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,\ldots$, 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 $W^{\mu}_{s}$ 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 $W^{\mu}_{s}$ and use the smallest amount of input information in order.