On the relationship of a convolution type transformation and the best approximation of periodic functions

Yu. Kh. Khasanov ; Y. F. Kasimova

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 24 (2021) no. 1, pp. 5-15.

Résumé

The space $L_p$ is understood as the collection of $2\pi$ periodic functions $f(x)\in L_p$, for which $|f(x)|^p$ is Lebesgue integral with the norm $$ \|f\|_{L_p}=\left(\int_0^{2\pi}|f(x)|^pdx\right)^{1/p}\,\,\,(1\leq p\infty), $$ for $p=\infty$ $$\|f\|_{L_\infty}=vrai\sup_{x\in[0,2\pi]}|f(x)|.$$ For spaces, consider $L_p\,\,\,(1\leq p\leq\infty)$ transformations of the type $$ F(f;y;x;h)=\int_{-\infty}^\infty f(x-uh)dy(u), $$ where $f(x)\in L_p, \,\,\,h$ is an arbitrary parameter, and $y(x)$ is an arbitrary function of bounded variation at $(-\infty, \infty)$ is not identically zero and satisfies the conditions $$ \int_{-\infty}^\infty dy(u)=0,\,\,\,V(y)=\int_{-\infty}^\infty|dy(u)|\infty. $$ For transformations $F(f;y;x;h)$ in the works of H.S. Shapiro and J. Boman the general problem of possible dependencies is considered $$ G(f;y;h;p)=\|\int_{-\infty}^\infty f(x-uh)dy(u)\|_{L_p} $$ for two different values $y_1(x)$ and $y_2(x)$. In particular, if $f(x)\in L_p$ $(1\leq p\leq\infty)$, and $y_1(x)$, $y_2(x)$ are two finite-dimensional on $(-\infty,\infty)$ functions for which, for $l>0$ the condition $\widehat{y}_2(x)=x^lF(x)$, where $F(x)$ is the Fourier transform of some measure $\rho(x)$, then the inequality \begin{multline*} W(y_2;f;h)_{L_p}\leq M(y_1,y_2,\rho)\left\{\int_0^hW^\gamma(y_1;f;t)_{L_p}\frac{dt}{t}+\right. \\ +\,\left.h^\gamma\int_h^\infty W^\gamma(y_1;f;Bt)_{L_p}\frac{dt}{t^{\gamma l+1}}\right\}^{\frac{1}{\gamma}}, \end{multline*} where $$ W(y;f;h)_{L_p}=\sup_{|t|\leq h}\|G(f;y;h;\rho)\|_{L_p}, $$ $\gamma=\min(2,\rho)$ for $1\leq p\infty$ and $\gamma=1$, if $p=\infty$, and $M(y_1,y_2,\rho)$ и $B$ — are some constants. Continuing and refining the results of H.S. Shapiro and J. Boman, M.F. Timan obtained order-sharp estimates both from above and below for the value $F(f;y;x;h)$ depending on the values of the best approximations of the function $f(x)\in L_p$ $(1\leq p\leq\infty)$. In this work, a number of results were obtained, which complement in some cases and refine the results of the above works of the authors. In order to generalize some of the results, the question of the relationship between the quantity $F(f;y;x;h)$ and best approximations $E_n(f)_{L_p}$ is studied.

Détail

Keywords: periodic function, Fourier series, best approximations, trigonometric polynomials, bounded variation functions.
Mots-clés : convolution type transformation, Fourier transform, Fourier coefficients