Subdivision schemes on the dyadic half-line

Abstract

We consider subdivision schemes, which are used for the approximation of functions and generation of curves on the dyadic half-line. In the classical case of functions on the real line, the theory of subdivision schemes is widely known because of its applications in constructive approximation theory and signal processing as well as for generating fractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-line endowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role of exponentials is played by Walsh functions. We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectral properties of matrices and in terms of the smoothness of solutions of the corresponding refinement equation. We also investigate the problem of convergence of subdivision schemes with non-negative coefficients. We obtain an explicit criterion for the convergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-line and prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.