Spectral Radius of Refinement and Subdivision Operators with Power Diagonal Dilations

Abstract

Two types of estimate for the spectral radius of the multivariate refinement operator with power diagonal dilations are presented. One type contains multiplicator norm of number matrices generated by the symbol of the corresponding operator and by specific subsets of repeating fractions. These subsets are used together with the little Fermat theorem to establish estimates that comprise integrals over tori of various dimensions. Moreover, we note certain classes of symbols when the exact value of the spectral radius of refinement operator can be found. For the spectral radius of subdivision operators point value estimates are established. Mathematics Subject Classification (2000). Primary 47A10, 47B33; Secondary 42C40.

Ont-entropy and variational principle for the spectral radii of transfer and weighted shift operators

“…Now letting n in equation 23go to infinity and using equation (24) and the arbitrariness of ε > 0, we obtain the estimate of the form required:…”

“…(2) The expression (24) for the constant that satisfies the estimate (25) is not the best one. One can improve it by means of the following consideration.…”

“…In the paper by Didenko [24], a series of lower estimates for the spectral radius of the subdivision operator was obtained. In fact, some of these estimates have been derived with the help of the usage of periodic points of the mapping α.…”

“…We dwell on this in such detail to emphasize the fact that such estimates follow directly from equation (49). To be true, we have to note that in [24] there was considered a more complicated situation as well, namely the subdivision operators with matrix coefficients in the space of vector functions. The whole of the theory presented in this paper is not applicable readily to these operators.…”

The spectral radius of matrix continuous refinement operators

“…Usually this is a very demanding task; and as mentioned in [2], at present there are only a few analytic formulas for the spectral radii of general weighted shift operators with matrix coefficients. Moreover, even for discrete refinement operators there are no such formulas and all the results known depend upon the structure of both the dilation and symbol matrices M and a (see [6,7] and references there). However, in the case at hand, the matrix a has remarkable properties, so the expression in the right-hand side of (7) can be treated effectively.…”