On Some Fast Implementations of Fourier Interpolation

Nersessian, Anry

doi:10.1007/978-3-030-77493-6_27

Cited by 4 publications

(5 citation statements)

References 6 publications

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“…The universality of our approach was confirmed in [13] in the case of an unusual interpolation on a finite grid. Here, too, a phenomenon of over-convergence was discovered, and the corresponding traditional and adaptive algorithms were proposed.…”

Section: Remark 14mentioning

confidence: 57%

“…The theoretical substantiation of this hypothesis (about the existence of "phenomenon of over-convergence") was obtained and applied in the case of Fourier series quite recently in [9][10][11][12]. In [13], a similar phenomenon was established in a discrete case when the convergence of a finite Fourier series was accelerated.…”

Section: Over-convergence Phenomenonmentioning

confidence: 89%

“…The studies published in papers [3][4][5][6][7][8][9][10][11][12][13] as well as the above results make it possible to predict their natural development. Here we note the following directions:…”

Section: About Nearest Development and Generalizationsmentioning

confidence: 99%

“…Above approximations was proposed earlier (see [9], Remark 3), but were not implemented numerically. Such an approximation has already been successfully applied in the discrete case for a finite Fourier series (see [13], formula (26)).…”

Section: Convergence Acceleration Formulasmentioning

confidence: 99%

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

Nersessian

2022

Armen.J.Math.

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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.

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Section: Remark 14mentioning

confidence: 57%

Section: Over-convergence Phenomenonmentioning

confidence: 89%

Section: About Nearest Development and Generalizationsmentioning

confidence: 99%

Section: Convergence Acceleration Formulasmentioning

confidence: 99%

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

Nersessian

2022

Armen.J.Math.

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“…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]. Furthermore, some simplifications and applications of Eckhoff algorithm have been studied by A.…”

Section: Introductionmentioning

confidence: 99%

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

Páez-Rueda,

Fajardo,

Pérez

et al. 2023

MCA

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This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach.

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Super-Fast Approximation Algorithms Using Classical Fourier Tools

Nersessian

2024

APM

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On Some Fast Implementations of Fourier Interpolation

Cited by 4 publications

References 6 publications

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

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

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

Super-Fast Approximation Algorithms Using Classical Fourier Tools

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