ACTA issues

A class of summability methods of the inverse Fourier integral

Hubert Berens , Wolfgang zu Castell

Acta Sci. Math. (Szeged) 70:3-4(2004), 683-693

Abstract. The authors introduce a class of summability methods of the inverse Fourier integral on ${\msbm R}^d$, which grew out of investigations known as $\ell $-1 summability. There is a basic kernel $F_d(\rho; \cdot )$, $\rho >0$, $\rho\to \infty $, on ${\msbm R}^d$: Starting with $F_1(\rho; \cdot )=2\cos\rho (\cdot )$, on ${\msbm R}$, the kernel $F_d$ is defined inductively as Laplace convolution w.r.t. the parameter $\rho $; i.e., $F_d(\cdot;{\bf x})=F_{d-1}(\cdot;{\bf x}')*_L F_1(\cdot;x_d)$, ${\bf x}=({\bf x}',x_d)=(x_1,\ldots,x_d)$ in ${\msbm R}^d$. It defines the Dirichlet kernel of the inverse Fourier integral w.r.t. the $l_1$-norm on ${\msbm R}^d$. Here we investigate an extension of the kernel $F_d$ by introducing a set of parameters $({\bf a};{\bf b})\in{\msbm R}_+^d\times{\msbm R}^d$ and by further taking means, see the definitions below. We are interested in basic properties of the summability kernels, especially in convergence of the methods. The paper extends and, in a way, completes previous work of Y. Xu, Zh.-K. Li and the authors.

AMS Subject Classification (1991): 42B08, 26A33, 40A10

Keyword(s): summability, radial functions, fractional derivatives, Ces?ro means, Riesz means, Abel means, convolution kernels

Received July 21, 2003, and in revised form April 20, 2004. (Registered under 5838/2009.)