On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

Abstract

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial

“…The first is the standard partial Fourier series using the coefficients (92), which have a decreasing rate O(|k| −1 ), and they produce ringing artifacts, as shown in Figure 10. The second is a mixed Fourier series using particular coefficients with x(0) = 0 and x(T) = 1 obtained from the BVP by ( 88)- (91). The mixed series includes Fourier coefficients with a decreasing rate O(|k| −3 ), and they do not produce ringing artifacts, as shown in Figure 10.…”

“…Over the last two decades, several Armenian researchers have made significant contributions to the Lanczos approach. For instance, the works of A. Nersessian and A. Poghosyan addressed the main issue of some alternatives to the quasi-Bernoulli series in [67,[87][88][89][90][91][92], such as the quasi-polynomial series, the Fourier-Pade series, the trigonometric interpolations series, and the quasi-polynomial Pade series. Similarly, A. Nersessian studied a framework based on a biorthogonal system and adaptive algorithms with a strong potential for accelerating the convergence of Fourier series due to an over-convergence phenomenon [93][94][95][96].…”