On Summation of Nonharmonic Fourier Series

Summation of Fourier Series

A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A s...

Some Stability Theorems for Nonharmonic Fourier Series

The theory of nonharmonic Fourier series in L2(-ir,tr) is concerned with the completeness and expansion properties of sets of complex exponentials {e'x"'}. It is well known, for example, that the completeness of the set {e'x"'} ensures that of {e'^"'} whenever 2 lA„ ~~ M»l < oo. In this note we establish two results which guarantees that if {elX"'} is a Schauder basis for l}(—n, it), then [e'^"...

Pointwise and directional regularity of nonharmonic Fourier series

We investigate how the regularity of nonharmonic Fourier series is related to the spacing of their frequencies. This is obtained by using a transform which simultaneously captures the advantages of the Gabor and Wavelet transforms. Applications to the everywhere irregularity of solutions of some PDEs are given. We extend these results to the anisotropic setting in order to derive directional ir...

Summation of Multiple Fourier Series by Spherical Means.

Summation of Fourier