On the Fourier orthonormal bases of a class of self-similar measures on ℝ n

Abstract

Abstract Let μ M , D {\mu_{M,D}} be a self-similar measure generated by an n × n {n\times n} expanding real matrix M = ρ - 1 ⁢ I {M=\rho^{-1}I} and a finite digit set D ⊂ ℤ n {D\subset{\mathbb{Z}}^{n}} , where 0 < | ρ | < 1 {0<\lvert\rho\rvert<1} and I is an n × n {n\times n} unit matrix. In this paper, we study the existence of a Fourier basis for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} , i.e., we find a discrete set Λ such that E Λ = { e 2 ⁢ π ⁢ i ⁢ 〈 λ , x 〉 : λ ∈ Λ } {E_{\Lambda}=\{e^{2\pi i\langle\lambda,x\rangle}:\lambda\in\Lambda\}} is an orthonormal basis for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . Under some suitable conditions for D, some necessary and sufficient conditions for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} to admit infinite orthogonal exponential functions are given. Then we set up a framework to obtain necessary and sufficient conditions for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} to have a Fourier basis. Finally, we demonstrate how these results can be applied to self-similar measures.