Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
HTML articles powered by AMS MathViewer
- by Knut S. Eckhoff PDF
- Math. Comp. 64 (1995), 671-690 Request permission
Abstract:
Knowledge of a truncated Fourier series expansion for a $2\pi$-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 $2\pi$-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.References
- Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480, DOI 10.1007/978-3-642-84108-8 J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart, LINPACK user’s guide, SIAM, Philadelphia, PA, 1979.
- Knut S. Eckhoff, Accurate and efficient reconstruction of discontinuous functions from truncated series expansions, Math. Comp. 61 (1993), no. 204, 745–763. MR 1195430, DOI 10.1090/S0025-5718-1993-1195430-1 A. Erdélyi, W. Magnus, F. Oberhettinger, and F. C. Tricomi, Higher transcendental functions, McGraw-Hill, New York, 1953.
- Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
- David Gottlieb, Issues in the application of high order schemes, Algorithmic trends in computational fluid dynamics (1991), ICASE/NASA LaRC Ser., Springer, New York, 1993, pp. 195–218. MR 1295637
- David Gottlieb, Liviu Lustman, and Steven A. Orszag, Spectral calculations of one-dimensional inviscid compressible flows, SIAM J. Sci. Statist. Comput. 2 (1981), no. 3, 296–310. MR 632901, DOI 10.1137/0902024
- David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR 0520152
- F. B. Hildebrand, Introduction to numerical analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0347033
- Peter D. Lax, Accuracy and resolution in the computation of solutions of linear and nonlinear equations, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 107–117. MR 519059 J. N. Lyness, The calculation of trigonometric Fourier coefficients, J. Comput. Phys. 54 (1984), 57-73.
- George Majda, Walter A. Strauss, and Mu Sheng Wei, Computation of exponentials in transient data, IEEE Trans. Antennas and Propagation 37 (1989), no. 10, 1284–1290. MR 1018729, DOI 10.1109/8.43537
- M. J. D. Powell, Approximation theory and methods, Cambridge University Press, Cambridge-New York, 1981. MR 604014
- William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical recipes, Cambridge University Press, Cambridge, 1986. The art of scientific computing. MR 833288 R. Prony, Essai experimental et analytic ..., J. L’Ecole Polytech. (Paris), vol. 1, cahier 2, 1795, pp. 24-76.
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- © Copyright 1995 American Mathematical Society
- 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