On the representation of operators in bases of compactly supported wavelets. (English) Zbl 0766.65007

Beylkin, G.

SIAM J. Numer. Anal. 29, No. 6, 1716-1740 (1992).

The author finds exact and explicit representations of several basic operators (derivatives, Hilbert transform, convolution operators, shifts, etc.) in orthonormal bases of compactly supported wavelets. The method of computing these representations can be directly applied to multidimensional convolution operators.
As an application in the numerical analysis the author presents an \(O(N\log N)\) algorithm for computing the wavelet coefficients of all \(N\) circulant shifts of a vector of the length \( N = 2^ n \) . Using 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 may be reduced from \(O(N\log N)\) to \( O(\log^ 2 N) \) significant entries.

Reviewer: G.Grozev (Montreal)

MSC:

65D15	Algorithms for approximation of functions
65R10	Numerical methods for integral transforms
44A15	Special integral transforms (Legendre, Hilbert, etc.)
35S30	Fourier integral operators applied to PDEs

Keywords:

derivatives; Hilbert transform; convolution operators; shifts; orthonormal bases; wavelets; fast algorithm; pseudodifferential operator