Optimal stability of regularized spectral differentiation in Sobolev spaces

Abstract

We study the problem of stable spectral differentiation of functions in Sobolev spaces from noisy data. We introduce a class of admissible Fourier multipliers under simple and directly verifiable conditions and show that the corresponding regularized differentiation operators achieve minimax optimal stability rates. The results extend the previous $L^2$ based results to Sobolev spaces $H^{s,p}({\mathbb{R}}^n)$, $1<p<\infty$. The analysis relies on multiplier estimates and applies to a wide class of multipliers, including Gaussian, spectral cutoff, and Tikhonov-type regularizations. Numerical examples demonstrate the behavior of several admissible spectral multipliers.

21 pages, 3 figures