Reduction of the Gibbs phenomenon for smooth functions with jumps by the ε-algorithm

Recently, Brezinski has proposed to use Wynn's @e-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Pade-Fourier and Pade-Chebyshev approximants, including those recently studied by Kaber and Maday.

[1]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[2]  Allan Pinkus,et al.  Progress in Approximation Theory , 1991 .

[3]  Claude Brezinski,et al.  Extrapolation Algorithms for Filtering Series of Functions, and Treating the Gibbs Phenomenon , 2004, Numerical Algorithms.

[4]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[5]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[6]  Anne Gelb,et al.  Detection of Edges in Spectral Data II. Nonlinear Enhancement , 2000, SIAM J. Numer. Anal..

[7]  G. Páris,et al.  The Gibbs phenomenon in generalized Padé approximation , 1985 .

[8]  Sergey Pavlovich Suetin,et al.  On the Rate of Convergence of Padé Approximants of Orthogonal Expansions , 1992 .

[9]  Thomas Hagstrom,et al.  Locating Discontinuities of a Bounded Function by the Partial Sums of Its Fourier Series , 1999, J. Sci. Comput..

[10]  Knut S. Eckhoff.  Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions , 1995 .

[11]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[12]  W. Rudin.  Real and complex analysis , 1968 .

[13]  George Kvernadze.  Determination of the Jumps of a Bounded Function by Its Fourier Series , 1998 .

[14]  Stefan Paszkowski.  Approximation uniforme des fonctions continues par les fonctions rationnelles , 1962 .

[15]  P. Wynn,et al.  TRANSFORMATIONS TO ACCELERATE THE CONVERGENCE OF FOURIER SERIES. , 1966 .

[16]  E. Christiansen,et al.  Handbook of Numerical Analysis , 1996 .

[17]  M. Anshelevich,et al.  Introduction to orthogonal polynomials , 2003 .

[18]  Yvon Maday,et al.  Analysis of Some Padé-Chebyshev Approximants , 2005, SIAM J. Numer. Anal..

[19]  Ramón A. Orive Rodríguez,et al.  Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour , 2005, J. Approx. Theory.

[20]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[21]  Jochem Fleischer,et al.  Nonlinear Padé approximants for Legendre series , 1973 .

[22]  C. Brezinski.  Padé-type approximation and general orthogonal polynomials , 1980 .

[23]  Knut S. Eckhoff.  Accurate and efficient reconstruction of discontinuous functions from truncated series expansions , 1993 .

[24]  George Kvernadze.  Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients , 2004, Math. Comput..

[25]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[26]  Tobin A. Driscoll,et al.  A Padé-based algorithm for overcoming the Gibbs phenomenon , 2004, Numerical Algorithms.

[27]  V. N. Sorokin,et al.  Rational Approximations and Orthogonality , 1991 .

[28]  Knut S. Eckhoff.  On a high order numerical method for functions with singularities , 1998, Math. Comput..

[29]  Anne Gelb,et al.  Detection of Edges in Spectral Data , 1999 .

[30]  E A Rahmanov,et al.  Convergence of Diagonal PADÉ Approximants , 1977 .