Discrete orthogonal transforms with bases generated by self-similar sequences

New bases of discrete orthogonal transforms associated with some recursive processes and possessing a property of self-similarity are introduced and investigated in the paper. Sufficient conditions of orthogonality of a system of basic functions are proved. For transforms with the introduced bases, fast algorithms of the transforms are synthesized. The relationship between the considered bases and the analytic properties of generating Dirichlet series is discussed.