Sbornik : Mathematics 197:10 1529–1558 c 2006 RAS(DoM) and LMS Matematicheski˘ ı Sbornik 197:10 129–160 DOI 10.1070/SM2006v197n10ABEH003811 Dyadic wavelets and refinable functions on a half-line V.Yu. Protasov and Yu.A. Farkov Abstract. For an arbitrary positive integer n refinable functions on the pos- itive half-line R+ are defined, with masks that are Walsh polynomials of order 2 n - 1. The Strang-Fix conditions, the partition of unity property, the linear independence, the stability, and the orthonormality of integer translates of a solution of the corresponding refinement equations are studied. Necessary and sufficient conditions ensuring that these solutions generate multiresolu- tion analysis in L 2 (R+) are deduced. This characterizes all systems of dyadic compactly supported wavelets on R+ and gives one an algorithm for the con- struction of such systems. A method for finding estimates for the exponents of regularity of refinable functions is presented, which leads to sharp esti- mates in the case of small n. In particular, all the dyadic entire compactly supported refinable functions on R+ are characterized. It is shown that a refinable function is either dyadic entire or has a finite exponent of regularity, which, moreover, has effective upper estimates. Bibliography: 13 items. Introduction Throughout this paper we use the following notation: R + = [0, +∞) is the posi- tive half-line, {w j } is the Walsh system on R + , ⊕ and  are the dyadic operations in R + ,  f is the Walsh-Fourier transform of a function f (see § 1 and also [1], and [2]). As usual, we denote by N and Z + the sets of positive and of non-negative integers, respectively. Basic facts about orthogonal wavelets and refinable functions on the real line R can be found in [3]. In this paper, for an arbitrary positive integer n we study solutions ϕ of the refinement equation ϕ(x)= 2 n -1  k=0 c k ϕ(2x  k), x ∈ R + , (0.1) generating multiresolution analyses in L 2 (R + ). The coefficients c k of equation (0.1) are arbitrary complex numbers. We focus on compactly supported non-trivial solu- tions ϕ ∈ L 2 (R + ) of this equation. If such a solution ϕ exists, then it is unique up to multiplication by a constant and, moreover,  ϕ(0) = 0. In § 2 we show that The research of the first author was carried out with the support of the Russian Foundation for Basic Research (grant no. 05-01-00066) and the Programme of Support of Leading Scientific Schools of RF (grant no. NSh 304.2003.1). The research of the second author was carried out with the support of the Russian Foundation for Basic Research (grant no. 02-01-00386). AMS 2000 Mathematics Subject Classification. Primary 42C40; Secondary 43A70.