Abstract
In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.
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References
Battle, G., A block spin construction of ondelettes. Part I. Lemarié functions, Comm. math. Phys. 110(1987), 601–615.
C. de Boor, DeVore, R. and Ron, A., Approximation from shift—invariant subspaces ofL 2(Rs), Trans. Amer. Math. Soc. 341(1994), 787–806.
Chui, C. K. An introduction to wavelets, Acad. Press, San Diego CA, 1992.
Kamada, M., Toraici, K. and Riochi Mori, Periodic splines orthonormal bases. J. Approx. Th. 55(1988), 27–34.
Koh, Y. W., Lee, S. L., and Tan, H. H., Periodic orthogonal splines and wavelets, Appl. and Comp. Harm. Anal., 2(1995), 201–218.
Lemarié, P. G. Ondelettes à localization exponentielle, J. de Math. Pure et Appl., 67(1988), 227–236.
Neittaanmäki, P., Rivkind V. J. and Zheludev, V. A., Periodic spline wavelets and representation of integral operators, Jyväkylä University, Finland, Dept. of Math., Preprint 177, 1995.
Schoenberg, I. J., Cardinal spline interpolation, CBMS, 12, SIAM, Philadelphia, 1973.
Schoenberg, I. J., Contribution to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., 4(1946), 45–99, 112–141.
Schoenberg, I. J., Cardinal interpolation and spline functions I, J. Approx. Theory, 6(1972), 404–420.
Subbotin, Yu. N., On the relation between finite differences and the corresponding derivatives, Proc. Steklov Inst. Math., 78(1965), 24–42.
Zemanian, A. H., Distribution theory and transform analysis, McGraw-Hill, New York, 1965.
Zheludev, V. A., Spline Harmonic Analysis and Wavelet Bases, In Proc. Symp. Appl. Math., 48(ed. W. Gautcshi), AMS 1994, 415–419.
Zheludev, V. A., Periodic splines and wavelets, in Contemporary Mathematics, 190, Mathematical Analysis, Wavelets & Signal Processing, (Ismail, M. E. H., Nashed, M. Z., Zayed, A. I., Ghaleb, A. F. Eds.), Amer. Math. Soc., Providence, RI, 339–354, 1995.
Zheludev, V. A., Asymptotic formulas for local spline approximation on a uniform mesh. Soviet. Math. Dokl., 27(1983), 415–419.
Zheludev, V. A., Local splines of defect 1 on a uniform mesh. Siberian Journal of Computer Math., 1(1992), 123–156.
Zheludev, V. A., Integral representation of slowly growing equidistant splines and spline wavelets, Tel Aviv University, The School of Mathmeatical Sciences, Technical Report 5-96, 1996.
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Zheludev, V.A. Integral representation of slowly growing equidistant splines. Approx. Theory & its Appl. 14, 66–88 (1998). https://doi.org/10.1007/BF02856150
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DOI: https://doi.org/10.1007/BF02856150