On the fourier extension of nonperiodic functions

SIAM Journal on Numerical Analysis

Volumn 47, Issue 6, 2010, Pages 4326-4355

  Huybrechs, Daan ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Fourier series; Frames; Least squares; Orthogonal polynomials

Indexed keywords

CHEBYSHEV POLYNOMIALS; COMPLETE CONVERGENCE; FOURIER EXTENSION; LEAST SQUARE; LEAST SQUARES; ORTHOGONAL POLYNOMIAL; PERIODIC FUNCTION;

FOURIER SERIES; HARMONIC ANALYSIS; NUMBER THEORY; NUMERICAL METHODS; POLYNOMIALS;

FOURIER ANALYSIS;

EID: 77649098509     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/090752456     Document Type: Article

References (32)

1. M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1965.
2. S. N. BERNSTEIN, Collected Works, Vol. 2, Akademii nauk SSSR, Moscow, 1954.
3. J. P. BOYD, Chebyshev and Fourier Spectral Methods, Courier Dover Publications, Mineola, NY, 2001.
4. J. P. BOYD, A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds, J. Comput. Phys., 178 (2002), pp. 118-160.
5. J. P. BOYD, Fourier embedded domain methods: Extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and C∞, Appl. Math. Comput., 161 (2005), pp. 591-597.
6. J. P. BOYD, Asymptotic Fourier coefficients for a C∞ bell (smoothed-top-hat) and the Fourier extension problem, J. Sci. Comput., 29 (2006), pp. 1-24.
7. O. P. BRUNO, Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics, in Topics in Computational Wave Propagation: Direct and Inverse Problems, M. Ainsworth, P. Davies, D. Duncan, P. Martin, and B. Rynne, eds., Lecture Notes Comput. Sci. Engrg. 31, Springer, New York, 2003, pp. 43-82.
8. O. P. BRUNO, Y. HAN, AND M. M. POHLMAN, Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis, J. Comput. Phys., 227 (2007), pp. 1094-1125.
9. I. DAUBECHIES, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Ser. Appl. Math. 61, SIAM, Philadelphia, 1992.
10. P. DEIFT AND X. ZOU, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Math., 137 (1993), pp. 295-368.
11. J. R. DRISCOLL AND D. HEALY, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15 (1994), pp. 202-250.
12. R. J. DUFFIN AND A. C. SCHAEFFER, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
13. M. ELGHAOUI AND R. PASQUETTI, A spectral embedding method applied to the advectiondiffusion equation, J. Comput. Phys., 125 (1996), pp. 464-476.
14. C. FEFFERMAN, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoamericana, 21 (2005), pp. 313-348.
15. W. GAUTSCHI, Orthogonal Polynomials: Computation and Approximation, Clarendon Press, Oxford, UK, 2004.
16. G. H. GOLUB AND J. H. WELSCH, Calculation of Gauss quadrature rules, Math. Comput., 23 (1969), pp. 221-230.
17. D. GOTTLIEB AND C.-W. SHU, On the Gibbs phenomenon and its resolution, SIAM Rev., 39 (1997), pp. 644-668.
18. M. R. HESTENES, Extension of the range of a differentiable function, Duke Math. J., 8 (1941), pp. 183-192.
19. D. HUYBRECHS, A. ISERLES, AND S. P. NøRSETT, From high oscillation to rapid approximation IV: Accelerating convergence, IMA J. Numer. Anal., (2009), to appear.
20. D. HUYBRECHS AND S. VANDEWALLE, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44 (2006), pp. 1026-1048.
21. A. ISERLES AND S. P. NøRSETT, From High Oscillation to Rapid Approximation II: Expansions in Polyharmonic Eigenfunctions, Technical Report 2006/NA07, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK, 2006.
22. A. ISERLES AND S. P. NøRSETT, From high oscillation to rapid approximation I: Modified Fourier expansions, IMA J. Numer. Anal., 28 (2008), pp. 862-887.
23. A. ISERLES AND S. P. NøRSETT, From high oscillation to rapid approximation III: Multivariate expansions, IMA J. Numer. Anal., 29 (2009), pp. 882-916.
24. A. B. J. KUIJLAARS, K. T. R. MCLAUGHLIN, W. VAN ASSCHE, AND M. VANLESSEN, The Riemann-Hilbert approach to strong asymptotics of orthogonal polynomials on [-1, 1], Adv. Math., 188 (2004), pp. 337-398.
25. C. L. LAWSON AND R. J. HANSON, Solving Least Squares Problems, Classics in Appl. Math. 15, SIAM, Philadelphia, 1995.
26. J. C. MASON AND D. C. HANDSCOMB, Chebyshev Polynomials, Chapman and Hall/CRC, London/Boca Raton, FL, 2003.
27. D. POTTS, G. STEIDLS, AND M. TASCHE, Fast algorithms for discrete polynomial transforms, Math. Comput., 67 (1998), pp. 1577-1590.
28. E. TADMOR, Filters, mollifiers and the computation of the Gibbs phenomenon, Acta Numer., 16 (2007), pp. 305-378.
29. L. N. TREFETHEN, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 50 (2008), pp. 67-87.
30. J. WALDVOGEL, Fast construction of the Fejér and Clenshaw-Curtis quadrature rules, BIT, 46 (2006), pp. 195-202.
31. J. A. C. WEIDEMAN AND L. N. TREFETHEN, The kink phenomenon in Fejér and Clenshaw-Curtis quadrature, Numer. Math., 107 (2007), pp. 707-727.
32. H. WHITNEY, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), pp. 63-89.