Positive definite functions, completely monotone functions, the Bernstein-Widder theorem, and Schoenberg s theorem

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1 Positive definite functions, completely monotone functions, the Bernstein-Widder theorem, and Schoenberg s theorem Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto June 6, 15 1 Linear operators For a comple Hilbert space H let L H) be the bounded linear operators H H. It is a fact that A L H) is self-adjoint if and only if Ah, h R for all h H. 1 For a bounded self-adjoint operator A it is a fact that A sup Ah, h. h 1 A L H) is called positive if it is self-adjoint and Ah, h, h H; because we have taken H to be a comple Hilbert space, for A to be positive it suffices that the inequality is satisfied. For A, B L C n ), we define their Hadamard product A B L H) by So, A B)e i Ae i, e j Be i, e j e j. j1 A B)e i, e j Ae i, e j Be i, e j. The Schur product theorem states that if A, B L C n ) are positive then their Hadamard product A B is positive. 3 1 John B. Conway, A Course in Functional Analysis, second ed., p. 33, Proposition.1. John B. Conway, A Course in Functional Analysis, second ed., p. 34, Proposition Ward Cheney and Will Light, A Course in Approimation Theory, p. 81, chapter 1. 1

2 Positive definite functions Let X be a real or comple linear space, let f : X C be a function, and for 1,..., n X, define F f;1,..., n L C n ) by F f;1,..., n e i f i j )e j, j1 where {e 1,..., e n } is the standard basis for C n. Thus for u n i1 u ie i C n, n n F f;1,..., n u, u f i j )e j, u k e k u i i1 j1 n u i i1 j1 i1 j1 f i j ) u i u j f i j ). e j, k1 u k e k k1 We call f positive definite if for all 1,..., n X, F f;1,..., n operator, i.e. for u C n, is a positive F f;1,..., n u, u. We call f strictly positive definite for all distinct 1,..., n X and nonzero u C n, F f;1,..., n u, u >. 3 Completely monotone functions A function f : [, ) R is called completely monotone if 1. f C[, ). f C, ) 3. 1) k f k) ) for k and, ) Because a completely monotone function is continuous, f) tends to f) as. Because a completely monotone function is nonincreasing and conve, f) has a limit, which we call f ), as. The Bernstein-Widder theorem states that a function f satisfying f) 1 is completely monotone if and only if it is the Laplace transform of a Borel probability measure on [, ). 4 4 Peter D. La, Functional Analysis, p. 138, chapter 14, Theorem 3; the-bernstein-theorem-on-completely-monotone-functions/

3 Theorem 1 Bernstein-Widder theorem). A function f : [, ) R satisfies f) 1 and is completely monotone if and only if there is a Borel probability measure µ on [, ) such that f) e t dµt), [, ). Proof. If f is the Laplace transform of some probability measure µ on B [, ), then using the dominated convergence theorem yields that f is continuous and by induction that f C, ). For k and for, ), f k) ) t) k e t dµt), as t k e t dµt) so 1) k f k) ). Hence f is completely monotone, and f) dµt) 1. If f satisfies f) 1 and is completely monotone, then for each k, the function 1) k f k) :, ) R is nonnegative and is nonincreasing, so for k 1 and t, ), using that 1) k f k) is nondecreasing and that 1) k 1 f k 1) is nonnegative, t 1) k f k) t) 1) k f k) u)du t t/ ) t 1)k f k 1) t) f k 1) t/) Doing induction, for any k 1, 1) k f k) t) t 1)k 1 f k 1) t/). k 1 j1 k 1 j1 Because f) f) as, t f k 1 j t j t ) ) t f k 1 t k kk 1)/ f ) f and because f) f ) as, ) ) t t f k 1 f k ) k t f t k 1 t k 1 ) t f ) f ) t k, t, 3, t. t k k )) )).

4 Hence for each k 1, and ft) o k t k ), t, 1) ft) o k t k ), t. ) Furthermore, for any, ), f k) t) f k) ) as t, so it is immediate that t ) k f k) t), t. 3) For and k 1, integrating by parts, using ) and 1) or 3) respectively as or >, f) f ) f t)dt t )f t) Hence for and n, Define + f t)t )dt f t)t )dt t ) f t) f t ) t) dt f t ) t) dt 1) k f k) t )k 1 t) dt. k 1)! f) f ) 1)n+1 n! For n 1, by change of variables, For t, define f) f ) 1)n+1 n! φ n y) 1 y/n) n 1 [,n] y). 1)n+1 n 1)! 1)n+1 n 1)! 1)n+1 n 1)! s n t) 1)n+1 n 1)! /n /n f n+1) t)t ) n dt. f n+1) nu)nu ) n ndu f n+1) nu)nu) n 1 ) n du nu nu) n φ n /u)f n+1) nu)du n/t) n φ n t)f n+1) n/t)t dt. 1/t 4 nu) n f n+1) nu)du,

5 where s n ), and for t < let s n t). s n is continuous and because f is completely monotone, s n is nondecreasing, so there is a unique positive measure σ n on B R such that 5 σna, b]) snb) sna), a b. On the other hand, s n is absolutely continuous, so σ n is absolutely continuous with respect to Lebesgue measure λ 1, and for λ 1 -almost all t R, 6 dσ n dλ 1 t) s nt). Now for t >, by the fundamental theorem of calculus and the chain rule, and therefore s nt) 1)n+1 n 1)! n/t)n f n+1) n/t) t, f) f ) φ n t)s nt)dλ 1 t) φ n t) dσ n dλ 1 t)dλ 1 t) φ n t)dσ n t). The total variation of σ n is equal to the total variation of s n, and because s n is nondecreasing, which is σ n s nt) dt σ n 1)n+1 n 1)! s nt)dt s n ) s n ) s n ), nu) n f n+1) nu)du f) f ), showing that {σ n : n 1} is bounded for the total variation norm. We claim that {σ n : n 1} is tight: for each ɛ > there is a compact subset K ɛ of R such that σ n Kɛ c ) < ɛ for all n. Taking this for granted, Prokhorov s theorem 7 states that there is a subsequence σ kn of σ n that converges narrowly to some positive measure σ on B R. Finally, the sequence t φ n t) tends in C b [, )) to t e t, and it thus follows that 8 φ n t)dσ n t) e t dσt), 5 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 393, Theorem H. L. Royden, Real Analysis, third ed., p. 33, Eercise V. I. Bogachev, Measure Theory, volume II, p., Theorem cf. Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 511, Corollary

6 so Let with which hence f) f ) e t dµt) µ σ + f )δ, f) e t dσt). e t dσt) + f ), e t dµt). Because f) 1, dµt) 1, showing that µ is a probability measure. 4 Fourier transforms For a topological space X and a positive Borel measure µ on X, F X is called a support of µ if i) F is closed, ii) µf c ), and iii) if G is open and G F then µg F ) >. If F 1 and F are supports of µ, it is straightforward that F 1 F. It is a fact that if X is second-countable then µ has a support, which we denote by supp µ. 9 Lemma. If µ is a Borel measure on a topological space X and µ has a support supp µ, if f : X [, ) is continuous and fdµ then f) for all X supp µ. Proof. Let F supp µ and let E { X : f) }. E is an open subset of X. Suppose by contradiction that there is some E F, i.e. that E F. Because f is continuous and f) >, there is some open neighborhood G of for which fy) > f)/ for y U. Then G F, so G F and because F is the support of µ, µg F ) > and a fortiori µg) >. Then X fdµ G fy)dµy) G f) f) dµy) µg) >, a contradiction. Therefore E F, i.e. for all F, f). The following lemma asserts that a certain function is nonzero λ d -almost everywhere, where λ d is Lebesgue measure on R d. 1 9 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 44, Theorem Ward Cheney and Will Light, A Course in Approimation Theory, p. 91, chapter 13, Lemma 6. 6

7 Lemma 3. Let 1,..., n be distinct points in R d, let u C n not be the zero vector, and define gy) u j e πij y, y R d. j1 For λ d -almost all y R d, gy). The following theorem gives conditions under which the Fourier transform of a Borel measure on R d is strictly positive definite. 11 Theorem 4. If µ is a finite Borel measure on R d and λ d supp µ) >, then ˆµ : R d C is strictly positive definite. Proof. For distinct 1,..., n R d and for nonzero u C n, u j u k ˆµ j k ) R d u j u k e R πij k) y dµy) d n ) u j e πij y u k e πi k y dµy) j1 k1 u j e πij y dµy) R d j1 gy) dµy). R d It is apparent that this is nonnegative. If it is equal to then because g is continuous we obtain from Lemma that gy) for all y supp µ, i.e. gy) for all y supp µ. In other words, supp µ {y R d : gy) }. But by Lemma 3, λ d {y R d : gy) }), so λ d supp µ), contradicting the hypothesis λ d supp µ) >. Therefore R d gy) dµy) >, which shows that ˆµ is strictly positive definite. 5 Schoenberg s theorem Let X,, ) be a real inner product space. We call a function F : X R radial when y implies that F ) F y). 11 Ward Cheney and Will Light, A Course in Approimation Theory, p. 9, chapter 13, Theorem 3. 7

8 An identity that is worth memorizing is that for y R, e π e πiy d e πy. R Using this and Fubini s theorem yields, y R d, e π,y e π y. R d e π Lemma 5. For α > and y R d, π ) ) d/ ep π R α α e πi,y d e α y. d Proof. Define T : R d R d by T ) π α, Rd. T ) π α I L Rd ) and J T ) det T ) π d/. α) Let u R d and define f) e π e πi,u. By the change of variables formula, 1 f T ) J T dλ d fdλ d, R d T R d ) and because T is self-adjoint this is π T ) e πi,t u π ) d/ d α R d e and therefore R d π α e π R d e πi,u d, ) ) d/ ep π α e πi,t u d e π u. For u T 1 y) α π y this is π ) ) d/ ep π R α α e πi,y d e α y, d proving the claim. We now prove that on a real inner product space, e α positive definite whenever α >. 13 is strictly 1 Charalambos D. Aliprantis and Owen Burkinshaw, Principles of Real Analysis, third ed., p. 393, Theorem Ward Cheney and Will Light, A Course in Approimation Theory, p. 14, chapter 15, Theorem. 8

9 Theorem 6. Let X,, ) be a real inner product space. If α >, then is radial and strictly positive definite. e α, X, Proof. Let 1,..., n be distinct points in X. There is an n-dimensional linear subspace V of X that contains 1,..., n. By the Gram-Schmidt process, V has an orthonormal basis {v 1,..., v n }. Define T : V R n by T v j e j, where {e 1,..., e n } is the standard basis for R n, which is an orthogonal transformation, and define fu) e α u, u R d. For u C n, u, u j u k e α j k u j u k ep α T j k ) ) u j u k ft j T k ). Now, let µ be the Borel measure on R d whose density with respect to λ d is ) π d/ y ep α) π α y. Because µ is absolutely continuous with respect to λ d, λ d supp µ) >, so Theorem 4 states that the Fourier transform ˆµ : R d C is strictly positive definite. Applying Lemma 5, the Fourier transform of µ is π ) ) d/ ˆµu) ep π R α α y e πi y,u dy e α u fu), d so f is strictly positive definite. Because T is an orthogonal transformation it is in particular one-to-one, so T 1,..., T n are distinct points in R d. Thus the fact that f is strictly positive definite means that u j u k e α j k u j u k ft j T k ) >, which establishes that e α is strictly positive definite. The following is Schoenberg s theorem Ward Cheney and Will Light, A Course in Approimation Theory, p. 11, chapter 15, Theorem 1; René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein Functions: Theory and Applications, p. 14, Theorem 1.14; William F. Donoghue Jr., Distributions and Fourier Transforms, p. 5, 41. 9

10 Theorem 7 Schoenberg s theorem). Let X,, ) be a real inner product space. If f : [, ) R is completely monotone, f) 1, and f is not constant, then f ), X [, ), is radial and strictly positive definite. Proof. Because f is completely monotone, the Bernstein-Widder theorem Theorem 1) tells us that there is a Borel probability measure µ on [, ) such that ft) e st dµs), t [, ), that is, f is the Laplace transform of µ. Now, the Laplace transform of δ is t 1, and because f is not constant, the Laplace transform of µ is not equal to the Laplace transform of δ, which implies that µ δ. 15 Therefore µ, )) >. Let 1,..., n be distinct points in X and let u C n, u. Then, because n n u ju k, u j u k f j k ) + u j u k u j u k µ{}) 1, ) s) 1, ) s) gs)dµs). ep s j k ) dµs) u j u k ep s j k ) dµs) u j u k ep s j k ) dµs) u j u k ep s j k ) dµs) Assume by contradiction that gs)dµs). Because g, this implies that µ{s [, ) : gs) > }). 16 By Theorem 6, for each s >, u j u k ep s j k ) >, 15 Bert Fristedt and Lawrence Gray, A Modern Approach to Probability Theory, p. 18, 13.5, Theorem Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 411, Theorem

11 so gs) > when s >. Thus µ, )), a contradiction. Therefore, u j u k f j k ) gs)dµs) >, which shows that f ) is strictly positive definite. 11