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Year 2023, Volume: 6 Issue: 1, 55 - 76, 15.03.2023
https://doi.org/10.33205/cma.1251068

Abstract

References

  • J. P. Berrut, L. N. Trefethen: Barycentric lagrange interpolation, SIAM REVIEW, 46 (3) (2004), 501–517.
  • M. M. Džrbašjan: Expansions in systems of rational functions with fixed poles, Izv. Akad. Nauk Arm. SSR, Ser. Mat., 2 (1) (1967), 3–51.
  • M. M. Džrbašjan: Biorthogonal Systems of Rational Functions and best Approximation of the Cauchy Kernel on the Real Axis, Sbornik: Mathematics, 24 (3) (1974), 409–433.
  • T. Eisner, M. Pap: Discrete orthogonality of the Malmquist-Takenaka system of the upper half plane and rational interpolation, J. Fourier Anal. Appl., 20 (4) (2014), 1–16.
  • S. Fridli, P. Kovács, L. Lócsi and F. Schipp: Rational modeling of multi-lead QRS complexes in ECG signals, Annales Univ. Sci, Budapest., Sect. Comp., 36 (2012), 145–155.
  • S. Fridli, L. Lócsi and F. Schipp: Rational function systems in ECG processing, in R. Moreno-Diaz et al. (Eds.) EUROCAST 2011 (2011), 88–95.
  • S. Fridli, F. Schipp: Discrete rational biorthogonal systems on the disc, Annales Univ. Sci. Budapest., Sect. Comp., 50 (2020), 127–134.
  • M. Gaál, B Nagy, Zs. Nagy-Csiha and Sz. Gy. Révész: Minimal energy point system on the unit circle and the real line, SIAM J. Math. Anal., 52 (6) (2020), 6281–6296.
  • J. B. Garnett: Bounded Analytic Functions, Academic Press (1981).
  • K. Hoffman: Banach spaces of analytic functions, Prentice Hall Inc. (1962).
  • A. C. Ionita: Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems, Rice University’s digital scholarship archive (2013).
  • G. G. Lorentz, M. Golitschek and Y. Makovoz: Constructive Approximation: Advanced Problems, Springer Berlin Heidelberg (1996).
  • F. Malmquist: Sur la détermination d’une classe functions analytiques par leurs dans un esemble donné de points, Compute Rendus Six. Cong. math. scand. Kopenhagen, Denmark (1925), 253–259.
  • Zs. Nagy-Csiha, M. Pap: Discrete biorthogonal systems and equilibrium condition in the Hardy space of unit disc and upper half-plane, Mathematical Methods for Engineering Applications. ICMASE 2021. Springer Proceedings in Mathematics and Statistics, vol 384. Springer (2022), 291–301.
  • M. Pap: Hyperbolic wavelets and multiresolution in H2(T), Journal of Fourier Analysis and Applications, 17 (2011), 755–776.
  • M. Pap, F. Schipp: Malmquist-Takenaka systems and equilibrium conditions, Math. Pannon., 12 (2) (2001), 185-194.
  • M. Pap, F. Schipp: Equilibrium conditions for the Malmquist-Takenaka systems, Acta Sci. Math. (Szeged), 81 (3-4) (2015), 469–482.
  • Z. Szabó: Rational orthonormal functions and applications. PhD Thesis, Eötvös Loránd University, Budapest (2001).
  • Z. Szabó: Interpolation and quadrature formulae for rational systems on the unit circle, Annales Univ. Sci. Budapest., Sect. Comp., 21 (2004), 41–56.
  • S. Takenaka: On the orthogonal functions and a new formula of interpolation, Japanese J. Math. II. (1925), 129–145.

Construction of rational interpolations using Mamquist-Takenaka systems

Year 2023, Volume: 6 Issue: 1, 55 - 76, 15.03.2023
https://doi.org/10.33205/cma.1251068

Abstract

Rational functions have deep system-theoretic significance. They represent the natural way of modeling linear dynamical systems in the frequency (Laplace) domain. Using rational functions, the goal of this paper to compute models that match (interpolate) given data sets of measurements. In this paper, the authors show that using special rational orthonormal systems, the Malmquist-Takenaka systems, it is possible to write the rational interpolant $r_{(n, m)}$, for $n=N-1, m=N$ using only $N$ sampling nodes (instead of $2N$ nodes) if the interpolating nodes are in the complex unit circle or on the upper half-plane. Moreover, the authors prove convergence results related to the rational interpolant. They give an efficient algorithm for the determination of the rational interpolant.

References

  • J. P. Berrut, L. N. Trefethen: Barycentric lagrange interpolation, SIAM REVIEW, 46 (3) (2004), 501–517.
  • M. M. Džrbašjan: Expansions in systems of rational functions with fixed poles, Izv. Akad. Nauk Arm. SSR, Ser. Mat., 2 (1) (1967), 3–51.
  • M. M. Džrbašjan: Biorthogonal Systems of Rational Functions and best Approximation of the Cauchy Kernel on the Real Axis, Sbornik: Mathematics, 24 (3) (1974), 409–433.
  • T. Eisner, M. Pap: Discrete orthogonality of the Malmquist-Takenaka system of the upper half plane and rational interpolation, J. Fourier Anal. Appl., 20 (4) (2014), 1–16.
  • S. Fridli, P. Kovács, L. Lócsi and F. Schipp: Rational modeling of multi-lead QRS complexes in ECG signals, Annales Univ. Sci, Budapest., Sect. Comp., 36 (2012), 145–155.
  • S. Fridli, L. Lócsi and F. Schipp: Rational function systems in ECG processing, in R. Moreno-Diaz et al. (Eds.) EUROCAST 2011 (2011), 88–95.
  • S. Fridli, F. Schipp: Discrete rational biorthogonal systems on the disc, Annales Univ. Sci. Budapest., Sect. Comp., 50 (2020), 127–134.
  • M. Gaál, B Nagy, Zs. Nagy-Csiha and Sz. Gy. Révész: Minimal energy point system on the unit circle and the real line, SIAM J. Math. Anal., 52 (6) (2020), 6281–6296.
  • J. B. Garnett: Bounded Analytic Functions, Academic Press (1981).
  • K. Hoffman: Banach spaces of analytic functions, Prentice Hall Inc. (1962).
  • A. C. Ionita: Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems, Rice University’s digital scholarship archive (2013).
  • G. G. Lorentz, M. Golitschek and Y. Makovoz: Constructive Approximation: Advanced Problems, Springer Berlin Heidelberg (1996).
  • F. Malmquist: Sur la détermination d’une classe functions analytiques par leurs dans un esemble donné de points, Compute Rendus Six. Cong. math. scand. Kopenhagen, Denmark (1925), 253–259.
  • Zs. Nagy-Csiha, M. Pap: Discrete biorthogonal systems and equilibrium condition in the Hardy space of unit disc and upper half-plane, Mathematical Methods for Engineering Applications. ICMASE 2021. Springer Proceedings in Mathematics and Statistics, vol 384. Springer (2022), 291–301.
  • M. Pap: Hyperbolic wavelets and multiresolution in H2(T), Journal of Fourier Analysis and Applications, 17 (2011), 755–776.
  • M. Pap, F. Schipp: Malmquist-Takenaka systems and equilibrium conditions, Math. Pannon., 12 (2) (2001), 185-194.
  • M. Pap, F. Schipp: Equilibrium conditions for the Malmquist-Takenaka systems, Acta Sci. Math. (Szeged), 81 (3-4) (2015), 469–482.
  • Z. Szabó: Rational orthonormal functions and applications. PhD Thesis, Eötvös Loránd University, Budapest (2001).
  • Z. Szabó: Interpolation and quadrature formulae for rational systems on the unit circle, Annales Univ. Sci. Budapest., Sect. Comp., 21 (2004), 41–56.
  • S. Takenaka: On the orthogonal functions and a new formula of interpolation, Japanese J. Math. II. (1925), 129–145.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Zsuzsanna Nagy-csıha This is me 0000-0001-7491-6888

Margıt Pap This is me 0000-0001-5097-3757

Ferenc Weisz 0000-0002-7766-2745

Publication Date March 15, 2023
Published in Issue Year 2023 Volume: 6 Issue: 1

Cite

APA Nagy-csıha, Z., Pap, M., & Weisz, F. (2023). Construction of rational interpolations using Mamquist-Takenaka systems. Constructive Mathematical Analysis, 6(1), 55-76. https://doi.org/10.33205/cma.1251068
AMA Nagy-csıha Z, Pap M, Weisz F. Construction of rational interpolations using Mamquist-Takenaka systems. CMA. March 2023;6(1):55-76. doi:10.33205/cma.1251068
Chicago Nagy-csıha, Zsuzsanna, Margıt Pap, and Ferenc Weisz. “Construction of Rational Interpolations Using Mamquist-Takenaka Systems”. Constructive Mathematical Analysis 6, no. 1 (March 2023): 55-76. https://doi.org/10.33205/cma.1251068.
EndNote Nagy-csıha Z, Pap M, Weisz F (March 1, 2023) Construction of rational interpolations using Mamquist-Takenaka systems. Constructive Mathematical Analysis 6 1 55–76.
IEEE Z. Nagy-csıha, M. Pap, and F. Weisz, “Construction of rational interpolations using Mamquist-Takenaka systems”, CMA, vol. 6, no. 1, pp. 55–76, 2023, doi: 10.33205/cma.1251068.
ISNAD Nagy-csıha, Zsuzsanna et al. “Construction of Rational Interpolations Using Mamquist-Takenaka Systems”. Constructive Mathematical Analysis 6/1 (March 2023), 55-76. https://doi.org/10.33205/cma.1251068.
JAMA Nagy-csıha Z, Pap M, Weisz F. Construction of rational interpolations using Mamquist-Takenaka systems. CMA. 2023;6:55–76.
MLA Nagy-csıha, Zsuzsanna et al. “Construction of Rational Interpolations Using Mamquist-Takenaka Systems”. Constructive Mathematical Analysis, vol. 6, no. 1, 2023, pp. 55-76, doi:10.33205/cma.1251068.
Vancouver Nagy-csıha Z, Pap M, Weisz F. Construction of rational interpolations using Mamquist-Takenaka systems. CMA. 2023;6(1):55-76.