Spectral property of a class of Moran measures on R

Abstract

Let b ≥ 2 be a positive integer. Let D be a finite subset of Z and {n k } ∞ k=1 ⊆ N be a sequence of strictly increasing numbers. A Moran measure μ b,D,{nk } is a Borel probability measure generated by the Moran iterated function system (Moran IFS)In this paper we study one of the basic problems in Fourier analysis associated with μ b,D,{nk } . More precisely, we give some conditions under which the measure μ b,D,{nk } is a spectral measure, i.e., there exists a discrete subset Λ ⊆ R such that E(Λ) = {e 2πiλx : λ ∈ Λ} is an orthonormal basis for L 2 (μ b,D,{nk } ).

Spectrality of a class of Moran measures on with consecutive digit sets

“…Recently, Dutkay, Hausserman, and Lai [10] generalized the result to higher dimensions, thus settling a long-standing conjecture proposed by Jorgensen and Pedersen. Meanwhile, many interesting spectral measures have been found (see, e.g., [1–7, 9, 12, 13, 15, 16, 23–26, 29] and the references therein for recent advances), but there are only a few classes.…”

“…In the last decade, many researchers studied the spectrality of the Moran measures, which are the nonself-similar generalization of the Cantor measures through the infinite convolution. Note that most results of the known cases are concentrated on one-dimensional Moran measures (see, e.g., [1,2,12,13,15] and the references therein). There are few studies involving Moran measures of higher dimension other than [8,26,29].…”

Non-spectral problem for the planar self-affine measures with decomposable digit sets

“…In recent years, there has been a wide range of interests in expanding the classical Fourier analysis to fractal or more general probability measures [14]. One of the central themes of this area of research involves finding Spectral measures and nonspectral measures.…”

Spectrality of the planar Sierpinski family

“…It relates analysis, geometry and topology (see, e.g. [7,14,16,[26][27][28]30] and references therein), in which the good functions are complex exponentials. The best approximation appears when L 2 (μ) has a basis consisting of complex exponentials (Fourier basis).…”

“…For example, in 1998 Jorgensen and Pedersen [16] found the first singular, non-atomic, self-similar spectral measure supported on 1 4 -Cantor set. Nowadays, there is a large literature on this topic [1][2][3][4][5][6][7][8]10,[13][14][15][16]19,20,26,28,29,31]. Most of this literature deals with the issue in one dimension.…”