On spectral eigenvalue problem of a class of generalized Cantor measures

[摘要] Let q = 4r with integer r >= 1. Let a = a(1)a(2) center dot center dot center dot, b = b(1)b(2) center dot center dot center dot epsilon {0,1, 2,3}(infinity) with a < b, i.e., a(n) < b(n) for all n >= 1. We consider the following generalized Cantor measure mu(q,a,b) = delta(1/q{a1,b1}) * delta (1/q2{a2,b2}) * center dot center dot center dot, where delta(E) = 1/#E Sigma(a epsilon E) delta(a) denotes the uniformly discrete probability on E. In this paper we investigate the spectral eigenvalues (of the second type) of the measure mu(q,a,b) (the spectrality of mu(q,a,b) is settled by He et al. (2017) [17]). More precisely, we characterize the possible real numbers t satisfying that there exists a countable set Lambda subset of R such that A and to are spectra of mu(q,a,b) simultaneously, i.e., the sets E(Lambda) := {e(-2 pi i lambda x): lambda epsilon Lambda and E(t Lambda) := {e(-2 pi i lambda x): lambda epsilon Lambda} are both orthonormal basis for L-2 (mu(q,a,b)). (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-01

[关键词] Generalized Cantor measures;Orthonormal basis of exponential;Spectral measures;Spectral eigenvalue problem