Super-fast Computer Implementations of the Classical Fourier Series | Chapter 9| Mathematics and Computer Science: Contemporary Developments Vol. 9

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 coefficients of the function f from an infinite dimensional 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 efficiency of these convergence accelerations for sufficiently smooth functions. In conclusion, the main results are summarized, and some prospects for developing and generalizing the proposed approaches are discussed.

 

Author (s) Details

Anry Nersessian
Institute of Mathematics, National Academy of Sciences of Armenia. Yerevan, Republic of Armenia.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v9/2784