Compactly supported wavelets

The paper is devoted to one-dimensional compactly supported wavelets which are of the greatest interest for applications because of the simplest numerical realization of expansion and synthesis algorithms. It contains the review of papers (known to the author) about compactly supported wavelets and some new results of the author on the topic. The paper consists of 7 sections. In the second section the problem of existence of scaling function for wavelet bases is considered. Sections 3 and 4 are devoted to a brief account of the multiresolution analysis and the theory of compactly supported wavelets. Section 5 presents results about regularity of compactly supported wavelets in Sobolev and Holder spaces. The final two sections are devoted to localization of wavelets in time and in frequency.