Asymptotics of Series Arising from the Approximation of Periodic Functions by Riesz and Cesàro Means

V. P. Zastavnyi

Donetsk National University

Abstract: Asymptotic expansions in powers of $\delta$ as $\delta\to+\infty$ of the series
$$ \sum_{k=0}^\infty(-1)^{(\beta+1)k}\frac{Q((\delta^\alpha-(ak+b)^\alpha)_+)}{(ak+b)^{r+1}}, $$
where $\beta\in\mathbb Z$, $\alpha,a,b>0$, and $r\in\mathbb C$, while $Q$ is an algebraic polynomial satisfying the condition $Q(0)=0$, are obtained. In special cases, these series arise from the approximation of periodic differentiable functions by the Riesz and Cesàro means.

Keywords: Riesz mean, Cesàro mean, periodic differentiable function, approximation of periodic functions, algebraic polynomial, Hurwitz function, Euler gamma function, Bernoulli spline, Euler spline, Bernoulli polynomial.

UDC: 517.518.83+517.15

Received: 12.09.2010

DOI: 10.4213/mzm9109

 English version: Mathematical Notes, 2013, 93:1, 58–68

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