On the Representation of Operators in Bases of Compactly Supported Wavelets

Abstract

International audienceThis paper describes exact and explicit representations of the differential operators, $d^n /dx^n$, $n = 1, 2,\ldots$, 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 $N = 2^n$ 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 may be reduced from $O(N)$ to $O(\log^2 N)$ significant entries