Spectrality of a class of infinite convolutions

Abstract

For a finite set D ⊂ Z and an integer b ≥ 2, we say thata∈E δ a denote the uniformly discrete probability measure on E. We prove that the class of infinite convolution (Moran measure) μ b,{Dk } = δ b −1 D1 * δ b −2 D2 * · · · is a spectral measure provided that there is a common C ⊂ Z + compatible to all the (b, D k ) and C + C ⊆ {0, 1, . . . , b − 1}. We also give some examples to illustrate the hypotheses and results, in particular, the last condition on C is essential.

Spectral property of a class of Moran measures on R

“…We will omit the proof, one can see Lemma 3.1 and Lemma 4.3 in [2]. We would like to point out that the proof of Lemma 4.3 in [2] is easier with the help of Lemma 2.4 in our setting.…”

“…We would like to point out that the proof of Lemma 4.3 in [2] is easier with the help of Lemma 2.4 in our setting. The reader may refer to [2] for more detail. 2…”

“…Before stating our main result, we first introduce and not include the proof of the following technique lemma, since it follows readily from Lemma 3.1 and Lemma 4.3 in [2]. …”

“…Also, we obtain that μ b,D,{n k } is a spectral measure for certain 'lacunary' sequence {n k } ∞ k=1 in Section 3. We then get a sufficient condition such that Conjecture 1.1 holds in Section 4, in which the condition is inspired by [2]. In the final section, we get that the measure μ b,D,{n k } is a spectral measure when the sequence {n k } ∞ k=1 is ultimately periodic (see Definition 5.1).…”

“…In recent years, there has been a wide range of interests in expanding the classical Fourier analysis to fractal or more general probability measures after the pioneer work of Jorgensen and Pedersen [16] in 1998 [1][2][3][4][5][6][7][9][10][11]14,15,[19][20][21][22][23][24][27][28][29]31]. One of the central themes of this area of research involves constructing Fourier bases in L 2 (μ), where μ is a measure which is generated by the iterated function systems.…”