Uniform convergence of Fourier series via deferred Cesàro mean and its applications

Abstract

In this article, we found an undesirable feature in the theory of summability; that is, the Fourier series of 2π$$ 2\pi $$‐periodic functions is uniformly convergent to the functions via the Ceàsro mean. However, it does not preserve the uniform convergence for the arbitrary periodic functions. To overcome this limitation, the objective of the paper is to introduce and study the notion of deferred Ceàsro mean based on the q$$ q $$ integers for the Fourier series of arbitrary periodic functions. We further study the behavior of our proposed method under both ordinary and statistical versions of convergence and accordingly establish two Korovkin‐type approximation theorems for trigonometric test functions. Finally, we also provide some examples with geometrical illustrations in support of the effectiveness of our results established in this work.

Approximation and simulation of signals via harmonic Banach summable factors of Fourier series

“…Subsequently, in the same year, Weisz [5] discussed the unrestricted Cesàro summability of $$$ d $$$‐dimensional Fourier series and Lebesgue points. Very recently, Jena et al [6] studied and proved uniform convergence of Fourier series via deferred Cesàro mean and its applications. The approximation of signals using various summability means has been studied by various researchers like Das et al [7], Jena et al [8], Lal and Mishra [9], Mittal et al [10, 11], Móricz and Németh [12], Mursaleen and Alotaibi [13], Sharma and Rao [14], and Singh et al [15].…”