TITLE:
Super-Fast Approximation Algorithms Using Classical Fourier Tools
AUTHORS:
Anry Nersessian
KEYWORDS:
Fourier Series, Acceleration of Convergence, Parametric Biorthogonalization, Spectral Methods, Over-Convergence Phenomenon
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.7,
July
29,
2024
ABSTRACT: In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.