NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS
|
Abstract:
Abstract. The recursive algorithm of a (fast) discrete wavelet transform, as well as its generalizations, can be described as repeated applications of block-Toeplitz operators or, in the case of periodized wavelets, multiplications by block circulant matrices. Singular values of a block circulant matrix are the singular values of some matrix trigonometric series evaluated at certain points. The norm of a block-Toeplitz operator is then the essential supremum of the largest singular value curve of this series. For all reasonable wavelets, the condition number of a block-Toeplitz operator thus is the lowest upper bound for the condition of corresponding block circulant matrices of all possible sizes. In the last section, these results are used to study conditioning of biorthogonal wavelets based on B-splines.
