The spaces of dyadic distributions

TL;DR

This paper studies subdivision schemes on a dyadic half-line, establishing convergence conditions, analyzing smoothness, and exploring fractal curves, with implications for approximation theory and signal processing.

Contribution

It introduces and analyzes subdivision schemes on a dyadic half-line, providing convergence criteria and smoothness results, extending classical theory to a new setting.

Findings

Convergence conditions are characterized by spectral properties of matrices.

Explicit convergence criterion for schemes with four coefficients is derived.

Fractal curves on a dyadic half-line are constructed and their smoothness analyzed.

Abstract

In this paper subdivision schemes, which are used for functions approximation and curves generation, are considered. In classical case, for the functions defined on the real line, the theory of subdivision schemes is widely known due to multiple applications in constructive approximation theory, signal processing as well as for generating fractal curves and surfaces. Subdivision schemes on a dyadic half-line, which is the positive half-line, equipped with the standard Lebesgue measure and the digitwise binary addition operation, where the Walsh functions play the role of exponents, are defined and studied. Necessary and sufficient convergence conditions of the subdivision schemes in terms of spectral properties of matrices and in terms of the smoothness of the solution of the corresponding refinement equation are proved. The problem of the convergence of subdivision schemes with non-negative coefficients is also investigated. Explicit convergence criterion of the subdivision schemes with four coefficients is obtained. As an auxiliary result fractal curves on a dyadic half-line are defined and the formula of their smoothness is proved. The paper contains various illustrations and numerical results.