On approximate properties of Fourier series by systems of rational functions on the real axis and some applications (Q2703340)

The article deals with the approximate properties of the segments of the Fourier series by the systems of functions \(\omega_{k}(x)=({x-i\over x+ i})^{k}\) and \(\psi_{k}(x)=({x-i\over x+i})^{k} {2i\over x+i}\), \(k=0,\pm 1,\ldots\); \(x\in\mathbb{R}\). For example the following results are presented. If \((U_{n}f)(x)=\sum_{k=-n}^{n}f_{k}\psi_{k}(x)\), where \(f_{k}={1\over 4\pi}\int_{\mathbb{R}}f(x)\overline{\psi_{k}(x)} dx\), \(f(x)\in L_2\), then \(\|f(x)-(U_{n}f)(x)\|_{L_2}\to 0\), as \(n\to\infty\). If \(f(x)\in L_{2}^{(r)}\), \(r\geq 1\), then \(\|f(x)-(U_{n}f)(x) \|_{L_2}\leq \sqrt{2\pi}K_{r}(n+1)^{-r}\|f^{(r)}(x) \|_{L_2}\). The approximate properties of the Fourier series by the Laguerre functions \(l_{k}(x)=e^{-x}\sum_{j=0}^{k}c_{k}^{j}(-1)^{j} {2^{j}\over j!}x^{j}\), \(x>0\), are considered, too.