Summary of Sinc numerical methods. (English) Zbl 0964.65010

This article attempts to summarize the existing numerical methods based on Sinc approximation. Starting with a comparison of polynomial and Sinc approximation, basic formulas for the latter in the one-dimensional case are given. The author also covers the following:
(i) Explicit spaces of analytic functions for one dimensional Sinc approximation,
(ii) applications of Sinc indefinite integration and collocation to the solution of ordinary differential equation initial and boundary value problems,
(iii) results obtained for solution of partial differential equations, via Sinc approximation of the derivatives,
(iv) some results obtained on the solutions of integral equations,
(v) use of Sinc convolution, a technique for evaluating one and multi-dimensional convolution-type integrals.
A list of some existing computer algorithms based on Sinc methods is also given.


65D15 Algorithms for approximation of functions
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
65T40 Numerical methods for trigonometric approximation and interpolation
Full Text: DOI


[2] Andersson, J.-E.; Bojanov, B. D., A note on the optimal quadrature in
((H^p\), Numer. Math., 44, 301-308 (1984) · Zbl 0546.41026
[3] Ang, D. D.; Lund, J. R.; Stenger, F., Complex variable and regularization methods of inversion of the Laplace transform, Math. Comp., 51, 589-608 (1989) · Zbl 0676.65136
[4] Anselone, P., Collectively Compact Operator Approximation Theory and Applications to Integral Equations (1971), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0228.47001
[5] Bialecki, B.; Stenger, F., Sinc-Nyström method for numerical solution of one-dimensional cauchy singular integral equations given on a smooth arc in the complex plane, Math. Comp., 51, 133-165 (1988) · Zbl 0662.65120
[6] Bojanov, B., Best Quadrature Formula for a Certain Class of Analytic Functions, Zastosowania Matematyki Appl. Mat., XIV, 441-447 (1974) · Zbl 0299.65012
[7] Borel, E., Sur l’interpolation, C.R. Acad. Sci. Paris, 124, 673-676 (1897) · JFM 28.0225.04
[8] Burchard, H. G.; Höllig, K., \(N\)-width and entropy of \(H^p classes in L^q(−1,1)\), SIAM J. Math. Anal., 16, 405-421 (1985) · Zbl 0554.41030
[10] Davis, P. J., Errors of numerical approximation for analytic functions, J. Rational Mech. Anal., 2, 303-313 (1953) · Zbl 0050.13005
[13] Ikebe, Y.; Li, T. Y.; Stenger, F., Numerical solution of Hilbert’s problem, (Law, A. G.; Sahney, B., Theory of Approximation with Applications (1976), Academic Press: Academic Press NewYork), 338-358 · Zbl 0356.65018
[14] Jarratt, M.; Lund, J.; Bowers, K. L., Galerkin schemes and the Sinc-Galerkin method for singular Sturm-Liouville problems, J. Comput. Phys., 89, 41-62 (1990) · Zbl 0702.65078
[15] Jerri, A., The Shannon sampling theorem – its various extensions and applications: a tutorial review, Proc. IEEE, 65, 1565-1596 (1977) · Zbl 0442.94002
[16] Johnson, S. A.; Zhou, Y.; Tracy, M. L.; Berggren, M. J.; Stenger, F., Inverse scattering solutions by Sinc basis, multiple source moment method, - Part III: fast algorithms, Ultrasonic Imaging, 6, 103-116 (1984)
[17] Keinert, F., Uniform approximation to |\(x|^β\) by Sinc functions, J. Approx. Theory, 66, 44-52 (1991) · Zbl 0738.41023
[19] Kowalski, M.; Sikorski, K.; Stenger, F., Selected Topics in Approximation and Computation (1993), Oxford University Press: Oxford University Press Oxford
[21] Lippke, A., Analytic solution and Sinc function approximation in thermal conduction with nonlinear heat generation, J. Heat Transfer (Trans. ASME), 113, 5-11 (1991)
[22] Lund, J., Symmetrization of the Sinc-Galerkin method for boundary value problems, Math. Comput., 47, 571-588 (1986) · Zbl 0629.65085
[23] Lund, J.; Bowers, K. L., Sinc Methods for Quadrature and Differential Equations (1992), SIAM: SIAM Philadelphia, PA · Zbl 0753.65081
[24] Lundin, L., A Cardinal Function Method of Solution of the Equation Δ \(u=u\) − \(u^3\), Math. Comput., 35, 747-756 (1980) · Zbl 0445.65086
[25] Lundin, L.; Stenger, F., Cardinal-type approximation of a function and its derivatives, SIAM J. Numer. Anal., 10, 139-160 (1979) · Zbl 0399.41018
[26] McArthur, K.; Bowers, K.; Lund, J., Numerical implementation of the Sinc-Galerkin method for second-order hyperbolic equations, Numer. Methods Partial Differential Equations, 3, 169-185 (1987) · Zbl 0698.65069
[27] McNamee, J.; Stenger, F.; Whitney, E. L., Whittaker’s Cardinal function in retrospect, Math. Comp., 25, 141-154 (1971) · Zbl 0216.48502
[28] Moran, P. A.P., Numerical integration in the presence of singularities, Acta Math. Sci., 1, 83-85 (1981) · Zbl 0519.65016
[34] O’Reilly, M.; Stenger, F., Computing solutions to medical problems, IEEE Trans. Automat. Control, 43, 843-846 (1998)
[37] Plana, G., Sur une nouvelle expression analytique des nombres Bernoulliens, Academia di Torino, 25, 403-418 (1820)
[38] Rahman, Q. I.; Schmeisser, G., The summation formulæ of Poisson, Plana, Euler-Maclaurin and their relationship, J. Math. Sci., 28, 151-171 (1994) · Zbl 1019.65500
[41] Schwing, J.; Sikorski, K.; Stenger, F., ALGORITHM 614, A FORTRAN subroutine for numerical integration in
((H^p\), ACM TOMS, 10, 152-160 (1984)
[45] Stenger, F., An analytic function which is an approximate characteristic function, SIAM J. Numer. Anal., 12, 239-254 (1975) · Zbl 0274.65010
[46] Stenger, F., The approximate solution of convolution-type integral equations, SIAM J. Math. Anal., 4, 536-555 (1973) · Zbl 0258.45005
[47] Stenger, F., Integration formulas based on the trapezoidal formula, J. Inst. Math. Appl., 12, 103-114 (1973) · Zbl 0262.65011
[48] Stenger, F., Approximations via the Whittaker Cardinal Function, J. Approx. Theory, 17, 222-240 (1976) · Zbl 0332.41013
[49] Stenger, F., A Sinc-Galerkin method of solution of boundary value problems, Math. Comp., 33, 85-109 (1979) · Zbl 0402.65053
[50] Stenger, F., Numerical methods based on the Whittaker Cardinal, or Sinc functions, SIAM Rev., 23, 165-224 (1981) · Zbl 0461.65007
[52] Stenger, F., Numerical Methods Based on Sinc and Analytic Functions (1993), Springer: Springer NewYork · Zbl 0803.65141
[53] Stenger, F., Collocating convolutions, Math. Comp., 64, 211-235 (1995) · Zbl 0828.65017
[54] Stenger, F.; Barkey, B.; Vakili, R., Sinc convolution method of solution of burgers’ equation, (Bowers, K.; Lund, J., Proceedings of Computation and Control III (1993), Birkhäuser: Birkhäuser Basel) · Zbl 0822.65073
[60] Takahasi, H.; Mori, M., Error estimation in the numerical integration of analytic functions, Report Comput. Centre, Univ. Tokyo, 3, 41-108 (1970)
[61] Takahasi, H.; Mori, M., Quadrature formulas obtained by variable transformation, Numer. Math., 21, 206-219 (1973) · Zbl 0267.65016
[62] Takahasi, H.; Mori, M., Double exponential formulas for numerical integration, Publ. RIMS Kyoto Univ., 9, 721-741 (1974) · Zbl 0293.65011
[63] Whittaker, E. T., On the functions which are represented by expansion of the interpolation theory, Proc. Roy. Soc. Edinburgh, 35, 181-194 (1915) · JFM 45.1275.02
[64] Wilderotter, K., \(n\)-Widths of \(H^p-spaces in Lq(−1,1)\), J. Complexity, 8, 324-335 (1992) · Zbl 0803.46056
[65] Yee, K., Numerical solution of boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas and Propagation, AP-16, 302-307 (1966) · Zbl 1155.78304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.