Pointwise Convergence of Fourier-type Series with Exponential Weights

TL;DR

This paper proves a pointwise convergence theorem for Fourier-type series associated with orthonormal polynomials under exponential weights on the real line, advancing understanding of their convergence behavior.

Contribution

It establishes a new pointwise convergence result for Fourier-type series with exponential weights, extending previous theories to a broader class of weights.

Findings

Proves pointwise convergence of Fourier-type series with exponential weights.

Extends classical convergence results to weighted orthogonal polynomial series.

Provides theoretical foundation for approximation using weighted Fourier series.

Abstract

Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow[0,\infty)$ be an even function. We consider the exponential weights $w(x)=e^{-Q(x)}$, $x\in \mathbb{R}$. In this paper we obtain a pointwise convergence theorem for the Fourier-type series with respect to the orthonormal polynomials $\left\{p_n(w^2;x)\right\}$.