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Pointwise error bounds for orthogonal cardinal spline approximation. (English) Zbl 1019.41007
Summary: For orthogonal cardinal spline approximation, closed form expressions of the reproducing kernel and the Peano kernels in terms of exponential splines are proved. Concrete and sharp pointwise error bounds are deduced for low degree splines.
41A15 Spline approximation
65D07 Numerical computation using splines
Full Text: DOI
[1] Battle, G., A block spin construction of ondelettes part I: lemarié functions, Comm. math. phys., 110, 601-615, (1987)
[2] C. de Boor, I. Schoenberg, Cardinal interpolation and spline functions VIII. The Budan-Fourier theorem for splines and applications, in: K. Böhmer, G. Meinardus, W. Schempp (Eds.), Spline Functions, Lecture Notes in Mathematics, vol. 501, Springer, Berlin, 1976 · Zbl 0319.41010
[3] Braß, H.; Förster, K.J., On the application of the Peano kernel representation of linear functionals in numerical analysis, ()
[4] Butzer, P.L.; Stens, R.L., Sampling theory for not necessarily band-limited functions: a historical overview, SIAM rev., 34, 40-53, (1992) · Zbl 0746.94002
[5] Chui, C.K.; Wang, J.Z., On compactly supported spline wavelets and a duality principle, Trans. am. math. soc., 330, 2, 903-915, (1992) · Zbl 0759.41008
[6] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Conf. Ser. Appl. Math., vol. 61, SIAM, 1992
[7] R. DeVore, G.G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften, vol. 303, Springer, Berlin, 1993
[8] S. Ehrich, On localised error bounds for orthogonal approximation from shift invariant spaces, Rend. Circ. Mat. Palermo. Ser. II (52) (1998) 303-319 · Zbl 0919.42023
[9] Ehrich, S., Sard-optimal prefilters for the fast wavelet transform, Numer. alg., 16, 303-319, (1997) · Zbl 0902.65097
[10] Ehrich, S., Error bounds for linear approximations on the real line, Analysis, 20, 51-63, (2000) · Zbl 0980.41032
[11] Günttner, R., Exact bounds for the uniform approximation by cardinal spline interpolants, Numer. math., 68, 263-267, (1994) · Zbl 0812.41007
[12] N. Korneichuk, Exact constants in approximation theory, Encycl. Math. Appl., vol. 38, Cambridge University Press, Cambridge, 1991
[13] Lemarié, P.G., Ondelettes à localisation exponentielles, J. math. pures et appl., 67, 227-236, (1988) · Zbl 0758.42020
[14] Meinardus, G., Über die norm des operators der kardinalen spline interpolation, J. approx. th., 16, 289-298, (1976) · Zbl 0325.41001
[15] Reimer, M., Best constants occuring with the modulus of continuity in the error estimate for spline interpolants of odd degree on equidistant grids, Numer. math., 44, 407-415, (1984) · Zbl 0523.41009
[16] I.J. Schoenberg, Cardinal spline interpolation, CBMS-NSF Conf. Ser. Appl. Math., vol. 12, SIAM, 1973
[17] Schoenberg, I.J., On the remainders and the convergence of cardinal spline interpolation for almost periodic functions, (), 277-303 · Zbl 0338.41007
[18] L. Schumaker, Spline functions, Basic Theory, Wiley-Interscience, New York, 1981 · Zbl 0449.41004
[19] M. Unser, Ten good reasons for using spline wavelets, Wavelet Applications in Signal and Image Processing V, Proc. SPIE, vol. 3169, 1997, pp. 422-431
[20] Unser, M.; Aldroubi, A.; Eden, M., A family of polynomial spline wavelet transforms, Signal process., 30, 141-162, (1993) · Zbl 0768.41012
[21] S. Wolfram, The Mathematica Book, Wolfram Media Inc./Cambridge University Press, Cambridge, 1996 · Zbl 0878.65001
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