From cardinal spline wavelet bases to highly coherent dictionaries

Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\em{dictionary}}, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterise the correlation of the dictionary elements by measuring their `coherence' and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation.