Super-fast Computer Implementations of the Classical Fourier Series

In recent series of publications by the author, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system on a finite segment has proven unusually effective. On its basis, it was first discovered that knowledge of a finite number of Fourier coeffcients of the function f from an infinitedimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Then the corresponding parametric biorthogonal systems for trigonometric Fourier series were constructed and the corresponding over-convergence phenomena were confirmed. As a result, some very fast algorithms for reconstructing a function from its truncated series are proposed. These algorithms for accelerating the convergence of classical Fourier series are convenient for implementation in computer systems that allow extensive symbolic computations.
These results are presented below as a general scheme for all types of classical Fourier series, with an emphasis on the details of computer implementation.The presented numerical experiments confirm the high effciency of these convergence accelerations for suffciently smooth functions. In conclusion, the main results are summarized, and some prospects for developing and generalizing the proposed approaches are discussed.