Periodic cardinal interpolatory wavelets (Q1387498)

In this paper the authors construct a multiresolution analysis for periodic functions of period \(2\pi\) with a basis which is interpolatory, in the sense that the elements of the space \(V_j\) can be recovered from the scaling function \(\phi_j\) and the values of the function at \(2\pi m/2^j\), where \(m=0,\ldots,2^{j-1}\). To be more precise, one starts with a periodic function \(g\) with positive Fourier coefficients and defines \(V_j\) to be the span of the functions \(g(\cdot - 2\pi m/2^j)\). A scaling function \(\phi_j\) is then constructed so that \(\phi_j(\cdot - 2\pi m/2^j)\) is a basis for \(V_j\), and moreover one has the reproducing formula \[ f(x) = \sum_{m=0}^{2^j-1} f(2\pi m/2^j) \phi_j(x - 2\pi m/2^j). \] The corresponding wavelets, and dual scaling functions and wavelets, are also constructed. The wavelet basis is symmetric but not orthogonal.