Near optimality of the sinc approximation

Sugihara, Masaaki

doi:10.1090/S0025-5718-02-01451-5

Near optimality of the sinc approximation
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Math. Comp. 72 (2003), 767-786 Request permission

Abstract:

Near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Jan-Erik Andersson, Optimal quadrature of $H^{p}$ functions, Math. Z. 172 (1980), no. 1, 55–62. MR 576296, DOI 10.1007/BF01182779
S. N. Bernstein, Sur la meilleure approximation de $|x|^{p}$ par des polynômes de degrées très élevés, Bull. Acad. Sci. USSR, Cl. Sci. Math. Nat. 2 (1938), 181–190.
H. G. Burchard and K. Höllig, $N$-width and entropy of $H_p$-classes in $L_q(-1,1)$, SIAM J. Math. Anal. 16 (1985), no. 2, 405–421. MR 777476, DOI 10.1137/0516029
R. DeVore and K. Scherer, Variable knot, variable degree spline approximation to $x^{\beta }$, Quantitative approximation (Proc. Internat. Sympos., Bonn, 1979) Academic Press, New York-London, 1980, pp. 121–131. MR 588175
Tord Ganelius, Rational approximation in the complex plane and on the line, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 129–145. MR 0499176, DOI 10.5186/aasfm.1976.0211
K. L. Bowers and J. Lund (eds.), Computation and control. II, Progress in Systems and Control Theory, vol. 11, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1140009, DOI 10.1007/978-1-4612-0427-5
Fritz Keinert, Uniform approximation to $|x|^\beta$ by sinc functions, J. Approx. Theory 66 (1991), no. 1, 44–52. MR 1113577, DOI 10.1016/0021-9045(91)90054-E
Marek A. Kowalski, Krzysztof A. Sikorski, and Frank Stenger, Selected topics in approximation and computation, Oxford University Press, New York, 1995. MR 1418861, DOI 10.1093/oso/9780195080599.001.0001
John Lund and Kenneth L. Bowers, Sinc methods for quadrature and differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1171217, DOI 10.1137/1.9781611971637
Masatake Mori and Masaaki Sugihara, The double-exponential transformation in numerical analysis, J. Comput. Appl. Math. 127 (2001), no. 1-2, 287–296. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808579, DOI 10.1016/S0377-0427(00)00501-X
Donald J. Newman, Quadrature formulae for $H^{p}$ functions, Math. Z. 166 (1979), no. 2, 111–115. MR 525614, DOI 10.1007/BF01214036
John R. Rice, On the degree of convergence of nonlinear spline approximation, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 349–365. MR 0267324
Frank Stenger, Optimal convergence of minimum norm approximations in $H_{p}$, Numer. Math. 29 (1977/78), no. 4, 345–362. MR 483329, DOI 10.1007/BF01432874
Frank Stenger, Numerical methods based on Whittaker cardinal, or sinc functions, SIAM Rev. 23 (1981), no. 2, 165–224. MR 618638, DOI 10.1137/1023037
Frank Stenger, Explicit, nearly optimal, linear rational approximation with preassigned poles, Math. Comp. 47 (1986), no. 175, 225–252. MR 842132, DOI 10.1090/S0025-5718-1986-0842132-0
Frank Stenger, Numerical methods based on sinc and analytic functions, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR 1226236, DOI 10.1007/978-1-4612-2706-9
Frank Stenger, Summary of Sinc numerical methods, J. Comput. Appl. Math. 121 (2000), no. 1-2, 379–420. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780056, DOI 10.1016/S0377-0427(00)00348-4
Masaaki Sugihara, Optimality of the double exponential formula—functional analysis approach, Numer. Math. 75 (1997), no. 3, 379–395. MR 1427714, DOI 10.1007/s002110050244
Hidetosi Takahasi and Masatake Mori, Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci. 9 (1973/74), 721–741. MR 0347061, DOI 10.2977/prims/1195192451
Guang Gui Ding, On almost isometries from $L^1(\mu )$ into $L^\infty (\nu )$ or $C_b(\Delta )$, Acta Math. Sci. (English Ed.) 12 (1992), no. 3, 308–311. MR 1207442, DOI 10.1016/S0252-9602(18)30164-4