Fractional derivatives and the inverse Fourier transform of ℓ1-radial functions

Abstract

In [Berens, H. and Xu, Y. (1997). ℓ − 1 summability of multiple Fourier integrals and positivity. Math. Proc. Camb. Phil. Soc., 122, 149–172] and [Berens, H. and zu Castell, W. (1998). Hypergeometric functions as a tool for summability of the Fourier integral. Result. Math., 34, 69–84], H. Berens, Y. Xu, and the author proved that the inverse Fourier integral of ℓ1-radial functions, i.e., functions which are radial w.r.t. the ℓ1-norm on ℝ d , can be decomposed into a multi-dimensional integral with B-spline kernel, which is independent of the function, and a transformation on ℝ+ the kernel of which is given by a Meijer G-function. The latter can be seen as an analog of the well-known Fourier–Bessel transform. Here, we associate weights to the underlying transform. There is an analog development where the B-spline has to be replaced by a Dirichlet spline, while on ℝ+ the related transforms can again be defined via G-function kernels. We further show that the underlying transform can be interpreted as a fractional derivative.