On the Representation of Operators in Bases of Compactly Supported Wavelets

This paper describes exact and explicit representations of the differential operators, dn/dxn, n = 1, 2, ⋯, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is direct...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1992-12, Vol.29 (6), p.1716-1740
Main Author: Beylkin, G.
Format: Article
Language:English
Subjects:
Absolute convergence
Algebra
Algorithms
Applied mathematics
Autocorrelation
Coefficients
Exact sciences and technology
Linear algebra
Mathematical vectors
Mathematics
Null condition
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Polynomials
Sciences and techniques of general use
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Summary:This paper describes exact and explicit representations of the differential operators, dn/dxn, n = 1, 2, ⋯, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators. Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring O(N log N) operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length N = 2n is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141-183]) may be reduced from O(N) to O(log2 N) significant entries.
ISSN:0036-1429
1095-7170
DOI:10.1137/0729097