## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 48, Number 2 (2019), 263-280.

### Uniform convergence of orthogonal polynomial expansions for exponential weights

Kentaro ITOH, Ryozi SAKAI, and Noriaki SUZUKI

#### 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.

#### Article information

**Source**

Hokkaido Math. J., Volume 48, Number 2 (2019), 263-280.

**Dates**

First available in Project Euclid: 11 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1562810508

**Digital Object Identifier**

doi:10.14492/hokmj/1562810508

**Mathematical Reviews number (MathSciNet)**

MR3980942

**Zentralblatt MATH identifier**

07080094

**Subjects**

Primary: 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol s kii-type inequalities)

Secondary: 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10}

**Keywords**

uniformly convergence of Fourier series weighted polynomial approximation Erdős type weight de la Vallée Poussin mean Nikolskii-type inequality

#### Citation

ITOH, Kentaro; SAKAI, Ryozi; SUZUKI, Noriaki. Uniform convergence of orthogonal polynomial expansions for exponential weights. Hokkaido Math. J. 48 (2019), no. 2, 263--280. doi:10.14492/hokmj/1562810508. https://projecteuclid.org/euclid.hokmj/1562810508