On convergence almost everywhere of multiple Fourier Integrals

TL;DR

This paper proves that partial sums of multiple Fourier integrals of functions in L2(R^N) converge to zero almost everywhere, extending the understanding of convergence in spectral expansions of polyharmonic operators.

Contribution

It establishes the almost everywhere convergence of multiple Fourier integrals for functions in L2(R^N), advancing spectral expansion theory for polyharmonic operators.

Findings

Partial sums of multiple Fourier integrals converge almost everywhere.

Convergence applies to functions in L2(R^N).

Results extend the principle of localisation for spectral expansions.

Abstract

In this paper we investigate the principle of the generalised localisation for spectral expansions of the polyharmonic operator, which coincides with the multiple Fourier integrals summed over the domains corresponding to the surface levels of the polyharmonic polynomials. It is proved that the partial sums of the multiple Fourier integrals of a function f\inL_2 (R^N ) converge to zero almost-everywhere on R^N\supp(f).