Über ein Turánsches problem für image radiale, positiv definite Funktionen, II Original Research Article

Turánʹs problem is to determine the greatest possible value of the integral image for positive definite functions image, image, supported in a given convex centrally symmetric body image, image. We consider the problem for positive definite functions of the form image, image, with image supported in image, 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, image summability of the inverse Fourier integral on image. Their investigations gave rise to a pair of transformations image on image which they studied using special functions, in particular spherical Bessel functions. To study the d-dimensional Turán 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 Turánʹs problem.