Non-saturated estimates of the Kotelnikov formula error

O. L. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part I, Tome 499 (2021), pp. 22-37

Résumé

We estimate the error of approximation by Kotelnikov sums $$U_Tf(x)= \sum_{j\in\Bbb Z}f\left(\frac{j}{T}\right)\mathrm{sinc}(Tx-j),\quad T>0,\quad \mathrm{sinc}{z}=\frac{\sin{\pi z}}{\pi z}.$$ Let $f\in\mathbf{A}$, i.e. $f(x)=\int_{\Bbb R}g(y)e^{ixy}\,dy$, $g\in L_1(\Bbb R)$, and let $\|f\|_\mathbf{A}=\int_{\Bbb R}|g|$ iz Wiener norm of $f$. Then the sharp inequality $$\|f-U_Tf\|_{\mathbf A}\leqslant 2A_{T\pi}(f)_{\mathbf A}$$ holds, where $A_{\sigma}(f)_{\mathbf{A}}$ is the best approximation of $f$ in the Wiener norm by entire functions of type not exceeding $\sigma$. We also establish non-saturated uniform estimates.