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Trigonometric wavelets for Hermite interpolation

Author(s): Ewald Quak.
Journal: Math. Comp. 65 (1996), 683-722.
MSC (1991): Primary 42A15, 41A05, 65D05
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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:

[1]
P. Auscher, Wavelets with boundary conditions on the interval, Wavelets: a Tutorial in Theory and Applications (C.K. Chui, ed.), Academic Press, Boston, 1992, pp. (237--256). MR 93c:42029

[2]
A. S. Cavaretta, A. Sharma and R. S. Varga, Lacunary trigonometric interpolation on equidistant nodes, Quantitative Approximation (R. A. DeVore and K. Scherer, eds.), Academic Press, New York, 1980, pp. (63--80). MR 81k:42004

[3]
C. K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992. MR 93f:42055

[4]
C. K. Chui and H. N. Mhaskar, On trigonometric wavelets, Constr. Approx. 9 (1993), 167--190. MR 94c:42002

[5]
C. K. Chui and J. Z. Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), 903--915. MR 92f:41020

[6]
P. J. Davis, Circulant matrices, Wiley Interscience, New York, 1979. MR 81a:15003

[7]
T. N. T. Goodman, Interpolatory Hermite spline wavelets, J. Approx. Theory 78 (1994), 174--189. CMP 94:15

[8]
Y. W. Koh, S. L. Lee and H. H. Tan, Periodic orthogonal splines and wavelets, Applied and Computational Harmonic Analysis 2 (1995), 201--218. CMP 95:15

[9]
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. CMP 95:05

[10]
D. Offin and K. Oskolkov, A note on orthonormal polynomial bases and wavelets, Constr. Approx. 9 (1993), 319--325. MR 94f:42047

[11]
J. Prestin and E. Quak, Trigonometric interpolation and wavelet decompositions, Numerical Algorithms 9 (1995), 293--317. CMP 95:15

[12]
------, 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). CMP 95:03

[13]
------, 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).

[14]
A. A. Privalov, On an orthogonal trigonometric basis, Mat. Sbornik 182 (3) (1991), 384--394. MR 92f:42005

[15]
A. F. Timan, Theory of approximation of functions of a real variable, Pergamon Press, Oxford, 1963. MR 33:465

[16]
Y. Xu, The generalized Marcinkiewicz-Zygmund inequality for trigonometric polynomials, J. Math. Anal. Appl. 161 (1991), 447--456. MR 93f:42008

[17]
A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1959. MR 21:6498

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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

DOI: 10.1090/S0025-5718-96-00719-3
PII: S 0025-5718(96)00719-3
Keywords: Trigonometric wavelets, Hermite interpolation, decomposition and reconstruction sequences and algorithms
Received by editor(s): May 4, 1994
Received by editor(s) in revised form: October 23, 1994
Copyright of article: Copyright 1996, American Mathematical Society

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