# On a convergence of the Fourier-Pade interpolation

## Abstract

We investigate convergence of the rational-trigonometric-polynomial interpolation that performs convergence acceleration of the classical trigonometric interpolation by sequential application of polynomial and rational correction functions. Unknown parameters of the rational corrections are determined along the ideas of the Fourier-Pade approximations. The resultant interpolation we call as Fourier-Pade interpolation and investigate its convergence in the regions away from singularities. Comparison with other rational-trigonometric-polynomial interpolations outlines the convergence properties of the Fourier-Pade interpolation.

## References

[1] B. Adcock, Modified Fourier expansions: theory, construction and applications. PHD thesis, Trinity Hall, University of Cambridge, 2010.

[2] G.A. Baker, and P. Graves-Morris, Pad˙e Approximants, Encyclopedia of mathematics and its applications. Vol. 59, 2nd ed., Cambridge Univ. Press, Cambridge, 1966.

[3] G. Baszenski, F.-J. Delvos, and M. Tasche. A united approach to accelerating trigonometric expansions, Comput. Math. Appl. 30(36)(1995), 33–49.

[4] D. Batenkov, and Y. Yomdin. Algebraic Fourier reconstruction of piecewise smooth functions, Mathematics of Computation, 81(277)(2012), 277–318.

[5] J. P. Boyd, Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of 1/n, J. Comp. Phys. 228(5)(2009), 1404-1411.

[6] A. Krylov, On approximate calculations. Lectures delivered in 1906, Tipolitography of Birkenfeld, Petersburg, 1907.

[7] C. Lanczos, Evaluation of noisy data, J. Soc. Indust. Appl. Math., Ser. B Numer. Anal., vol. 1, 76-85, 1964.

[8] C. Lanczos, Discourse on Fourier Series, Oliver and Boyd, Edinburgh, 1966.

[9] A. Poghosyan, On a fast convergence of the rational-trigonometric-polynomial interpolation, Submitted to Analysis and Applications, 2012.

[10] A. Poghosyan, Asymptotic behavior of the Krylov-Lanczos interpolation, Analysis and Applications 7(2) (2009), 199–211.

[11] A. Poghosyan, On a linear rational-trigonometric interpolation of smooth functions, Reports of NAS RA, 106(1) (2006), 13–19.

[12] A. Poghosyan, On a convergence acceleration of trigonometric interpolation, Submitted to Reports of NAS RA, 2012.

[13] Poghosyan A., Barkhudaryan A., Mkrtchyan S., Accelerating the convergence of trigonometric interpolation, Proceedings of the Third Russian-Armenian workshop on mathematical physics, complex analysis and related topics, October 4 - 8, 2010, Tsaghkadzor, Armenia, 133–137.

[2] G.A. Baker, and P. Graves-Morris, Pad˙e Approximants, Encyclopedia of mathematics and its applications. Vol. 59, 2nd ed., Cambridge Univ. Press, Cambridge, 1966.

[3] G. Baszenski, F.-J. Delvos, and M. Tasche. A united approach to accelerating trigonometric expansions, Comput. Math. Appl. 30(36)(1995), 33–49.

[4] D. Batenkov, and Y. Yomdin. Algebraic Fourier reconstruction of piecewise smooth functions, Mathematics of Computation, 81(277)(2012), 277–318.

[5] J. P. Boyd, Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of 1/n, J. Comp. Phys. 228(5)(2009), 1404-1411.

[6] A. Krylov, On approximate calculations. Lectures delivered in 1906, Tipolitography of Birkenfeld, Petersburg, 1907.

[7] C. Lanczos, Evaluation of noisy data, J. Soc. Indust. Appl. Math., Ser. B Numer. Anal., vol. 1, 76-85, 1964.

[8] C. Lanczos, Discourse on Fourier Series, Oliver and Boyd, Edinburgh, 1966.

[9] A. Poghosyan, On a fast convergence of the rational-trigonometric-polynomial interpolation, Submitted to Analysis and Applications, 2012.

[10] A. Poghosyan, Asymptotic behavior of the Krylov-Lanczos interpolation, Analysis and Applications 7(2) (2009), 199–211.

[11] A. Poghosyan, On a linear rational-trigonometric interpolation of smooth functions, Reports of NAS RA, 106(1) (2006), 13–19.

[12] A. Poghosyan, On a convergence acceleration of trigonometric interpolation, Submitted to Reports of NAS RA, 2012.

[13] Poghosyan A., Barkhudaryan A., Mkrtchyan S., Accelerating the convergence of trigonometric interpolation, Proceedings of the Third Russian-Armenian workshop on mathematical physics, complex analysis and related topics, October 4 - 8, 2010, Tsaghkadzor, Armenia, 133–137.

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## Published

2013-07-17

## How to Cite

*Armenian Journal of Mathematics*,

*5*(1), 1-25. Retrieved from http://armjmath.sci.am/index.php/ajm/article/view/14

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Articles