Trigonometric wavelets for Hermite interpolation
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Abstract:
The aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials. The pair of scaling functions which span the sample spaces are fundamental functions for Hermite interpolation on a dyadic partition of nodes on the interval $[0,2\pi )$. Two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions. Together with the corresponding dual functions, these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences. Consequently, this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment.References
- Pascal Auscher, Wavelets with boundary conditions on the interval, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 217–236. MR 1161253
- A. S. Cavaretta Jr., A. Sharma, and R. S. Varga, Lacunary trigonometric interpolation on equidistant nodes, Quantitative approximation (Proc. Internat. Sympos., Bonn, 1979) Academic Press, New York-London, 1980, pp. 63–80. MR 588171
- Charles K. Chui, An introduction to wavelets, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR 1150048
- C. K. Chui and H. N. Mhaskar, On trigonometric wavelets, Constr. Approx. 9 (1993), no. 2-3, 167–190. MR 1215768, DOI 10.1007/BF01198002
- Charles K. Chui and Jian-zhong Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), no. 2, 903–915. MR 1076613, DOI 10.1090/S0002-9947-1992-1076613-3
- Philip J. Davis, Circulant matrices, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 543191
- T. N. T. Goodman, Interpolatory Hermite spline wavelets, J. Approx. Theory 78 (1994), 174–189.
- Y. W. Koh, S. L. Lee and H. H. Tan, Periodic orthogonal splines and wavelets, Applied and Computational Harmonic Analysis 2 (1995), 201–218.
- R. A. Lorentz and A. A. Sahakian, Orthogonal trigonometric Schauder bases of optimal degree for $C(K)$, Journal of Fourier Analysis and Applications 1 (1994), 103–112.
- D. Offin and K. Oskolkov, A note on orthonormal polynomial bases and wavelets, Constr. Approx. 9 (1993), no. 2-3, 319–325. MR 1215775, DOI 10.1007/BF01198009
- J. Prestin and E. Quak, Trigonometric interpolation and wavelet decompositions, Numerical Algorithms 9 (1995), 293–317.
- —, A duality principle for trigonometric wavelets, Wavelets, Images, and Surface Fitting (P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds.), A K Peters, Boston, 1994, pp. (407–418).
- —, Decay properties of trigonometric wavelets, Proceedings of the Cornelius Lanczos International Centenary Conference (J. D. Brown, M. T. Chu, D. C. Ellison and R. J. Plemmons, eds.), SIAM, Philadelphia, 1994, pp. (413–415).
- A. A. Privalov, An orthogonal trigonometric basis, Mat. Sb. 182 (1991), no. 3, 384–394 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 363–372. MR 1110072, DOI 10.1070/SM1992v072n02ABEH002143
- A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
- Yuan Xu, The generalized Marcinkiewicz-Zygmund inequality for trigonometric polynomials, J. Math. Anal. Appl. 161 (1991), no. 2, 447–456. MR 1132120, DOI 10.1016/0022-247X(91)90344-Y
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- Ewald Quak
- Affiliation: Center for Approximation Theory, Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: quak@math.tamu.edu
- Received by editor(s): May 4, 1994
- Received by editor(s) in revised form: October 23, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 683-722
- MSC (1991): Primary 42A15, 41A05, 65D05
- DOI: https://doi.org/10.1090/S0025-5718-96-00719-3
- MathSciNet review: 1333324