On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits
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(2020) 26:82
On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits Cong Wang1 · Zhi-Yi Wu2 Received: 5 January 2020 / Revised: 2 June 2020 / Accepted: 18 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let μ p,q be a self-similar spectral measure with consecutive digits generated by an q−1 iterated function system { f i (x) = xp + qi }i=0 , where 2 ≤ q ∈ Z and q| p. It is known that for each w = w1 w2 · · · ∈ {−1, 1}∞ := {i 1 i 2 · · · : all i k ∈ {−1, 1}}, the set
w =
n
ajwj p
j−1
: a j ∈ {0, 1, . . . , q − 1}, n ≥ 1
j=1
is a spectrum of μ p,q . In this paper, we study the possible real number t such that the set tw are also spectra of μ p,q for all w ∈ {−1, 1}∞ . Keywords Self-similar measures · Spectral eigenvalue problems · Spectral measures Mathematics Subject Classification Primary 42A65 · 42B05; Secondary 28A78 · 28A80
Communicated by Dorin Dutkay. This work was supported by the NSF of China (Grant Nos. 11971194 and 11771457), the FRF for Central Universities (No. 2019YBZZ059), the Sun Yat-Sen University Youth Teacher Cultivation Project (20lgpy142), China Postdoctoral Science Foundation (2020M672928) and by Guangdong Province Key Laboratory of Computational Science at the Sun Yat-Sen University.
B
Zhi-Yi Wu [email protected] Cong Wang [email protected]
1
School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, People’s Republic of China
2
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China 0123456789().: V,-vol
82
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Journal of Fourier Analysis and Applications
(2020) 26:82
1 Introduction Let μ be a compactly supported Borel probability measure on R. We say that μ is a spectral measure if there exists a countable set ⊆ R such that {e2πiλx : λ ∈ } forms an orthonormal basis for L 2 (μ). The set is then called a spectrum of μ. The theory of spectral measures could date back to 1974, when Fuglede [17] proposed the famous spectral set conjecture and the study entered into the realm of fractals when Jorgensen and Pederson [20] discovered the first example of a singular and nonatomic spectral measure μ4,2 (see (1.2)), i.e., the standard one-fourth Cantor measure, with a spectrum
4,2 = {0, 1} + 4{0, 1} + 42 {0, 1} + · · ·
(all finite sums).
(1.1)
Since then, the theory of singular spectral measures has been extensively studied, which involves constructing new examples [1–3,6,7,13,23] and classifying their possible spectra [4,5,9,11,14,22,24,25]. At the same time, many differences between singular spectral measures and absolutely continuous spectral measures have been discovered. For example, a typical difference is that there is only one spectrum = Z containing 0 for L 2 ([0, 1]), while there are uncountable spectra (up to translations) for L 2 (μ4,2 ) [11,16]. Moreover, for the spectrum 4,2 given by (1.1) there exist infinitely many t such that each t4,2 is also a spectrum of μ4,2 [8,9]. Another surpr
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