On the convergence rate of a modified Fourier series

The rate of convergence for an orthogonal series that is a minor modification of the Fourier series is proved. This series converges pointwise at a faster rate than the Fourier series for nonperiodic functions. We present the error as an asymptotic expansion, where the lowest term in this expansion is of asymptotic order two. Subtracting out the terms from this expansion allows us to increase the order of convergence, though the terms of this expansion depend on derivatives. Alternatively, we can employ extrapolation methods which achieve higher convergence rates using only the coefficients of the series. We also present a method for the efficient computation of the coefficients in the series.

[1]  Daan Huybrechs,et al.  On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation , 2006, SIAM J. Numer. Anal..

[2]  David Levin,et al.  Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations , 1982 .

[3]  F. Olver.  Asymptotics and Special Functions , 1974 .

[4]  L. Filon.  III.—On a Quadrature Formula for Trigonometric Integrals. , 1930 .

[5]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[6]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[7]  John P. Boyd,et al.  Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations , 2005 .

[8]  Daisuke Fujiwara,et al.  On a special class of pseudo-differential operators , 1967 .

[9]  Sheehan Olver,et al.  Moment-free numerical integration of highly oscillatory functions , 2006 .

[10]  Arieh Iserles,et al.  From high oscillation to rapid approximation II: Expansions in polyharmonic eigenfunctions , 2006 .

[11]  A. G. Greenhill,et al.  Handbook of Mathematical Functions with Formulas, Graphs, , 1971 .

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

[13]  Peter J. Mohr,et al.  Application of the Combined Nonlinear-Condensation Transformation to Problems in Statistical Analysis and Theoretical Physics , 2003 .

[14]  M. Powell,et al.  Approximation theory and methods , 1984 .

[15]  Hari M. Srivastava,et al.  Series Associated with the Zeta and Related Functions , 2001 .

[16]  A. Iserles,et al.  Efficient quadrature of highly oscillatory integrals using derivatives , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  Arieh Iserles,et al.  From high oscillation to rapid approximation I: Modified Fourier expansions , 2008 .