On &agr;-symmetric multivariate characteristic functions
Journal of Multivariate Analysis
On a theorem of T. Gneiting on &agr;-symmetric multivariate characteristic functions
Journal of Multivariate Analysis
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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Turan's problem is to determine the greatest possible value of the integral @!"R"^"df(x)dx/f(0) for positive definite functions f(x), x@?R^d, supported in a given convex centrally symmetric body D@?R^d, d@?N. We consider the problem for positive definite functions of the form f(x)=@f(@?x@?"1), x@?R^d, with @f supported in [0,@p], extending results of our first paper from two to arbitrary dimensions. Our two papers were motivated by investigations of Professor Y. Xu and the 2nd named author on, what they called, @?-1 summability of the inverse Fourier integral on R^d. Their investigations gave rise to a pair of transformations (h"d,m"d) on R"+ which they studied using special functions, in particular spherical Bessel functions. To study the d-dimensional Turan problem, we had to extend relevant results of B. & X., and we did so using again Bessel functions. These extentions seem to us to be equally interesting as the application to Turan's problem.