Uniform convergence of orthogonal polynomial expansions for exponential weights

Abstract

We consider an exponential weight $w(x) = \exp(-Q(x))$ on ${\mathbb R} = (-\infty,\infty)$, where $Q$ is an even and nonnegative function on ${\mathbb R}$. We always assume that $w$ belongs to a relevant class $\mathcal{F}(C^2+)$. Let $\{p_n\}$ be orthogonal polynomials for a weight $w$. For a function $f$ on ${\mathbb R}$, $s_n(f)$ denote the $(n-1)$-th partial sum of Fourier series. In this paper, we discuss uniformly convergence of $s_n(f)$ under the conditions that $f$ is continuous and has a bounded variation on any compact interval of ${\mathbb R}$. In the proof of main theorem, Nikolskii-type inequality and boundedness of the de la Vall{\'{e}}e Poussin mean of $f$ play important roles.