HARMONIC-FUNCTIONS FOR A TRANSITION OPERATOR AND APPLICATIONS

Let u be a continuous non-negative function, defined on [0, 1], such that u(x) + u(x + 1/2) = 1, FOR-ALL x is-a-member-of 0, 1/2] and P(u) be the transition operator defined by P(u)f(x) = u(x/2)f(x/2) + u(x/2 + 1/2)f(x/2 + 1/2), f continuous on [0, 1]. We study the asymptotic behaviour of the iterates (P(u)n, n is-a-member-of N), and give some applications to functional equations, wavelet theory and filtering.