Mean convergence of Fourier-Akhiezer-Chebyshev series

TL;DR

This paper proves the mean convergence of Fourier series based on Akhiezer-Chebyshev polynomials in L^p spaces for p>1, utilizing a weighted Hilbert transform inequality on the unit circle.

Contribution

It introduces a new proof of mean convergence for these Fourier series using a weighted Hilbert transform inequality.

Findings

Established mean convergence in L^p for p>1

Applied weighted Hilbert transform inequality on the unit circle

Extended convergence results to Akhiezer-Chebyshev polynomial series

Abstract

We prove mean convergence of the Fourier series in Akhiezer-Chebyshev polynomials in $L^p$, $p>1$, using a weighted inequality for the Hilbert transform in an arc of the unit circle.