A NOTE ON POSITIVE DEFINITE NORM DEPENDENT FUNCTIONS

A NOTE ON POSITIVE DEFINITE NORM DEPENDENT FUNCTIONS ALEXANDER KOLDOBSKY Abstract. Let K be an origin symmetric star body in R n. We prove, under very mild conditions on the function f : [, ) R, that if the function f( x K ) is positive definite on R n, then the space (R n, K ) embeds isometrically in L. This generalizes the solution to Schoenberg s problem and leads to progress in characterization of n-dimensional versions, i.e. random vectors X = (X 1,...,X n ) in R n such that the random variables a i X i are identically distributed for all a R n, up to a constant depending on a K only. 1. Introduction In 1938, Schoenberg [S] posed the problem of finding the exponents < p < 2 for which the function exp( x p q) is positive definite on R n, where x q is the norm the space l n q with 2 < q. Recall that a complex valued function f defined on R n is called positive definite on R n if, for every finite sequence {x i } m i=1 in R n and every choice of complex numbers {c i } m i=1, we have m m c i c j f(x i x j ). i=1 j=1 For q =, the problem was solved in 1989 by Misiewicz [M2], and for 2 < q <, the answer was given in [K1] in 1991 (note that, for 1 p 2, Schoenberg s question was answered earlier by Dor [D], and the case n = 2, < p 1 was established in [F], [H], [L]). The answers turned out to be the same for both cases: the function exp( x p q) is not positive definite if the dimension of the space is greater than 2, and for n = 2 the function is positive definite if and only if < p 1. Different and independent proofs of Schoenberg s problems were given by Lisitsky [L??] and Zastavnyi [Z1], [Z2] shortly after the paper [K1] appeared. For an origin symmetric star body K in R n, let E K = (R n, K ) be the space whose unit ball is K, where x K = min{a : x ak} is the Minkowski functional of K. Note that the class of star bodies 1

2 ALEXANDER KOLDOBSKY includes convex bodies, and E K is a normed space if and only if K is convex (see [K2], p. 13). Denote by Φ(K) = Φ(E K ) the class of continuous functions f : [, ) R for which f( K ) is a positive definite function on R n. The classes Φ(K) admit an interesting probabilistic interpretation. Following Eaton [E], we say that a random vector X in R n is an n- dimensional version if all linear combinations of its coordinates have the same distribution, up to a constant, namely for any vector a R n the random variables n a i X i and a K X 1 i=1 are identically distributed. The result of Eaton is that a random vector is an n-dimensional version if and only if its characteristic functional has the form f( x K ). Hence, by Bochner s theorem, the problem of finding all n-dimensional versions is equivalent to characterizing the classes Φ(K). Note that, by the classical result of P.Lévy, if K is the unit ball of a finite dimensional subspace of L q, < q 2, then the function exp( t q ) Φ(K), and the corresponding n-dimensional versions are the classical q-stable vectors. The classes Φ(K) have been studied by a number of authors. Schoenberg [S??] proved that f Φ(B2 n ) if and only if f(t) = Ω n (tr) dλ(r) where B2 n is the unit Euclidean ball in R n, Ω n ( 2 ) is the Fourier transform of the uniform probability measure on the sphere S n 1, and λ is a finite measure on [, ). In the same paper, Schoenberg proved an infinite dimensional version of this result: f Φ(l 2 ) if and only if f(t) = exp( t 2 r 2 ) dλ(r). Bretagnolle, Dacunha-Castelle and Krivine [BDK] proved a similar result for the classes Φ(l q ) for all q (, 2) (one just has to replace 2 by q in the formula), and showed that for q > 2 the classes Φ(l q ) (corresponding to infinite dimensional l q -spaces) are trivial, i.e. contain constant functions only. Cambanis, Keener and Simons [CKS] obtained a similar representation for the classes Φ(B n 1 ). Richards [Ri] partially characterized the classes Φ(B n q ) for < q < 2. Aharoni, Maurey and Mityagin [AMM] proved that if E is an infinite dimensional

POSITIVE DEFINITE FUNCTIONS 3 Banach space with a symmetric basis {e n } n=1 such that e 1 + + e n lim =, n n 1/2 then the class Φ(E) is trivial. Misiewicz [M??] proved that for n 3 the classes Φ(l n ) are trivial, and Lisitsky [Lis] and Zastavnyi [Za1], [Za2] proved the same the classes Φ(l n q), q > 2, n 3. One can find more detailed information and references on isotropic random vectors in [MiS], [Mis3], [K2],[Gn]. In all the results mentioned above the classes Φ(K) appear to be nontrivial only if K is the unit ball of a subspace of L q with < q 2. It was conjectured by Misiewicz [M??] that the latter condition on K is necessary for Φ(K) to be non-trivial. In support of this conjecture, Misiewicz [M??] and Kuritsyn [Ku] proved that if f Φ(K) is a nonconstant function and its inverse Fourier transform ν (which is a finite measure on R, by Bochner s theorem) has a finite moment or the order q (, 2], then K is the unit ball of a subspace of L q. Lisitsky showed that if f Φ(K) is a non-constant function and log t dν(t) < R then, (R n, K ) embeds in L (the definition of embedding in L was given later in [KKYY]; see below), and formulated a weaker conjecture that if Φ(K) is non-trivial then K is the unit ball of a subspace of L. The purpose of this note is to provide simple conditions on the function f itself (rather than on its inverse Fourier transform) under which f( x K ) can be positive definite only if K is the unit ball of a subspace of L q, q 2. We prove that if f is a continuous non-constant function satisfying f() f(t) C t q in a neighborhood of the origin, where C >, q (, 2), and f( x K ) is positive definite, then K is the unit ball of a subspace of L q. We also prove that if lim t t ǫ f(t) < for some ǫ (, 1), and f( x K ) is positive definite, then K is the unit ball of a subspace of L. This shows that, in order to defy the conjectures of Misiewicz and Lisitsky, the function f must exhibit rather odd behaviour at both the origin and infinity. Finally, we combine these facts with known results about embedding in L q to further generalize the solution of Schoenberg s problem. 2. Proofs As usual, we denote by S(R n ) the space of infinitely differentiable rapidly decreasing functions on R n (Schwartz test functions), and by S (R n ) the space of distributions over S(R n ). If φ S(R n ) and f S (R n ) is a locally integrable function with power growth at infinity,

4 ALEXANDER KOLDOBSKY then f,φ = f(x)φ(x) dx. R n We say that a distribution is positive (negative) outside of the origin in R n if it assumes non-negative (non-positive) values on non-negative Schwartz s test functions with compact support outside of the origin. The Fourier transform of a distribution f is defined by ˆf,φ = f, ˆφ for every test function φ. We need a Fourier analytic criterion of embedability in L q that applies to every q > which is not an even integer; see [K, Th.??]. Proposition 1. Let K be an origin-symmetric star body in R n, and q > is not an even integer. Then the space (R n, K ) embeds isometrically in L q if and only if Γ( q/2)( q K ) is a positive distribution on R n \ {}. We now prove our first result. Theorem 1. Let K be an origin symmetric star body in R n, and f a non-constant continuous function on [, ). Suppose that there exist C >, < q < 2, u > such that f() f(t) Ct q (1) for every t (,u). If f( x K ) is a positive definite function, then the space (R n, K ) embeds isometrically in L q. Proof : A positive definite function f( x K ) has absolute maximum at zero (see [??]) and is bounded on R n, hence f() f(t) for every t > and f is bounded on [, ). Let < α < q. The condition (1) and the remark above imply that the integral c = t 1 α (f() f(t)) dt converges and is positive (f is not a constant). Making a change of variables u = t x, we see that for every x R n \ {} c x α = t 1 α (f() f(t x )) dt. (2) Let φ be an even non-negative test function supported outside of the origin. Then R n ˆφ(x) dx = (2π) n φ() =. (3)

POSITIVE DEFINITE FUNCTIONS 5 Applying the definition of the Fourier transform of a distribution and equalities (2) and (3), we get ( α K),φ = x α K, ˆφ = x α ˆφ(x) K dx R n = 1 c = 1 c ) t (R 1 α f(t x K )ˆφ(x) dx dt n t 1 α (f(t K )),φ dt, because f(t K ) is a positive definite function on R n for every fixed t R, and, by Bochner s theorem, (f(t K )) is a finite measure on R n. For every < α < q and x R n, we have x α K ˆφ(x) max(1, x q K ) ˆφ(x), where the function of x R n in the right-hand side is integrable, so by the dominated convergence theorem, ( q K ),φ = x q ˆφ(x) K dx R n = lim x α ˆφ(x) K dx = lim ( α α q R n α q K),φ. Now the result follows from Proposition 1 with < q < 2. The concept of embedding in L was introduced in [KKYY]. Definition 1. We say that a space (R n, K ) embeds in L if there exist a finite Borel measure µ on the sphere S n 1 and a constant C R so that, for every x R n, ln x K = ln (x,ξ) dµ(ξ) + C. (4) S n 1 Embedding in L also admits a Fourier analytic characterization, as established in [KKYY, Th.??]. Proposition 2. Let K be an origin symmetric star body in R n. The space (R n, K ) embeds in L if and only if the Fourier transform of ln x K is a negative distribution outside of the origin in R n. We use the latter statement to prove our next result.

6 ALEXANDER KOLDOBSKY Theorem 2. Let K be an origin symmetric star body in R n, and f a continuous function on [, ) such that lim t tǫ f(t) = (5) for some ǫ (, 1). If f( x K ) is a positive definite function, then the space (R n, K ) embeds in L. Proof : By the condition (5) and since f is a bounded function, for every < δ < ǫ, the integral c = t 1+δ f(t) dt converges absolutely. We need to show that c >. In fact, making a change of variables z = tu and expressing the resulting integral in terms of the Γ-function, we get u δ exp( t 2 u 2 /2) du = t 1+δ Γ((1 δ)/2). The function f( t ) is positive definite on R as the restriction to R of a positive definite function. By Bochner s theorem, f( ) = ˆν for some finite measure ν on R. We have 1 c = u δ f( t ), exp( t 2 u 2 /2) du Γ((1 δ)/2) 1 = u δ ν, (exp( t 2 u 2 /2)) du >, Γ((1 δ)/2) since ν is a non-negative measure and the Fourier transform of a gaussian density is also a gaussian density, up to a positive constant. Now for any x R n \ {}, we have c x δ K = t 1+δ f(t x K ) dt. For every even non-negative test function φ, x δ K, ˆφ = 1 c t 1+δ f(t K ), ˆφ dt = 1 t 1+δ (f(t K )),φ dt, c since the function f(t K ) is positive definite for any fixed t R. Suppose that, in addition, φ is supported outside of the origin, then by (3) x δ K 1, δ ˆφ = 1 δ x δ K, ˆφ.

POSITIVE DEFINITE FUNCTIONS 7 Sending δ to zero, we get that log x K, ˆφ = (log K ),φ, and by Proposition 2, the space (R n, K ) embeds in L. 3. Generalizations of Schoenberg s problem Combining Theorems 1 and 2 with known results about embedding of normed spaces in L p, we get several generalizations of the solution of Schoenberg s problem. First, let us note that the function exp( t q ), < q < 2 satisfies the conditions of Theorem 1, and the space l 3 p with p > 2 does not embed in L q ; see [K??]. By Theorem 1, the function exp( x q p) is not positive definite, which confirms the solution of Schoenberg s problem. Theorem 3. Let < p < n and let X be an n-dimensional normed space with a normalized basis e 1,...,e n, n 3 so that: (i) For every fixed (x 2,...,x n ) R n 1 \ {}, the function n x 1 x 1 e 1 + x i e i has a continuous second derivative everywhere on R, and i=2 x x 1 (,x 2,...,x n ) = x x 2 1(,x 2,...,x n ) =, where x x 1 and x stand for the first and second partial derivatives x 2 1 by x 1 of the norm x 1 e 1 + + x n e n. (ii) There exists a constant C so that, for every x 1 R and every (x 2,...,x n ) R n 1 with x 2 e 2 + + x n e n = 1, one has (iii) Convergence in the limit x x 2 1(x 1,x 2,...,x n ) C. lim x 1 x x 2 1(x 1,x 2,...,x n ) = is uniform with respect to (x 2,...,x n ) R n 1 with x 2 e 2 + +x n e n = 1. Then the function p represents a positive definite distribution if and only if p [n 3,n). In particular the convex set {x R n : n i=1 x ie i 1} is a k-intersection body if and only if k is one of the numbers n 3,n 2,n 1, k. Theorem 4. Let X be a three-dimensional normed space with a normalized basis {e 1,e 2,e 3 } so that

8 ALEXANDER KOLDOBSKY (i) For every fixed (x 2,x 3 ) R 2 \{}, the function x 1 x 1 e 1 + x 2 e 2 + x 3 e 3 has a continuous second derivative everywhere on R, and x x 1 (,x 2,x 3 ) = x x 2 1(,x 2,x 3 ) =, where x x 1 and x stand for the first and second derivatives by x x 2 1, 1 respectively. (ii) There exists a constant C so that for every x 1 R and every (x 2,x 3 ) R 2 with x 2 e 2 + x 3 e 3 = 1, one has x x1(x 2 1,x 2,x 3 ) C. Then, for every < p 2, the function exp( p ) is not positive definite, and the space X does not embed isometrically in L p, < p 2. Theorem 5. Let k N {}, q > 2, and let Y be a finite dimensional normed spaces with dim(y ) k + 3. Then the unit ball of the q-sum of R and Y is not a k-intersection body. Another example of a space satisfying the conditions of Theorem 3 is a finite dimensional Orlicz space l n M, whose Orlicz function M C 2 ([, )) and M () = M () =. Recall that an Orlicz function M is a non-decreasing convex function on [, ) such that M() = and M(t) > for every t >. The norm M of the n-dimensional Orlicz space l n M is defined implicitly by the equality n k=1 M( x k / x M ) = 1, x R n \ {}. Theorem 6. Suppose that M is an Orlicz function such that M C 2 ([, )) and M () = M () =. Then, for every k < n 3 the unit ball of the n-dimensional Orlicz space l n M is not a k-intersection body. References [AMM] I. Aharoni, B. Maurey and B. Mityagin, Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces, Israel J. Math. 52 (1985), 251 265. [As] R. Askey, Radial characteristic functions, Tech. Report no. 1262, University of Wisconsin, Madison, 1973. [BDK] J. Bretagnolle, D. Dacunha-Castelle and J. L. Krivine, Lois stables et espaces L p, Ann. Inst. H. Poincaré Probab. Statist. 2 (1966), 231 259. [CKS] S. Cambanis, R. Keener, and G. Simons, On α-symmetric multivariate distributions, J. Multivariate Analysis 13 (1983), 213 233. [DK] S. Dilworth and A. Koldobsky, The Fourier transform of order statistics with applications to Lorentz spaces, Israel J. Math. 92 (1995), 411 425. [Do] L. Dor, Potentials and isometric embeddings in L 1, Israel J. Math. 24 (1976), 26 268. [Ea] M. Eaton, On the projections of isotropic distributions, Ann. Stat. 9 (1981), 391 4.

POSITIVE DEFINITE FUNCTIONS 9 [Fe] T. S. Ferguson, A representation of the symmetric bivariate Cauchy distributions, Ann. Math. Stat. 33 (1962), 1256 1266. [GS] I.M. Gelfand and G.E. Shilov, Generalized functions, vol. 1. Properties and operations, Academic Press, New York, (1964). [Gn1] T. Gneiting, On α-symmetric multivariate characteristic functions, J. Multivariate Anal. 64 (1998), 131 147. [Gn2] T. Gneiting, Criteria of Polya type for radial positive definite functions, Proc. Amer. Math. Soc. 129 (21), 239 2318. [He] C. Herz, A class of negative definite functions, Proc. Amer. Math. Soc. 14 (1963), 67 676. [KKYY] N. J. Kalton, A. Koldobsky, V. Yaskin and M. Yaskina, The geometry of L, Canad. J. Math., to appear. [K1] A. Koldobsky, The Schoenberg problem on positive-definite functions, Algebra i Analiz 3 (1991), 78 85; translation in St. Petersburg Math. J. 3 (1992), 563 57. [K12] A. Koldobsky, Positive definite distributions and subspaces of L p with applications to stable processes, Canad. Math. Bull. 42 (1999), 344 353. [K1] A. Koldobsky, Fourier analysis in convex geometry, Amer. Math. Soc., Providence RI, 25. [KL] A. Koldobsky and Y. Lonke, A short proof of Schoenberg s conjecture on positive definite functions, Bull. London Math. Soc. 31 (1999), 693 699. [Kur] Yu. G. Kuritsyn, Multidimensional versions and two problems of Schoenberg, Problems of Stability of Stochastic Models, VNIISI, Moscow, 1989, 72 79. [KuS] Yu. G. Kuritsyn, and A. V. Shestakov, On α-symmetric distributions, Theory Probab. Appl. 29 (1984), 769 771. [Le] P. Lévy, Théorie de l addition de variable aléatoires, Gauthier-Villars, Paris, 1937. [Li] J. Lindenstrauss, On the extension of operators with finite dimensional range, Illinois J. Math. 8 (1964), 488 499. [Lis1] A. Lisitsky, One more proof of Schoenberg s conjecture, unpublished manuscript, 1991. [Lis2] A. Lisitsky, The Eaton problem and multiplicative properties of multivariate distributions, Theor. Probab. Appl. 42 (1997), 618 632. [Mis1] J. Misiewicz, On norm dependent positive definite functions, Bull. Acad. Sci. Georgian SSR 13 (1988), 253 256. [Mis2] J. Misiewicz, Positive definite functions on l, Stat. Probab. Let. 8 (1989), 255 26. [Mis3] J. Misiewicz, Substable and pseudo-isotropic processes connections with the geometry of subspaces of L α -spaces, Dissertationes Math. (Rozprawy Mat.) 358 (1996). [MisR] J. Misiewicz and Cz. Ryll-Nardzewski, Norm dependent positive definite functions, Lecture Notes in Math. 1391, Springer-Verlag, Berlin-Heidelberg- New York, 1987, 284 292. [MisS] J. Misiewicz, and C. L. Scheffer, Pseudo isotropic measures, Nieuw Arch. Wisk. 8 (199), 111 152. [Ri] D. St. P. Richards, Positive definite symmetric functions on finite dimensional spaces. 1.Applications of the Radon transform, J. Multivariate Analysis 19 (1986), 28 298.

1 ALEXANDER KOLDOBSKY [Sc1] I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522 536. [Sc2] I. J. Schoenberg, Metric spaces and completely monotone functions, Annals of Math. 39 (1938), 811 841. [Za1] V. Zastavnyi, Positive definite norm dependent functions, Dokl. Russian Acad. Nauk. 325 (1992), 91 93. [Za2] V. Zastavnyi, Positive definite functions depending on the norm, Russian J. Math. Phys. 1 (1993), 511 522. Department of Mathematics// University of Missouri-Columbia// Columbia, MO 65211 E-mail address: koldobskmath.missouri.edu