |p|2 )m |fˆ(p)|(1 + |p|2 )−m dp Rd = Z Rd (1 + |p|2 )2m |fˆ(p)|2 dp 1/2 where K = (2π)−d/2 (10.7) K hR by Cauchy-Schwartz dp Rd (1+|p|2 )2m i1/2 <∞ The function (1 + s)n /(1 + sn ), for s ≥ 0, attains a maximum value of 2n−1 , and so the inequality (1 + s)2m ≤ 22m−1 (1 + s2m ), which leads to (1 + |p|2 )2m ≤ 22m−1 (1 + |p|4m ) Then, from (10.7), we have (10.8) ||f ||2sup ≤ K 2 24m−2 ||f ||2L2 + ||∆m f ||2L2
This last quantity is clearly bounded above by a linear combination of ||f || 0,b,2 for certain multi-indices b. Thus ||f ||sup is bounded above by a linear combination of ||f ||0,b,2 for certain multi-indices b. It follows that ||xa Db f ||sup is bounded above by a linear combination of ||f ||a′ ,b′ ,2 for certain multi-indices a′ , b′ . 24 JEREMY J. BECNEL AND AMBAR N. SENGUPTA For the inequality going the other way, the reasoning used above for (10.2) generalizes readily, again with (1 + x2 ) replaced by (1 + |x|2 )d . Thus, on S(Rd ) the topology generated by the family of semi-norms || · ||a,b,2 coincides with the Schwartz topology. Now we return to (10.8) for some further observations. First note that d ∆= 1X (Cj − Aj )2 4 j=1 and so ∆m consists of a sum of multiples of (3d)m terms each a product of 2m elements drawn from the set {A1 , C1 , ..., Ad , Cd }. Consequently, ||∆m f ||2L2 ≤ c2d,m ||f ||2m , (10.9) for some positive constant cd,m . Combining this with (10.8), we see that for m > d/4, there is a constant kd,m such that ||f ||sup ≤ kd,m ||f ||m (10.10) holds for all f ∈ S(Rd ). Now consider f ∈ Sp (Rd ), with p > d/4. Let X fN = hf, φj iφj j∈W d ,|j|≤N Then fN → f in L2 and so a subsequence {fNk }k≥1 converges pointwise almost everywhere to f . It follows then that the essential supremum ||f ||∞ is bounded above as follows: ||f ||∞ ≤ lim sup |fN |sup N →∞ Note that fN → f also in the || · ||p –norm. It follows then from (10.10) that (10.11) d ||f ||∞ ≤ kd,p ||f ||p holding for all f ∈ Sp (R ) with p > d/4. Replacing f by the difference f − fN in (10.11), we see that f is the L∞ –limit of a sequence of continuous functions which, being Cauchy in the sup-norm, has a continuous limit; thus f is a.e. equal to a continuous function, and may thus be redefined to be continuous. 11. Identification of S(R) with a sequence space Suppose a0 , a1 , ... are a sequence of complex numbers such that X (11.1) (n + 1)m |an |2 < ∞, for every integer m ≥ 0. n≥0 We will show that the sequence of functions given by n X sn = aj φj j=0 converges in the topology of S(R) to a function f ∈ S(R) for which an = hf, φn i for every n ≥ 0. All the hard work has already been done. From (11.1) we see that (sn )n≥0 is Cauchy in each norm || · ||m . So it is Cauchy in the Schwartz topology of S(R), and hence convergent to some f ∈ S(R). In particular, sn → f in L2 . Taking inner-products with φj we see that aj = hf, φj i. THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 25 Thus we have Theorem 11.1. Let W = {0, 1, 2, ...}, and define F : L2 (R) → CW by requiring that F (f )n = hf, φn iL2 for all n ∈ W . Then the image of S(R) under F is the set of all a ∈ CW for def P m 2 which ||a||2m = n≥0 (n + 1) |an | < ∞ for every integer m ≥ 0. Moreover, if
F S(R) is equipped with the topology generated by the norms || · ||m then F is a homeomorphism. 12. Spectral Theory in Brief Let H be a complex Hilbert space. A linear operator on H is a linear map A : DA → H, where DA is a subspace of H. Usually, we work with densely defined operators, i.e. operators A for which DA is dense. 12.1. Graph and Closed Operators. The graph of the operator A is (12.1) Gr(A) = {(x, Ax) : x ∈ DA } Thus Gr(A) is A viewed as a set of ordered pairs, and is thus A itself taken as a mapping in the set-theoretic sense. The operator A is said to be closed if its graph is a closed subset of H ⊕ H; put another way, this means that if (xn )n≥1 is any sequence in H which converges to a limit x and if limn→∞ Axn = y also exists then x is in the domain of A and y = Ax. 12.2. The adjoint A∗ . If A is a densely defined operator on H then there is an adjoint operator A∗ defined as follows. Let DA∗ be the set of all y ∈ H for which the map fy : DA → C : x 7→ hAx, yi is bounded linear. Clearly, DA∗ is a subspace of H. The bounded linear functional fy extends to a bounded linear functional fy on H. So there exists a vector z ∈ H such that fy (x) = hz, xi for all x ∈ H. Since DA is dense in H, the element z is uniquely determined by x and A. Denote z by A∗ y. Thus, A∗ y is the unique vector in H for which (12.2) hx, A∗ yi = hAx, yi holds for all x ∈ DA . Using the definition of A∗ for a densely-defined operator A it is readily seen that A∗ is a closed operator. 12.3. Self-adjoint Operators. The operator A is self-adjoint if it is densely defined and A = A∗ . Thus, if A is self-adjoint then DA = DA∗ and (12.3) hx, Ayi = hAx, yi for all x, y ∈ DA . Note that a self-adjoint operator A, being equal to its adjoint A∗ , is automatically a closed operator. 26 JEREMY J. BECNEL AND AMBAR N. SENGUPTA 12.4. Closure, and Essentially Self-adjoint Operators. Consider a denselydefined linear operator S on H. Assume that the closure of the graph of S is the graph of some operator S. Then S is called the closure of S. We say that S is essentially self-adjoint if its closure is a self-adjoint operator. In particular, S must then be a symmetric operator, i.e. it satisfies hSx, yi = hx, Syi (12.4) for all x, y ∈ H. A symmetric operator may not, in general, be essentially selfadjoint. 12.5. The Multiplication Operator. Let us turn to a canonical example. Let (X, F, µ) be a sigma-finite measure space. Consider the Hilbert space L2 (µ). Let f : X → C be a measurable function. Define the operator Mf on L2 (µ) by setting (12.5) Mf g = f g, with the domain of Mf given by (12.6) D(Mf ) = {g ∈ L2 (µ) : f g ∈ L2 (µ)} Let us check that D(Mf ) is dense in L2 (µ). By sigma-finiteness of µ, there is an increasing sequence of measurable sets Xn such that ∪n≥1 Xn = X and µ(Xn ) < ∞. For any h ∈ L2 (µ) let hn = 1Xn ∩{|f |≤n} h. Then Z Z |f hn | dµ ≤ n |h|1Xn dµ ≤ nµ(Xn )1/2 ||h||L2 < ∞ and so hn ∈ D(Mf ). On the other hand, ||hn − h||2L2 → 0 by dominated convergence. So D(Mf ) is dense in H. It may be shown that Mf∗ = Mf (12.7) Thus Mf is self-adjoint if f is real-valued. A very special case of the preceding example is obtained by taking X to be a finite set, say X = {1, 2, ..., d}, and µ as counting measure on the set of all subsets of X. In this case, L2 (µ) = Cd , and the operator Mf , viewed as a linear map is given by the diagonal matrix  f1  0   0 (12.8)   ..  . 0 Mf : Cd → Cd 0 f2 0 .. . 0 0 f3 .. . ··· ··· ··· .. . 0 0 0 .. . 0 0 ··· fd        Now take the case where µ is counting measure on the sigma-algebra of all subsets of a countable set X. Let f be any real-valued function on X. Let Df0 be the subspace of L2 (µ) consisting of all functions g for which {g 6= 0} is a finite set, and let Mf0 be the restriction of Mf to Df0 . Then it is readily checked that Mf0 is essentially self-adjoint. Consequently, the restriction of Mf to any subspace of Df larger than Df0 is also essentially self-adjoint. THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 27 12.6. The Spectral Theorem. The spectral theorem for a self-adjoint operator A on a separable complex Hilbert space H says that there is a sigma-finite measure space (X, F, µ), a unitary isomorphism U : H → L2 (µ) and a measurable real-valued function f on X such that (12.9) A = U −1 Mf U Expressing A in this way is called a diagonalization of A (the terminology being motivated by (12.8)). 12.7. The Functional Calculus. If g is any measurable function on R we can then form the operator (12.10) def g(A) = U −1 Mg◦f U If g is a polynomial then g(A) works out to be what it should be, a polynomial in A. Another example, is the function g(x) = eikx , where k is any constant; this gives the operator eikA . 12.8. The Spectrum. The essential range
of f is the smallest closed subset of R whose complement U satisfies µ f −1 (U ) = 0. It consists of all λ ∈ R for which the operator Mf − λI = Mf −λ has a bounded inverse (which is M(f −λ)−1 ). This essential range forms the spectrum σ(A) of the operator A. Thus σ(A) is the set of all real numbers λ for which the operator A − λI has a bounded linear operator as inverse. 12.9. The Spectral Measure. Associate to each Borel set E ⊂ R the operator PE′ = M1f −1 (E) on L2 (X, µ). This is readily checked to be an orthogonal projection operator. Hence, so is the operator P A (E) = U −1 PE′ U Moreover, it can be checked that the association E 7→ P A (E) is a projection-valued measure, i.e. P A (∅) = 0, P A (R) = I, P A (E ∩ F ) = P A (E)P A (F ), and for any disjoint Borel sets E1 , E2 , ... and any vector x ∈ H we have X (12.11) P A (∪n≥1 En )x = P A (En )x n≥1 This is called the spectral measure for the operator A, and is uniquely determined by the operator A. 12.10. The Number Operator. Let us examine an example. Let W = {0, 1, 2, ...}, and let µ be counting measure on W . On W we have the function N ′ : W → R : n 7→ n Correspondingly we have the multiplication operator MN ′ on the Hilbert space L2 (W, µ). Now consider the Hilbert space L2 (R). We have the unitary isomorphism U : L2 (R) → L2 (W, µ) : f 7→ (hf, φn i)n≥0 28 JEREMY J. BECNEL AND AMBAR N. SENGUPTA Consider the operator N on L2 (R) given by N = U −1 MN ′ U Then Nf = X n∈W and the domain of N is DN = {f ∈ L2 (R) : Comparing with (8.3) we see that (N f )(x) =  − nhf, φn iφn X n∈W n2 |hf, φn i|2 < ∞} x2 1 d2 + − 2 dx 4 2  f (x) for every f ∈ S(R). x2 1 d2 Thus the self-adjoint operator N extends the differential operator − dx 2 + 4 − 2, and, notationally, we will often not make a distinction. In view of the observation d2 x2 1 made at the end of subsection 12.5, the differential operator − dx 2 + 4 − 2 on the domain S(R) is essentially self-adjoint, with closure equal to the operator N . The operator U above helps realize the operator N as the multiplication operator MN ′ , and is thus an explicit realization of the fact guaranteed by the spectral theorem. 13. Explanation of physics terminology In quantum theory, one associates to each physical system a complex Hilbert space H. Each state of the system is represented by a bounded self-adjoint operator ρ ≥ 0 for which tr(ρ) = 1. An observable is represented by a self-adjoint operator A on H. The relationship of the mathematical formalism with physics is obtained by declaring that tr(P A (E)ρ) is the probability that in state ρ the observable A has value in the Borel set E ⊂ R. Here, P A is the spectral measure for the self-adjoint operator A. The states form a convex set, any convex linear combination of any two states being also clearly a state. There are certain states which cannot be expressed as a convex linear combination of distinct states. These are called pure states. A pure state is always given by the orthogonal projection onto a ray (1–dimensional subspace of H). If φ is any unit vector on such a ray then the orthogonal projection onto the ray is given by: Pφ: ψ 7→ hψ, φiφ and then the probability of the observable A having value in a Borel set E in the state Pφ then works out to be hP A (E)φ, φi Suppose, for instance, the spectrum of A consists of eigenvalues λ1 , λ2 , ..., with Aun = λn un for an orthonormal basis {un }n≥1 of H. Then the probability that the observable represented by A has value in E in state Pφ is X |hun , φi|2 {n:λn ∈E} Thus the spectrum σ(A) here consists of all the possible values of A which could be realized. THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 29 To every system there is a special observable H called the Hamiltonian. The physical significance of this observable is that it describes the energy of the system. There is a second significance to this observable: if ρ is the state of the system at a given time then time t later the system evolves to the state t t ρt = e−i ~ H ρei ~ H , where ~ is Planck’s constant. A basic system considered in quantum mechanics is the harmonic oscillator. One may think of this crudely as a ball attached to a spring, but the model is used widely, for instance also for the quantum theory of fields. The Hilbert space for the harmonic oscillator is L2 (R). The Hamiltonian operator, up to scaling and addition of the constant − 12 , is d2 x2 1 + − . dx2 4 2 The energy levels are then the spectrum of this operator. In this case the spectrum consists of all the eigenvalues 0, 1, 2, .... The creation operator bumps an eigenstate of energy n up to a state of energy n + 1; an annihilation operator lowers the energy by 1 unit. In many applications, the eigenstates represent quanta, i.e. particles. Thus raising the energy by one unit corresponds to the creation of a particle, while lowering the energy by one unit corresponds to annihilating a particle. H =− 14. The abstract formulation As before, we use the notation W = {0, 1, 2, ...}. We work with a real separable Hilbert space H0 , and a positive Hilbert-Schmidt operator B on H0 . Thus H0 has an orthonormal basis {un }n∈W of eigenvectors of B, with Bun = λn un and P n≥0 |λn |2 < ∞, with each λn > 0. 2 d The example to keep in mind is H0 = L2 (R), and B = (− dx 2 + We have the coordinate map
I : H0 7→ RW : f 7→ hf, un i n∈W Let (14.1) F0 = I(H0 ) = {(xn )n∈W : X n∈W x2n < ∞} Now, for each p ∈ W , let (14.2) Fp = {(xn )n∈W : X n∈W 2 λ−2p n xn < ∞} On Fp we have the inner-product h·, ·ip given by X ha, bip = λ−2p n an bn n∈W x2 4 + 12 )−1 . 30 JEREMY J. BECNEL AND AMBAR N. SENGUPTA This makes Fp a real Hilbert space, unitarily isomorphic to L2 (W, µp ) where µp is the measure on W specified by µp ({n}) = λ−2p n . Moreover, we have (14.3) def F = ∩p∈W Fp ⊂ · · · F2 ⊂ F1 ⊂ F0 = L2 (W, µ0 ) Each inclusion Fp+1 → Fp is Hilbert-Schmidt. Now we pull all this back to L2 (R). First set X 2 (14.4) Hp = I −1 (Fp ) = {x ∈ H0 : λ−2p n |hx, un i| < ∞} n≥0 It is readily checked that Hp = B p (H0 ) (14.5) On Hp we have the pull back inner-product h·, ·ip , which works out to hf, gip = hB −p f, B −p gi (14.6) Then we have the chain (14.7) def H = ∩p∈W Hp ⊂ · · · H2 ⊂ H1 ⊂ H0 , with each inclusion Hp+1 → Hp being Hilbert-Schmidt. Equip H with the topology generated by the norms || · ||p (i.e. the smallest topology making all inclusions H → Hp continuous). Then H is, more or less by definition, a nuclear space. The vectors un all lie in H and the set of all rational-linear combinations of these vectors produces a countable dense subspace of H. Consider a linear functional on H which is continuous. Then it must be continuous with respect to some norm || · ||p . Thus the topological dual H ′ is the union of the duals Hp′ . In fact, we have: (14.8) H ′ = ∪p∈W Hp ⊃ · · · H2′ ⊃ H1′ ⊃ H0′ ≃ H0 , where in the last step we used the usual Hilbert space isomorphism between H 0 and its dual H0′ . Going over to the sequence space, Hp′ corresponds to X def 2 (14.9) F−p = {(xn )n∈W : λ2p n xn < ∞} n∈W The element y ∈ F−p corresponds to the linear functional on Fp given by X x 7→ xn yn n∈W which, by Cauchy-Schwartz, is well-defined and does define an element of the dual Fp′ with norm equals to ||y||−p . Consider now the product space RW , along with the coordinate projection maps X̂j : RW → R : x 7→ xj for each j ∈ W . Equip RW with the product σ–algebra, i.e. the smallest sigmaalgebra with respect to which each projection map X̂j is measurable. A fundamental result in probability measure theory (a special case of Kolmogorov’s theorem, for instance) says that there is a unique probability measure ν on the product σ–algebra THE SCHWARTZ SPACE: A BACKGROUND TO WHITE NOISE ANALYSIS 31 such that each function X̂j , viewed as a random variable, has standard Gaussian distribution. Thus, Z 2 eitX̂j dν = e−t /2 RW for t ∈ R, and every j ∈ W . The measure ν is the product of the standard Gaussian 2 measure e−x /2 (2π)−1/2 dx on each component R of the product space RW . Since, for any p ≥ 1, we have Z X 2p X 2p λj x2j dν(x) = λj < ∞, RW j∈W j∈W it follows that ν(F−p ) = 1 ′ for all p ≥ 1. Thus ν(F ) = 1. We can, therefore, transfer the measure ν back to H ′ , obtaining a probability measure µ on the sigma–algebra of subsets of H ′ generated by the maps ûj : H ′ → R : f 7→ f (uj ), where {uj }j∈W is the orthonormal basis of H0 we started with (note that each uj lies in H = ∩p≥0 Hp ). This is clearly the sigma–algebra generated by the weak topology on H ′ (which happens to be equal also to the sigma–algebras generated by the strong/inductive-limit topology). x2 1 −1 d2 , we have Specialized to the example H0 = L2 (R), and B = (− dx 2 + 4 + 2) ′ the standard Gaussian measure on the distribution space S (R). The above discussion gives a simple direct description of the measure µ. Its existence is also obtainable by applying the well–known Minlos theorem. To summarize, we can state the starting point of much of infinite-dimensional distribution theory (white noise analysis): Given a real, separable Hilbert space H 0 and a positive Hilbert-Schmidt operator B on H0 , we have constructed a nuclear space H and a unique probability measure µ on the Borel sigma–algebra of the dual H ′ such that there is a linear map satisfying H0 → L2 (H ′ , µ) : x 7→ x̂, Z 2 eitx̂ dν = e−t ||x||20 /2 , H′ for every real t and x ∈ H0 . This Gaussian measure µ is often called the white noise measure and forms the background measure for white-noise analysis. References [1] Ch. Hermite, Sur un Nouveau Développement en Série des Fonctions, Comptes rendus de l’Academie des Sciences14, 93-266 (1864); in Oeuvres de Charles Hermite, Tome II, GauthierVillars (1908) [2] T. Hida, H. -H. Kuo, J. Potthoff, L. Streit, White Noise : An Infinite Dimensional Calculus, Kluwer Academic Publishers (1993) [3] H.-H. Kuo, White Noise Distribution Theory, CRC Press (1996). [4] B. Simon, Distributions and Their Hermite Expansions, J. Math. Phys. 12(1) 140 (1971). Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA, e-mail: [email protected]