Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions

Author:
Knut S. Eckhoff

Journal:
Math. Comp. **64** (1995), 671-690

MSC:
Primary 65T20; Secondary 65D15

DOI:
https://doi.org/10.1090/S0025-5718-1995-1265014-7

MathSciNet review:
1265014

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Abstract: Knowledge of a truncated Fourier series expansion for a -periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree *M* is constructed for the *M* singularity locations in each period for the function in question. The *M* coefficients in this algebraic equation are obtained by solving an algebraic system of *M* equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the *M* unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the -periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiable up to the specified order.

**[1]**C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,*Spectral methods in fluid dynamics*, Springer-Verlag, New York, 1988. MR**917480 (89m:76004)****[2]**J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart,*LINPACK user's guide*, SIAM, Philadelphia, PA, 1979.**[3]**K. S. Eckhoff,*Accurate and efficient reconstruction of discontinuous functions from truncated series expansions*, Math. Comp.**61**(1993), 745-763. MR**1195430 (94a:65073)****[4]**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. C. Tricomi,*Higher transcendental functions*, McGraw-Hill, New York, 1953.**[5]**G. H. Golub and C. F. Van Loan,*Matrix computations*, 2nd ed., Johns Hopkins Univ. Press, Baltimore, MD, 1989. MR**1002570 (90d:65055)****[6]**D. Gottlieb,*Issues in the application of high order schemes*, Proc. Workshop on Algorithmic Trends in Computational Fluid Dynamics (Hampton, Virginia, U.S.A.), (M. Y. Hussaini, A. Kumar, and M. D. Salas, eds.), Springer-Verlag, ICASE/NASA LaRC Series, 1991, pp. 195-218. MR**1295637 (95f:76072)****[7]**D. Gottlieb, L. Lustman, and S. A. Orszag,*Spectral calculations of one-dimensional inviscid compressible flows*, SIAM J. Sci. Statist. Comput.**2**(1981), 296-310. MR**632901 (82m:76002)****[8]**D. Gottlieb and S. A. Orszag,*Numerical analysis of spectral methods*:*Theory and applications*, SIAM, Philadelphia, PA, 1977. MR**0520152 (58:24983)****[9]**F. B. Hildebrand,*Introduction to numerical analysis*, 2nd ed., Tata McGraw-Hill, New Delhi, 1974. MR**0347033 (49:11753)****[10]**P. D. Lax,*Accuracy and resolution in the computation of solutions of linear and nonlinear equations*, Recent Advances in Numerical Analysis, Proc. Sympos. Univ. of Wisconsin-Madison (C. de Boor and G. H. Golub, eds.), Academic Press, New York, 1978, pp. 107-117. MR**519059 (80b:65147)****[11]**J. N. Lyness,*The calculation of trigonometric Fourier coefficients*, J. Comput. Phys.**54**(1984), 57-73.**[12]**G. Majda, W. A. Strauss, and M. Wei,*Computation of exponentials in transient data*, IEEE Trans. Antennas and Propagation**37**(1989), pp. 1284-1290. MR**1018729 (90h:78003)****[13]**M. J. D. Powell,*Approximation theory and methods*, Cambridge Univ. Press, Cambridge, 1981. MR**604014 (82f:41001)****[14]**W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,*NUMERICAL RECIPES, the art of scientific computing*, Cambridge Univ. Press, Cambridge, 1989. MR**833288 (87m:65001a)****[15]**R. Prony,*Essai experimental et analytic ...*, J. L'Ecole Polytech. (Paris), vol. 1, cahier 2, 1795, pp. 24-76.**[16]**A. Zygmund,*Trigonometric series*, Vol. I, Cambridge Univ. Press, Cambridge, 1968. MR**0236587 (38:4882)**

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DOI:
https://doi.org/10.1090/S0025-5718-1995-1265014-7

Article copyright:
© Copyright 1995
American Mathematical Society