Limiting Means for Spherical Slices

arXiv:1903.07693v1 [math.PR] 18 Mar 2019 LIMITING MEANS FOR SPHERICAL SLICES AMY PETERSON* AND AMBAR N. SENGUPTA Abstract. We show that for a suitable class of functions of finitely-many variables, the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions. 1. Introduction In this paper we generalize a result of [10] showing that the large-N limit of√the integral of a function f over affine slices of a high dimensional sphere S N −1 ( N ) is Gaussian. In [10] this was proved for bounded f , and here we establish the result for f in a suitable Lp space, for any p > 1. 1.1. Notation, Definitions, and Background. Let A be a closed affine subspace of l2 of finite codimension m then there is Q : l 2 → Rm a continuous linear surjection and w0 ∈ Rm so that we can write A as a level set of Q, A = Q−1 (w0 ). (1.1) √ N −1 0 −1 Let z be the point ( N ) be the sphere √ on Q (w0 ) closest to the origin and S in RN of radius N centered at the origin. Now we are interested in√’circles’ formed by intersecting the part of A in RN , AN , with the sphere S N −1 ( N ): √ SAN := AN ∩ S N −1 ( N ). (1.2) Let Q N : RN → Rm defined by QN = QJN where JN is the inclusion map from RN to l2 . We can −1 0 write AN = QN (w0 ) and the point on AN closest to the origin p as zN . Thus the 0 0 |2 where 2 0 = a − |zN ’circle’ SAN is a sphere with center at zN and radius azN √ a = N . See Figure 1.1. 0 Note that zN converges weakly to z 0 and so 0 T zN → T z0 (1.3) for continuous linear T : H → X where X is finite dimensional. See [10] Proposition 4.1 for full detail. 2010 Mathematics Subject Classification. Primary 28C20, Secondary 44A12. Key words and phrases. Gaussian Radon transform, spherical mean. * Corresponding author. 1 2 AMY PETERSON AND AMBAR N. SENGUPTA 0 Q−1 N (w ) 0 zN 0 az N √ S N −1 ( N ) −1 0 Figure ) slices the sphere √ 1. The affine subspace QN (w N −1 0 0 . S ( N ) in a ‘circle’ with center zN and radius azN We are interested in integrals of a function φ over SAN with respect to the normalized surface area measure σ̄: Z Z φ(x1 , . . . , xk )dσ̄(x). (1.4) φ(x1 , . . . , xk )dσ̄(x) = √ SAN 0 S N −1 ( N )∩Q−1 N (w ) We will take φ to be a Borel function that only depends on the first k-coordinates for k < N . We will also need an important disintegration formula for the integral (1.4) and to that end we need the following projections. Let P(k) : l2 → Rk = X z 7→ (z1 , . . . , zk ) be the projection from l2 onto the first k-coordinates. Then let L be the restriction of P(k) to ker Q: L : ker Q → X (1.5) This is a surjection provided dim (ker(Q)) > dim(X). Further we define LN to be the restriction of P(k) to ker(QN ): LN : ker QN = ker Q ∩ ZN → X (1.6) for large enough N , LN is also surjective. (See [10] Proposition 6.2). Next we also want to restrict L and LN to be isomorphisms. We define L0 to be the restriction of L to ker Q ⊖ ker L which is the orthogonal complement of ker Q ∩ ker L within ker Q: L0 : ker Q ⊖ ker L → X. This is an isomorphism. Lastly let L0,N be the restriction of LN to ker QN ⊖ker LN L0,N : ker QN ⊖ ker LN → X for large N this is an isomorphism (See again [10] Prop. 6.2). LIMITING MEANS FOR SPHERICAL SLICES 3 Now we state without proof the disintegration formula we will need. See [10] Theorem 3.3 for full detail. Theorem 1.1. Borel function defined on √ Let f be a 0bounded, or 0non-negative, −1 m 0 0 SAN = S N −1 ( N )∩Q−1 N (w ) for some w ∈ R . Let zN be the point on QN (w ) 0 0 closest to 0. Let L0,N and LN be defined as above and let x = LN (zN ) ∈ X. Then Z f dσ √ S N −1 ( N)∩Q−1 (w 0 ) = Z x∈DN (Z f dσ SAN ∩LN −1 (x) ) 0 az N q a2z0 − L0,N −1 (x − x0 ) (1.7) dx , | det L0,N | 2 N p 0 0 |2 and D 0 = N − |zN where azN N consists of all x ∈ x + LN (ker QN ) ⊂ X for which the term under the square-root is positive: 0 }. DN = x0 + {y ∈ LN (ker QN ) : L0,N −1 (y) < azN (1.8) Taking φ to be a Borel function on Rk we let f be the function obtained by extending φ to RN by setting f (x) = φ(x1 , . . . , xk ) for all x ∈ RN √ and denote S N −1 ( N ) ∩ Q−1 (w0 ) ∩ LN −1 (x) = SAN ∩ LN −1 (x) then the disintegration formula (1.7) for this particular f is: Z f √ S N −1 ( N )∩Q−1 (w 0 ) = Z x∈DN = Z x∈DN (Z SAN ∩LN  dσ f dσ −1 (x) ) 0 az N q a2z0 − L0,N −1 (x − x0 ) 2 N 0 az N Vol(SAN ∩ LN −1 (x)) q a2z0 − L0,N −1 (x − x0 ) N dx | det L0,N | 2 (1.9) dx . | det L0,N | Let d = N − 1 then the volume of the sphere is: i d−k−m h
2 −1 −1 0 2 2 (x − x ) a − L Vol SAN ∩ LN (x) = cd−k−m z0 0,N N (1.10) where cd−k−m is the surface measure of the (d − k − m)-dimensional sphere given, for all j, by the formula: j+1 π 2 cj = 2 j+1
. (1.11) Γ 2 We can then rewrite (1.10) as Z Z dx IN (x) f dσ = cd−k−m , (1.12) √ | det L0,N | x∈DN S N −1 ( N )∩Q−1 (w 0 ) where h 0 IN (x) = φ(x)azN a2z0 − L0,N −1 (x − x0 ) N 2 i d−k−m−1 2 . (1.13) 4 AMY PETERSON AND AMBAR N. SENGUPTA √ The sphere S N −1 ( N ) ∩ Q−1 (w0 ) has dimension d − m and its volume is cd−m ad−m . z0 N √ So, using the normalized surface measure σ on the sphere S N −1 ( N ) ∩ Q−1 (w0 ), we have Z Z dx cd−k−m f dσ , IN (x) = (1.14) √ d−m | det L0,N | cd−m az0 S N −1 ( N )∩Q−1 (w 0 ) x∈DN N where IN (x) is as in (1.13). 1.2. Related literature. This paper is a generalization of work done in [10] where more detailed results are proved for a bounded Borel function φ. The connection between Gaussian measure and the uniform measure on high dimensional spheres appeared originally in the works of Maxwell [7] and Boltzmann [2, pages 549-553]. Later works included Wiener’s paper [12] on “differential space”, Lévy [6], McKean [8], and Hida [3]. The work of Mehler [9] is one example illustrating the classical interest in functions on high-dimensional spheres. For the theory of Gaussian measures in infinite dimensions we refer to the monographs of Bogachev [1] and Kuo [5]. This paper is the fourth in a series of papers. The first [4] develops the Gaussian Radon transform for Banach spaces, where a support theorem was established. The second [11] establishes the result for hyperplanes and the third [10] proves the result for the case of affine planes. 2. Limiting Results In this section we review our previous results and prove the main result of this paper. 2.1. Previous Results. From the previous paper [10] the main result was the following theorem: Theorem 2.1. Let A be a finite-codimension closed affine subspace in l2 , specified by (1.1). Let k be a positive integer; suppose that the image of A under the coordinate projection l2 → Rk : z 7→ z(k) = (z1 , . . . , zk ) is all of Rk . Let φ be a bounded Borel function on Rk . Then Z Z φ(z(k) ) dµ(z), (2.1) lim φ(x1 , . . . , xk ) dσ(x1 , . . . , xN ) = N →∞ SAN R∞ where σ is the normalized surface area measure on SAN , and µ is the probability measure on R∞ specified by the characteristic function   Z 1 2 for all t ∈ R∞ (2.2) exp (iht, xi) dµ(x) = exp iht, pA i − kP0 tk 0 , 2 R∞ where pA is the point on A closest to the origin and P0 is the orthogonal projection in l2 onto the subspace A − pA . LIMITING MEANS FOR SPHERICAL SLICES 5 We give a sketch of the proof here for full detail refer to [10] Theorem 2.1. The pushforward measure π(k) ∗ µ of µ to Rk is
−1 π(k) ∗ µ(S) = µ π(k) (S) , for all Borel S ⊂ Rk , (2.3) where π(k) : R∞ → Rk : z 7→ z(k) = (z1 , . . . , zk ) is the projection on the first k coordinates. Now define µ∞ on Rk by   dx 1 0 0 ∗ −1 −k/2 , dµ∞ (x) = (2π) exp − h(L0 L0 ) (x − z(k) ), x − z(k) i 2 | det L0 | (2.4) 0 where L0 is given in (1.7) and z(k) is the first k-coordinates of z 0 , the point on −1 A = Q (w0 ) closest to the origin. From their respective characteristic functions we can deduce that π(k) ⋆ µ(S) = µ∞ Now, using this and Theorem 2.2 below, we can conclude that lim N →∞ Z = Z Rk Z = Rk √ 0 S N −1 ( N )∩Q−1 N (w ) φ(x1 , . . . , xk ) dσ̄(x1 , . . . , xN ) (2.5) φ dµ∞ φ dπ(k) ∗ µ = Z R∞ φ ◦ π(k) dµ. For the first equality in (2.5) we need the following theorem (from [10] Theorem 4.1). Theorem 2.2. Let A be an affine subspace of l2 given by Q−1 (w0 ), where Q : l2 → Rm is a continuous linear surjection. Suppose that the projection P(k) : l2 → √ √ Rk : z 7→ z(k) maps ker Q onto Rk . Let S N −1 ( N ) be the sphere of radius N in the subspace RN ⊕ {0} in l2 . Let φ be a bounded Borel function on Rk and let f be the function obtained by extending φ to l2 by setting f (x) = φ(x1 , . . . , xk ) for all x ∈ l2 . (2.6) Then lim N →∞ Z √ 0 SZN ( N )∩Q−1 N (w ) −k/2 = (2π) f dσ̄   h(L0 L0 ∗ )−1 (x − z 0 (k) ), x − z 0 (k) i dx φ(x) exp − , 2 | det L0 | x∈Rk (2.7) Z where L0 is the restriction of the projection P(k) to ker Q ⊖ ker P(k) , and z 0 is the point on Q−1 (w0 ) closest to the origin. 6 AMY PETERSON AND AMBAR N. SENGUPTA Note that in order to extend Theorem 2.1 for a more general function φ we only need to extend Theorem 2.2. √ Again we give a sketch of the proof for Theorem 2.2 . Let a = N and d = N −1. From the disintegration formula above (1.14) we have Z Z dx cd−k−m lim IN f dσ̄ = lim k , (2.8) N →∞ S N −1 (√N )∩Q−1 (w 0 ) N →∞ a 0,N cd−m Rk | det L0,N | z N where n IN = φ(x) 1 − a−2 L0,N −1 (x − z 0,N (k) ) z 0,N 2 o d−k−m−1 2 1DN (x). (2.9) We state here the limits of the constant term outside the integral (in (2.8)), as well as those of the full integrand on the right hand side, including the determinant term without proof, for full detail refer to [10]: lim N →∞ cd−k−m k az0,N cd−m = (2π)−k/2 , lim | det L0,N | = lim det(L0,N L∗0,N ) = det(L0 L∗0 ) = | det L0 |, N →∞ N →∞ (2.10) (2.11) and lim N →∞ n 1 − a−2 L0,N −1 (x − z 0,N (k) ) z 0,N 2 o d−k−m−1 2 1DN (x) oN n 2 2 −1 0,N 1DN (x) (x − z ) L = lim 1 − a−2 0,N (k) z 0,N N →∞   1 0 0 = exp − h(L0 L0 ∗ )−1 (x − z(k) i . ), x − z(k) 2 (2.12) Therefore if φ is such that we can apply dominated convergence theorem in (2.8) we have, Z Z dx cd−k−m IN f dσ̄ = lim (2.13) lim N →∞ ak0,N cd−m Rk N →∞ S N −1 (√N )∩Q−1 (w 0 ) | det L0,N | z N Z dx −k/2 = (2π) lim IN (2.14) N →∞ | det L0,N | k R   Z dx 1 0 0 ), x − z(k) i = (2π)−k/2 φ(x) exp − h(L0 L0 ∗ )−1 (x − z(k) 2 | det L0 | k R (2.15)  Z 2 dx 1 0 (2.16) ) = (2π)−k/2 φ(x) exp − L0 −1 (x − z(k) 2 | det L0 | Rk which is the result in Theorem 2.2. LIMITING MEANS FOR SPHERICAL SLICES 7 2.2. The Main Result. We turn now to the main result of this paper, an extension of the previous result Theorem 2.1 to more general functions. We will show that if φ is a Borel function on Rk which is Lp , p > 1, with respect to the Gaussian measure with density proportional to e−kL0 −1 2 0 (x−z(k) )k /2 dx, then the conclusion of 2.1 still holds. To this end we state and prove a generalization of Theorem 2.2. Theorem 2.3. Let A be an affine subspace of l2 given by Q−1 (w0 ), where Q : l2 → Rm is a continuous linear surjection. Suppose that the projection P(k) : l2 → √ √ Rk : z 7→ z(k) maps ker Q onto Rk . Let S N −1 ( N ) be the sphere of radius N in the subspace RN ⊕ {0} in l2 . Let φ be a Borel function on Rk which is in Lp with respect to the Gaussian measure with density proportional to e−kL0 −1 2 0 (x−z(k) )k /2 dx, for some p > 1, and let f be the function obtained by extending φ to l2 by setting f (x) = φ(x1 , . . . , xk ) for all x ∈ l2 . (2.17) Then lim N →∞ Z √ 0 S N −1 ( N )∩Q−1 N (w ) = (2π)−k/2 f dσ̄   h(L0 L0 ∗ )−1 (x − z 0 (k) ), x − z 0 (k) i dx , φ(x) exp − 2 | det L0 | x∈Rk (2.18) Z where L0 is the restriction of the projection P(k) to ker Q ⊖ ker P(k) , and z 0 is the point on Q−1 (w0 ) closest to the origin. Proof. Utilizing the proof from Theorem 2.2 we need only show (2.14) still holds, that is, Z Z cd−k−m dx dx −k/2 lim IN = (2π) (2.19) lim IN N →∞ ak0,N cd−m Rk N →∞ | det L | | det L0,N | k 0,N R z o d−k−m−1 n 2 2 1DN (x) and DN is where IN = φ(x) 1 − az−2 L0,N −1 (x − z 0,N (k) ) 0,N all x ∈ Rk such that the square-root term is positive. First we have the following inequality, o n 2 d−k−m−1 L0,N −1 (x − z 0,N (k) ) 1 − a−2 z 0,N ( 2 )N −k−m−2 L0,N −1 (x − z 0,N (k) ) = 1− N − |z 0,N |2 ( 2 )N −k−m−2 L0,N −1 (x − z 0,N (k) ) . ≤ 1− N (2.20) 8 AMY PETERSON AND AMBAR N. SENGUPTA We observe that, for N > k + m + 2, the maximum of the function  y N −k−m−2 y e for all y ∈ (0, N ] 1− N occurs at y = k +m+2; this is seen by checking that the derivative d/dy is positive for y ∈ [0, k + m + 2) and negative for y ∈ (k + m + 2, N ]. Thus,  y N −k−m−2 y 1− e ≤ N N −k−m−2  k+m+2 1− ek+m+2 N 2 Taking y = L0,N −1 (x − z 0,N (k) ) , we have: 2 )N −k−m−2 L0,N −1 (x − z 0,N (k) ) 1− N N −k−m−2  2 −1 0,N k+m+2 ek+m+2 e−kL0,N (x−z (k) )k ≤ 1− N 2 −1 0,N ≤ ek+m+2 e−kL0,N (x−z (k) )k . ( Thus ( L0,N −1 (x − z 0,N (k) ) 1− N 2) N −k−m−2 2 ≤e k+m+2 2 e −1 2 (2.21) 2 0,N kL−1 (k) )k 0,N (x−z (2.22) Lemma 2.4 gives the bound: e− 2 kL0,N (x−z 1 Let −1 0,N (k) ) 2 2 0 k ≤ e− 12 kL−1 0 (x−z (k) )k . 0,N −1 0 aN (x) = L−1 0 (x − z(k) ) and a(x) = L0 (x − z(k) ). (2.23) (2.24) Then by (1.3), for any ǫ > 0 and large enough N kaN (x) − a(x)k < ǫ. Then ka(x)k2 − kaN (x)k2 = (ka(x)k − kaN (x)k) (ka(x)k + kaN (x)k) ≤ ka(x) − aN (x)k (ka(x)k + kaN (x)k) ≤ ǫ (ka(x)k + kaN (x) − a(x)k + ka(x)k) and so This gives us: (2.25) ≤ ǫ (2 ka(x)k + ǫ) 2 2 (2.26) . (2.27) − kaN (x)k ≤ ǫ2 + 2ǫ ka(x)k − ka(x)k . 2 e−kaN (x)k /2 ≤ eǫ 2 /2 ǫka(x)k−ka(x)k2 /2 e k p Now since φ is a Borel function on R which is in L with respect to the Gaussian measure with density proportional to 2 e−ka(x)k /2 dx, LIMITING MEANS FOR SPHERICAL SLICES 9 for some p > 1. We have then the bound |φ(x)e−kaN (x)k 2 /2 | ≤ |φ(x)|eǫ 2 /2 ǫka(x)k −ka(x)k2 /2 e The dominating function is integrable: Z Z ǫ2 /2 ǫka(x)k −ka(x)k2 /2 e φ(x)e e dx ≤ cǫ Rk e p −ka(x)k2 /2 φ(x) e Rk where cǫ = e ǫ2 /2 Z e ǫqka(x)k−ka(x)k2 /2 Rk 1/q 1/p , dx (2.28) (2.29) (2.30) and q is the conjugate to p as usual: p−1 + q −1 = 1. The integral in cǫ is finite because, after changing variables to y = a(x), Z ∞ Z 2 2 etkyk−kyk /2 dy = |S k−1 | etR−R /2 Rk−1 dR < ∞, (2.31) Rk 0 for any t ∈ R. Using the argument above we can conclude the dominated convergence in (2.19) holds: Z dx cd−k−m IN (x) lim k N →∞ a 0,N cd−m Rk | det L0,N | z Z n o d−k−m−1 1 dx 2 2 −1 −2 0,N = (x − z ) L lim φ(x) 1 − a 0,N 0,N (k) z k/2 N →∞ | det L0,N | 2π Rk and using the limits (2.10), (2.11), and (2.12) we have established: Z Z dx cd−k−m IN f dσ̄ = lim k lim N →∞ a 0,N cd−m Rk N →∞ S N −1 (√N )∩Q−1 (w 0 ) | det L0,N | z N Z d−k−m−1 o n dx 1 2 2 −1 −2 0,N (x − z ) L = lim φ(x) 1 − a 0,N 0,N (k) z | det L0,N | 2π k/2 Rk N →∞   Z h(L0 L0 ∗ )−1 (x − z 0 (k) ), x − z 0 (k) i dx −k/2 = (2π) φ(x) exp − . 2 | det L0 | x∈Rk  Lemma 2.4. With notation as above,     0,N 0,N x − z(k) L−1 ≥ L−1 0 0,N x − z(k) (2.32) Proof. Recall the definition of L, it is the projection on k coordinates restricted to the ker Q: L : ker Q → X = Rk (2.33) and L0 is the restriction of L to the orthogonal complement of ker L inside ker Q. Since L is surjective L0 is an isomorphism. Let x ∈ Rk and y0 = L−1 0 (x) then any vector y ∈ L−1 (x) can be written as y = y0 + v This means v ∈ ker L. kyk = ky0 + vk v ∈ ker L 10 AMY PETERSON AND AMBAR N. SENGUPTA −1 for all y ∈ L−1 (x) therefore y0 = L−1 (x) of smallest norm. 0 (x) is the point in L By the same argument, provided N is large enough for LN to be a surjection, for −1 x ∈ Rk , the point L−1 0,N (x) is the point on LN (x) of smallest norm. N Let y ∈ ker(QN ) then y ∈ R and QN y = 0. Taking RN to be contained in l2 as RN ⊕ {0} then for all y ∈ ker QN we take y = (y, 0). Now 0 = QN y = Q(JN y) therefore JN y ∈ ker Q and so (y, 0) ∈ ker Q. Thus ker QN is contained in ker Q. Now for y ∈ ker QN we have L(y) = LN (y) since both L and LN are the −1 projection onto the first k coordinates. Since LN (x) is all y ∈ ker QN such that −1 LN (y) = x it is contained in L (x). Now we have the inequality (2.32).  Let us look at an example that shows the necessity of the Lp , p > 1, condition and the difficult nature of the limit of Gaussian integrals above. In this context for the function 2 g(x) = ex /2 (1 + x2 )−1 for all x ∈ R, we have Z Z 2 −x2 /2 g(x)e−(x−xN ) /2 dx = ∞ for all xN 6= 0, g(x)e dx < ∞ but R R and so lim z→0 Z g(x)e−(x−z) R 2 /2 (2.34) dx 6= Z g(x)e−x 2 /2 dx. (2.35) R Acknowledgments. We would like to thank Irfan Alam for many helpful discussions on the subject. We would also like to thank the online TikZ community and Arthur Parzygnat for help with the figure. References 1. Vladimir I. Bogachev. Gaussian measures, volume 62 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998. 2. Ludwig Boltzmann. Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten. Kk Hof-und Staatsdruckerei, 1868. 3. Takeyuki Hida. Stationary stochastic processes. Mathematical Notes. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. 4. Irina Holmes and Ambar N. Sengupta. A Gaussian Radon transform for Banach spaces. J. Funct. Anal., 263(11):3689–3706, 2012. 5. Hui Hsiung Kuo. Gaussian measures in Banach spaces. Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975. 6. Paul Lévy. Leçons de Analyse fonctionnelle. Gauthier-Villars, 1922. 7. James Clerk Maxwell. Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres. Philosophical Magazine, 19(124):19–32, 1860. 8. H. P. McKean. Geometry of differential space. Ann. Probability, 1:197–206, 1973. 9. F. Mehler. Über die Entwicklungen einer Function von beliebig vielen Variabln nach Laplacehen Functionen höherer Ordnung. J. Reine u. Angewandte Math., 1866. 10. Amy Peterson and Ambar N. Sengupta. The Gaussian Limit for High-Dimensional Spherical Means. J. Funct. Anal., 2018. 11. Ambar N. Sengupta. The Gaussian Radon transform as a limit of spherical transforms. J. Funct. Anal., 271(11):3242–3268, 2016. LIMITING MEANS FOR SPHERICAL SLICES 11 12. Norbert Wiener. Differential-space. Journal of Mathematics and Physics, 2(1-4):131–174, 1923. Amy Peterson: Department of Mathematics, University of Connecticut, Storrs, CT 062569, USA E-mail address: [email protected] Ambar N. Sengupta: Department of Mathematics, University of Connecticut, Storrs, CT 062569 E-mail address: [email protected]