Subdivision schemes on the dyadic half-line

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 becauseof its applications in constructive approximation theory and signal processing as well as for generatingfractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-lineendowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role ofexponentials is played by Walsh functions.We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectralproperties 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 theconvergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-lineand prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.

subdivision schemes, dyadic half-line, fractal curves, smoothness of fractal curves, spectral properties of matrices