Some inverse theorems for approximation of functions by the Fourier-Laguerre sums
Let \(L_2:=L_2\left((0,+\infty)\times(0,+\infty),x^\alpha y^\beta e^{-x-y}\right)\) with \(\alpha,\beta>-\frac12\). For \(f\in L_2\), let \(\Omega_k(f;\delta)=\sup_{0<h\leq\delta}\|\Delta^k_h(f;x,y)\|\) be a generalized modulus of continuity of the \(k\)th order of \(f\) and let \(E_N(f)=\|f-S_N\|\), where \(S_N(f;x,y)\) is the triangular partial sum of the development \(f(x,y)=\sum_{n,m=0}^\infty c_{n,m}L_n^{(\alpha)}(x)L_m^{(\beta)}(y)\), \(\{L_n^{(\alpha)}\},\{L_m^{(\beta)}\}\) being normalized orthogonal systems of Laguerre polynomials. The author shows that 1. \(\Omega_k(f;\frac1N)\ll\left(\frac1{N^{2k}}\sum_{n=1}^Nn^{2k-1}E_n^2(f)\right)^{1/2}\;(k=1,2,\dots)\); 2. If the series \(\sum_{n=1}^\infty n^{r-1}E_n(f)\;(r\in\{1,2,\dots\})\) converges then \(f\) is a Levi \(r\)-differentiable function and \(\Omega_k(D^rf;\frac1N)\ll\left(\frac1{N^{2k}}\sum_{n=1}^Nn^{2(r+k)-1}E_n^2(f)\right)^{1/2}+\sum_{n=1}^\infty n^{r-1}E_n(f)\).
weighted \(L_2\) approximation in \(\mathbb R^2\)
scientific article; zbMATH DE number 5840957