On approximation of Stepanov's almost periodic functions by means of Marcinkiewicz

Yu. Kh. Khasanov ; E. Safarzoda

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 61-69

Résumé

We study some questions of approximation of Stepanovs almost-periodic functions of partial Fourier sums and means of Marcinkiewicz, when the Fourier exponents of functions under consideration have a limit point in infinity. Let $S_p$ ($p\geq1$) denote the class of Stepanovs almost-periodic functions, whose Fourier exponents take the following form: $$ \lambda_0=0,\,\,\,\lambda_{-n}=-\lambda_n,\,\,\,\lim_{n\rightarrow\infty}\lambda_n=\infty,\,\,\, \lambda_n<\lambda_{n+1}\,\,\,(n=1,2,\ldots). $$ Consider the Fourier series for a functions $f(x)\in S_p$ $$ f(x)\sim\sum\limits_{n = -\infty}^\infty A_n e^{i\lambda_nx}, $$ where $$ A_n=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^Tf(x)e^{-i\lambda_nx}dx $$ are Fourier coefficients of the function $f(x)\in S_p$ and $$ S_\sigma(f;x)=\sum_{|\lambda_n|\leq\sigma}A_n e^{i\lambda_nx}\,\,\,(\sigma>0) $$ is a partial sum of Fourier series. Let $\Phi_\sigma(t)$ is an arbitrary real continuous even function such that $$ 1) \Phi_{\sigma}(0)=1; \,\,\,2) \Phi_{\sigma}(t)=0\,\,\,(|t|\leq\sigma). $$ We set $$ U_{\sigma}(f;\varphi;x)=\sum_{|\lambda_{m}|\leq\sigma}A_m\Phi_{\sigma}(\lambda_{m})e^{i\lambda_{m}x}. $$ Let $S_p(R)$ stand for the space of bounded functions $f(x)\in S_p \,\,\,(p\geq 1)$ with the norm $$ \|f(x)\|_{S_p}=\sup_{-\infty<x<\infty}\left\{\frac{1}{l}\int_x^{x+l}|f(x)|^p dx\right\}^{\frac{1}{p}}. $$ Consider the value $$ R(f;x)=\left\|U_{\sigma}(f;\varphi;x)-f(x)\right\|_{S_p}, $$ where $$ U_{\sigma}(f;\varphi;x)=\int_{-\infty}^{\infty} f(x+t) \Phi_{\sigma}(t)dt, $$ $$ \Phi_{\sigma}(t)=\frac{1}{2\pi}\int_{0}^{\infty}\varphi_{\sigma}(u)K_{u}(t)du,\,\,\, K_{u}(t)=2\frac{\sin(ut)}{t}, $$ $\varphi_{\sigma}(u)$ is some even function absolutely integrable on the interval $(0;\infty)$ with each fixed $\sigma>0$. Theorem. If $f(x)\in S_p$, where Fourier exponents have no limit points at a finite distance, i.e. $\lambda_n\rightarrow\infty$, then the following bound is valid $$ R(f;\varphi_{\sigma,a})\leq M\frac{\sigma+a}{\sigma-a}E_{\Lambda}(f)_{S_p}, $$ and $$ \left\|f(x)-\frac{1}{n+1}\sum_{k=0}^{n}S_{k}(f;x)\right\|_{S_p}\leq\frac{M}{n+1}\sum_{k=0}^{n}E_{k}(f)_{S_p}, $$ where $M$—constant and $$ E_{\Lambda}(f)_{S_p}=\inf_{A_m}\left\|f(x)-\sum_{|\lambda_{m}|\leq\Lambda}A_m e^{i\lambda_{m}x}\right\|_{S_p}. $$

Détail

Keywords: Stepanovs almost periodic functions, Fourier series, Fourier exponents, limiting point in infinity, means of Marcinkievicz, trigonometric polynomial, best approximation.

Bibliographie

[1] B.\;M. Levitan, Almost-Periodic Functions, Gostekhizdat Publ., M.–L., 1953, 396 pp.

[2] M.\;F. Timan, V.\;G. Ponomarenko, “On Approximation of Periodic Functions of Two Variables by Sums of Marcinkiewicz Type”, Russian Mathematics, 1975, no. 9, 59–67

[3] V.\;G. Ponomarenko, “On Approximation of Functions of Two Uniformly Continuous on the Whole Real Plane”, Siberian Mathematical Journal, 16:1 (1975), 86–97 | Zbl

[4] Yu.\;Kh. Khasanov, “Approximation of Almost Periodic Functions of Two Variables”, Russian Mathematics, 2010, no. 12, 82–86