Journal of Mathematical Sciences

, Volume 188, Issue 2, pp 113–127

# On the nonclassical approximation method for periodic functions by trigonometric polynomials

Article

## Abstract

We study the approximation of functions by linear polynomial means of their Fourier series with a function-multiplier φ that is equal to 1 not only at zero, in contrast with classical methods of summability. The exact order of convergence to zero of the sequence
$$\mathop{\max}\limits_{{x\in \left[ {-\pi, \pi } \right]}}\left| {f(x)-\sum\limits_{{\left| k \right|\leq n}} {\varphi \left( {\frac{{k\pi }}{n}} \right){{\hat{f}}_k}{e^{ikx }}} } \right|$$
($${{\hat{f}}_k}$$ Fourier coefficients) as n→∞ is obtained. The answer is given in terms of the values of difference operators of a continuous function f and a special K-functional (step of $$\frac{\pi }{n}$$). In addition, we obtain not only the sufficient conditions for φ but the necessary ones as well.

## Keywords

Fourier series Fourier transformation of a measure multiplier principle of comparison of multipliers moduli of smoothness K-functional Wiener’s $$\frac{1}{f}$$ theorem

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