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Spectrality of self-affine measures on the three-dimensional Sierpinski gasket

Part of: Nontrigonometric harmonic analysis Normed linear spaces and Banach spaces; Banach lattices Classical measure theory

Published online by Cambridge University Press:  20 April 2012

Jian-Lin Li
Jian-Lin Li
Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China ([email protected])
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Abstract

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The self-affine measure μM, D corresponding to M = diag[p1, p2, p3] (pj ∈ ℤ \ {0, ± 1}, j = 1, 2, 3) and D = {0, e1, e2, e3} in the space ℝ3 is supported on the three-dimensional Sierpinski gasket T(M, D), where e1, e2, e3 are the standard basis of unit column vectors in ℝ3. We shall determine the spectrality and non-spectrality of μM, D, and show that if pj ∈ 2ℤ \ {0, 2} for j = 1, 2, 3, then μM, D is a spectral measure, and if pj ∈ (2ℤ + 1) \ {±1} for j = 1, 2, 3, then μM, D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM, D), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measures.

Keywords

iterated function systemself-affine measureorthogonal exponentialsspectrality

MSC classification

Secondary: 28A80: Fractals 42C05: Orthogonal functions and polynomials, general theory 46C05: Hilbert and pre-Hilbert spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 477 - 496
Copyright
Copyright © Edinburgh Mathematical Society 2012

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