Fourier Tools are Much More Powerful than Commonly Thought

Abstract

In the proposed paper, some last autor’s results of studies devoted to the acceleration of the convergence of truncated Fourier series is presented. The corresponding universal (traditional) and special adaptive algorithms are constructed. The main result (the phenomenon of over-convergence for an non-linear adaptive algorithm) states that the use of finite Fourier coefficients leads to an exact approximation for functions from certain infinite-dimensional spaces of quasipolynomials. The corresponding summation formula of truncated Fourier series for smooth functions has unprecedented accuracy.