Wavelets are new families of basis functions that yield the representation f (x) = b₉₊ W (2ʲ x - k). Their construction begins with the solution (x) to a dilation equation with coefficients cₖ. Then W comes from, and the basis comes by translation and dilation of W. It is shown in Part 1 how conditions on the cₖ lead to approximation properties and orthogonality properties of the wavelets. Part 2 describes the recursive algorithms (also based on the cₖ) that decompose and reconstruct f. The object of wavelets is to localize as far as possible in both time and frequency, with efficient algorithms
Gilbert Strang (1989) studied this question.