Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 13, Issue 1

Issues

Volume 13 (2015)

An extended Prony’s interpolation scheme on an equispaced grid

Dovile Karalienė
  • Department of Applied Mathematics, Kaunas University of Technology, Studentu 50-325, LT-51368, Kaunas, Lithuania
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Zenonas Navickas
  • Department of Applied Mathematics, Kaunas University of Technology, Studentu 50-325, LT-51368, Kaunas, Lithuania
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Raimondas Čiegis
  • Department of Mathematical Modelling, Vilnius Gediminas Technical University, Sauletekio 11, LT-10223, Vilnius, Lithuania
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Minvydas Ragulskis
  • Research Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-147, LT-51368, Kaunas, Lithuania
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/math-2015-0031

Abstract

An interpolation scheme on an equispaced grid based on the concept of the minimal order of the linear recurrent sequence is proposed in this paper. This interpolation scheme is exact when the number of nodes corresponds to the order of the linear recurrent function. It is shown that it is still possible to construct a nearest mimicking algebraic interpolant if the order of the linear recurrent function does not exist. The proposed interpolation technique can be considered as the extension of the Prony method and can be useful for describing noisy and defected signals.

Keywords: Interpolation; Prony method; The minimal order of linear recurrent sequence

MSC: 65D05, 65D10, 65D15

References

  • [1] Badeau R., David B., High-resolution spectral analysis of mixtures of complex exponentials modulated by polynomials, IEEE Trans. Signal Process., 2006, 54, 1341–1350. CrossrefGoogle Scholar

  • [2] Badeau R., Richard G., David B., Performance of ESPRIT for estimating mixtures of complex exponentials modulated by polynomials, IEEE Trans. Signal Process., 2008, 56, 492–504. CrossrefWeb of ScienceGoogle Scholar

  • [3] Ehlich H., Zeller K., Auswertung der Normen von Interpolationsoperatoren, Math. Ann., 1996, 164, 105–112. CrossrefGoogle Scholar

  • [4] Higham N.J., The numerical stability of barycentric Lagrange interpolation, IMA J. Numer. Anal., 2004, 24, 547–556. CrossrefGoogle Scholar

  • [5] Navickas Z., Bikulciene L., Expressions of solutions of ordinary differential equations by standard functions, Mathematical Modeling and Analysis, 2006, 11, 399–412. Google Scholar

  • [6] Peter T., Plonaka G., A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems, 2013, 29, 025001. CrossrefWeb of ScienceGoogle Scholar

  • [7] Platte R.B., Trefethen L.N., Kuijlaars A.B.J., Impossibility of fast stable approximation of analytic functions from equispaced samples, SIAM Review, 2011, 53, 308–314. CrossrefWeb of ScienceGoogle Scholar

  • [8] Ragulskis M., Lukoseviciute K., Navickas Z., Palivonaite R., Short-term time series forecasting based on the identification of skeleton algebraic sequences, Neurocomputing, 2011, 64, 1735–1747. Web of ScienceCrossrefGoogle Scholar

  • [9] Runge C., Uber empirische Funktionen and die Interpolation zwischen aquidistanten Ordinaten, Z. Math. Phys., 1901, 46 224– 243. Google Scholar

  • [10] Salzer H.E., Lagrangian interpolation at the Chebyshev points xn;υ = cos(υπ/n), υ = 0(1)n; some unnoted advantages, Computer J., 1972, 15, 156–159. Google Scholar

  • [11] Schonhage A., Fehlerfortpflanzung bei Interpolation, Numer. Math., 1961, 3, 62–71. CrossrefGoogle Scholar

  • [12] Trefethen L.N., Pachon R., Platte R.B., Driscoll T.A., Chebfun Version 2, http://www.comlab.ox.ac.uk/chebfun/, Oxford University, 2008. Google Scholar

  • [13] Turetskii A.H., The bounding of polynomials prescribed at equally distributed points, Proc. Pedag. Inst. CityplaceVitebsk, 1940, 3, 117–127. Google Scholar

  • [14] Osborne M.R., Smyth G.K., A Modified Prony Algorithm For Exponential Function Fitting, SIAM Journal of Scientific Computing, 1995, 16, 119–138. Google Scholar

  • [15] Martin C., Miller J., Pearce K., Numerical solution of positive sum exponential equations, Applied Mathematics and Computation, 1989, 34, 89–93. Google Scholar

  • [16] Fuite J., Marsh R.E., Tuszynski J.A., An application of Prony’s sum of exponentials method to pharmacokinetic data analysis, Commun. Comput. Phys., 2007, 2, 87–98. Google Scholar

  • [17] Giesbrecht M., Labahn G., Wen-shin Lee, Symbolic-numeric sparse interpolation of multivariate polynomials, Journal of Symbolic Computation, 2009, 44, 943–959. Google Scholar

  • [18] Steedly W., Ying C.J., Moses O.L., A modified TLS-Prony method using data decimation, IEEE Transactions on Signal Processing, 1992, 42, 2292–2303. CrossrefGoogle Scholar

  • [19] Kurakin V.L., Kuzmin A.S., Mikhalev A.V., Nechavev A.A., Linear recurring sequneces over rings and modules, Journal of Mathematical Sciences, 1995, 76, 2793–2915. Google Scholar

  • [20] Kurakin V., Linear complexity of polinear sequences, Disctrete Math. Appl., 2001, 11, 1–51. CrossrefGoogle Scholar

  • [21] Potts D., Tasche M., Parameter estimation for multivariate exponential sums, Electron. Trans. Numer. Anal., 2013, 40, 204–224. Google Scholar

  • [22] Kaltofen E., Villard G., On the complexity of computing determinants, Computers Mathematics Proc. Fifth Asian Symposium (ASCM 2001), Lecture Notes Series on Computing, 2001, 9, 13–27. Google Scholar

  • [23] Kaw A., Egwu K., Numerical Methods with Applications, Textbooks collection Book 11, 2010, ch. 5. Google Scholar

About the article

Received: 2014-05-29

Accepted: 2015-04-09

Published Online: 2015-05-21


Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0031.

Export Citation

©2015 Dovile Karalienė et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
TADAS TELKSNYS, ZENONAS NAVICKAS, MARTYNAS VAIDELYS, and MINVYDAS RAGULSKIS
Advances in Complex Systems, 2016, Volume 19, Number 04n05, Page 1650010

Comments (0)

Please log in or register to comment.
Log in