Summary
Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.
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References
Axelsson O., Lindskog G. (1986): On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math.48, 499–523
Calderón, A.P. (1964): Intermediate spaces and interpolation, the complex method. Studia Math.24, 113–190
Ciarlet, P.G. (1978): The finite element methods for elliptic problems. North-Holland Amsterdam
Coifman, R., Weiss, G. (1971): Analyse Harmonique non commutative sur certains espaces homogènes. Springer Berlin Heidelberg New York
Combes, J.M., Grossmann A. Tchamitchian, Ph. (eds.) (1990): Wavelets, time-frequency methods and phase space, 2nd ed. Springer, Berlin Heidelberg New York
Cortina, E. Gomes, S.M. (1989): A wavelet based numerical method applied to free boundary problems. IPE Technical Report, Sao Jose dos Campos, Brasil
Daubechies, I. (1988): Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.41, 909–996
Daubechies, I., Lagarias, J.: Two-scale difference equations. I. Global regularity of solutions. II. Infinite matrix products, local regularity and fractals. AT&T Bell Laboratories, preprint
Duffin, R.J., Schaeffer, A.C. (1952): A class of nonharmonic Fourier series Trans. Am. Math. Soc.72, 341–366
Glowinski, R., Lawton, W.M., Ravachol, M., Tenenbaum, E. (1990): Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. In: R. Glowinsky, A. Lichnewsky, eds., Computing Methods in Applied Sciences and Engineering. SIAM, Philadelphia, pp. 55–120
Golub, G.H., Van Loan, C.F. (1988): Matrix computations, 2nd ed. Johns Hopkins University Press, Baltimore
Goupillaud, P., Grossmann, A., Morlet, J. (1984/85): Cycle-octave and related transforms in seismic signal analysis. Geoexploration23, 85–102
Grossmann, A., Holschneider, M., Kronland-Martinet, R., Morlet, J. (1987): Detection of abrupt changes in sound signals with the help of wavelet transforms. In: Inverse problems: an interdisciplinary study. Academic Press, London Orlando, pp. 289–306
Grossmann, A., Morlet, J. (1984): Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math.15, 723–736
Haar, A. (1910): Zur Theorie der orthogonalen Funktionensysteme. Math. Ann.69, 331–371
Heil, C. (1990): Wavelets and frames. In: Signal processing, pt. 1, 2nd ed., Springer, Berlin Heidelberg New York, pp. 147–160
Kronland-Martinet, R., Morlet, J., Grossmann, A. (1992): Analysis of sound patterns through wavelet transforms. To appear
Latto, A., Tenenbaum, E. (1990): Les ondelettes á support compact et la solution numérique de l'équation de Burgers. Preprint, AWARE Cambridge
Liénard, J.S.: Speech analysis and reconstruction using short-time, elementary waveforms. LIMSI-CNRS, Orsay, France
Mallat, S. (1989): Multiresolution approximations and wavelet orthonormal bases ofL 2ℝ. Trans. Amer. Math. Soc.315, 69–87
Mallat, S. (1989): Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoust. Speech Signal Process37, 2091–2110
Meyer, Y. (1985): Principe d'incertitude, bases hilbertiennes et algebres d'operateurs. Bourbaki Seminar, no. 662
Meyer, Y. (1990): Ondelettes et opérateurs, I. Ondelettes. Hermann, Paris
Rodet, X. (1984): Time-domain formant-wave-function synthesis. Comput. Music J.3, 9–14
Strang, G. (1989): Wavelets and dilation equations: A brief introduction. SIAM Review31, 614–627
Stromberg, J. (1983): A modified Franklin system and higher-order systems of ℝn as unconditional bases for Hardy spaces. In: Conference on harmonic analysis in honor of Antoni Zygmund. Wadsworth, Belmont, pp. 475–493
Yserentant, H. (1986): On the multi-level splitting of finite element spaces. Numer. Math.49, 379–412
Zienciewicz, O.C., Kelly, D.W., Gago, J., Babuška, I. (1982): Hierarchical finite element approaches, error estimates and adaptive refinement. In: The mathematics of finite elements and applications IV. Academic Press, London, pp. 313–346
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This work was supported by National Science Foundation