Eine Verallgemeiherung der Hardy-Landauschen Identität

Abstract

Suppose g¹, g² are the generating vectors of a plane lattice Λ and

If R is a non-negative real number, then the identity of Hardy and Landau gives

where J¹ is the first order Bessel function. This paper describes general sums

and proves the identity

for a function f with continous second derivatives in the circle of radius R about the origin. The main difficulty is the proof of the convergence of the series on the right hand side. The paper makes use of an inversion formula for the twodimensional Fourier-transform which does not seem to be known in this form.