Acceleration of Convergence of Fourier Series Using the Phenomenon of Over-Convergence

In recent publications of the author, the phenomenon of over-convergence was discovered, and a spectral method has been presented for accelerating the convergence of truncated Fourier series for smooth functions. On this basis, a certain parametric system that is biorthogonal to the corresponding segment of the Fourier system turned out to be unusually effective. This article reconsiders some approaches and makes some adjustments to previous publications. As a result, two improved schemes for the recovery of a function based on a finite set of its Fourier coefficients are proposed. Numerical experiments confirm a significant increase in the efficiency of corresponding algorithms in typical classes of smooth functions. In conclusion, some prospects for the development and generalization of the above approaches are discussed.