Abstract
The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D). In the present paper, we establish some relations inside the zero set by the Fourier transform of the self-affine measure μM,D. Based on these facts, we show that 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 extends several known conclusions.
Highlights
Let M ∈ M n ( ) be an expanding integer matrix, that is to let one with all eigenvalues λi (M ) > 1 and D ⊂ n be a finite subset of cardinality D
One of the non-spectral problems on μM,D is to estimate the number of orthogonal exponentials in
The probability measure μM,D associated with an iterated function system
Summary
Let M ∈ M n ( ) be an expanding integer matrix, that is to let one with all eigenvalues λi (M ) > 1 and D ⊂ n be a finite subset of cardinality D. It is known that the non-spectral problem on self-affine measures consists of the following two classes:. The general case for the non-spectrality of the self-affine measure μM ,D is not known. For self-affine measure μM ,D corresponding to (1.2), if p=4 p=5 p and p1, p ∈ 2 +1\ {0, −1} , 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. Based on these established facts, we prove. Note that each typical case is concluded with a contradiction
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