On Approximation of Almost Periodic Functions by Some Sums

Khasanov, Yu. Kh.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 4.

Abstract:
In this paper we study some questions of approximating almost periodic functions of two variables by partial Fourier sums and Marcinkiewicz type sums in the uniform metric, provided that the Fourier exponents of the functions under consideration have a limit point at infinity. More precisely, we consider a uniform almost periodic function of two variables whose Fourier exponents have a unique limit point at infinity. It is proved that the partial sum of this series with the weight function \(\Phi_\sigma(t,z)\) \((\sigma>0)\) admits an integral representation. As a weight function, we take an arbitrary real continuous even function \(\Phi_\sigma(t,z)\) that takes the value 1 for \(t=0\) and \(z=0\) and vanishes when either \(|t|\geq\sigma\) or \(|z|\geq\sigma\). First, we prove almost periodicity of the function \(f(x,y)\) and using the Fourier inversion formula we define the Fourier coefficients of this function. Then, we examine the deviation of the given function \(f(x,y)\) from partial sums of its Fourier series, depending on the speed of tending to zero of value of the best approximation by trigonometric polynomial of limited degree. Similarly, we obtain the upper bound of the deviation value of uniform almost-periodic functions from sums of Marcinkiewicz type.

Keywords: almost periodic function, approximation of functions, sums of Marcinkiewicz type, Fourier coefficients, Fourier exponents, limit points in infinity

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