Numerical differentiation of functions on the half-line in a weighted uniform metric

Abstract

We consider the problem of numerical differentiation of functions defined on the half-line: the derivative $f^{(r)}$, $r=1,2,\dots$, is recovered from a finite set of perturbed Fourier--Laguerre coefficients of a function $f$, whose error is measured in ${\ell }_p$, while the accuracy of approximation is measured in the uniform metric with the weight $t^{\alpha }{e}^{-t}$. It is established that the Wiener class $W_s^{\mu }$ admits such a setting for $\mu >r+1-1/s$. A class of methods ${\mathcal{S}}^{\theta }$ is constructed that realize the optimal order of accuracy of numerical differentiation on $W_s^{\mu }$ and use the smallest amount of input information in order.