On an Over-Convergence Phenomenon for Fourier series. Basic Approach

This paper is devoted to the acceleration of convergence partial sums classical Fourier series for sufficiently smooth functions. Some universal and adaptive algorithms are constructed studied. It shown that use a finite number coefficients makes it possible exact approximation given function from an infinite-dimensional set quasi-polynomials. In this sense, we call corresponding essentially nonlinear as over-convergent. The proposed implemented using Wolfram Mathematica system. Numerical results demonstrate their effectiveness.