Spectral number of 3-Bernoulli convolutions on and Sierpinski-type measures on

Abstract

Let be a class of Sierpinski-type measures generated by a pair , where with and and . And let µ b be the 3-Bernoulli convolutions on determined by the pair with . It has been shown that and µ b admit an infinitely many exponential mutually orthogonal system if and only if with , and with respectively. In this paper, we will study the maximal number of exponentials of orthogonal sets of and which we call the spectral number of or µ b . In view of the connection of orthogonality between Sierpinski-type measures on and 3-Bernoulli convolutions µ b on , we study the spectral number of µ b according to the cut-off point . Based on the results for µ b , we give a classification on the spectral number of all Sierpinski-type measures except for the case that at least one and it is not in the form of with . In addition, we provide a structure theorem on the exponential orthogonal sets in for , and at least one and that in for and . To the end, we give an explicit representation on the maximal orthogonal set of exponentials for a class of Moran measures µ w by defining a mixed tree map over a symbol space. As an application, all maximal orthogonal sets of exponentials of with the rational can be explicitly expressed. This result improves the characterization of maximal orthogonal set of exponentials for the integral matrix to that for the rational matrix .