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

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 approximatio