The exact number of orthogonal exponentials on the spatial Sierpinski gasket

Abstract

Abstract Let μ M , D {\mu_{M,D}} be a self-affine measure associated with an expanding real matrix M = diag ⁡ [ ρ 1 , ρ 2 , ρ 3 ] {M=\operatorname{diag}[\rho_{1},\rho_{2},\rho_{3}]} and the digit set D = { 0 , e 1 , e 2 , e 3 } {D=\{0,e_{1},e_{2},e_{3}\}} in the space ℝ 3 {\mathbb{R}^{3}} , where | ρ 1 | , | ρ 2 | , | ρ 3 | ∈ ( 1 , ∞ ) {\lvert\rho_{1}\rvert,\lvert\rho_{2}\rvert,\lvert\rho_{3}\rvert\in(1,\infty)} and e 1 , e 2 , e 3 {e_{1},e_{2},e_{3}} is the standard basis of unit column vectors in ℝ 3 {\mathbb{R}^{3}} . In this paper, we mainly consider the case ρ 1 ∈ { p q : p ∈ 2 ⁢ ℤ , q ∈ 2 ⁢ ℤ - 1 } , ρ 2 , ρ 3 ∈ { p q : p , q ∈ 2 ⁢ ℤ - 1 } . \rho_{1}\in\Bigl{\{}\frac{p}{q}:p\in 2\mathbb{Z},\,q\in 2\mathbb{Z}-1\Bigr{\}}% ,\quad\rho_{2},\rho_{3}\in\Bigl{\{}\frac{p}{q}:p,q\in 2\mathbb{Z}-1\Bigr{\}}. We prove that if ρ 2 = ρ 3 {\rho_{2}=\rho_{3}} , then there exist at most 4 mutually orthogonal exponential functions in the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} , where the number 4 is the best upper bound. If ρ 2 = - ρ 3 {\rho_{2}=-\rho_{3}} , then there exist at most 8 mutually orthogonal exponential functions in L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} , where the number 8 is the best upper bound. If | ρ 3 | ≠ | ρ 2 | {\lvert\rho_{3}\rvert\neq\lvert\rho_{2}\rvert} , then there are any number of orthogonal exponentials in L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . This gives the exact number of orthogonal exponentials on the spatial Sierpinski gasket in the above case.