  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 a 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.
