Spectrum Of Self-affine Measure

The basic problem of Fourier analysis is the existence of orthogonal basis, for the Lebesgue measure, the problem is very complete. Considering the gen-eral compactly supported Borel measure μ on Rn, we say that μ is a spectral measure, if there exists a countable set ∧(?) Rn called spectrum of the mea-sure μ, so that the collection of exponential functions E∧={e2πi<λ,x>:λ∈∧} forms an orthogonal basis for L2(μ) where (·,·) denote the standard inner product on Rn. The existence of spectrum is closely linked to the famous Fu-glede’s conjecture [10] (High dimension is solved by Tao), if μ is a normalized Lebesgue measure. However, for the non-atomic singular measure, Jorgensen and Pedersen firstly found the measure μ, with exponential orthogonal basis on L2(μ) ([9] in 1998, also see [26]). This measure is a Cantor measure, which is a simple and special self-affine measure. To determine that whether a self-affine measure is the spectral measure or not, immediately become a hot topic in fractal geometry combined with harmonic analysis. And a group of high quality and interesting results were obtained [13]-[25] and [27].For the spectrum of one Sierpinski type self-affine measure, professor Li Jianlin [161 in 2009 proved that:for the three-elements digit set and for an expanding integer matrix M E M2(Z), if det(M)(?) 3Z, then the self-affine measure μM,D determined by D and M has at most 3 mutually orthogonal exponentials in L2(μM,D), and the number 3 is the best. So μ is not spectral measure.In this paper, we study the spectrum of another Sierpinski type self-affine measure:considering the three-elements digit set and for an expanding integer matrix M ∈M2(Z), we study the self-affine measure μM,D determined by D and M. We obtain μM,D has at most 3 mutually orthogonal exponentials in L2(μM,D) and the number 3 is the best, if det(M)(?)3Z and for part of the matrix M (see the first chapter in the second quarter). This conclusion not only illustrates the self-affine mea-sure μM,D not spectral measure, but also find the orthogonal exponentials in L2(μM,D), which is a meaningful work. Our method is not completely the same with professor Li’s. We apply the method of orthogonality similarity to prove our conclusion, which is one of the innovation of this paper point.We also popularize the above conclusion to the more general three-elements digit set For an expanding integer matrix M ∈M2(Z), we establish the conditions (see the first chapter in the second quarter). With these conditions, self-affine measure μM,D determined by D and M has at most 3 mutually orthogonal exponentials in L2(μM,D), and the number 3 is the best.