Spectral Properties of Sierpinski Measures on Rn

• Published in: Constructive approximation 2024, Vol.60 (1), p.165-196
• Main Authors: Dai, Xin-Rong, Fu, Xiao-Ye, Yan, Zhi-Hui
• Format: Article
• Language: English
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Numerical Analysis; Prime numbers
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-023-09654-0

Let R = ϱ I n and D = 0 , e 1 , … , e n , where ϱ > 1 and e i is the i -th coordinate vector in R n . The spectral properties of the n - dimensional Sierpinski measure μ R , D has been studied over two decades. In this paper, a special type of spectrum called a typical spectrum for μ R , D is considered. We show that ϱ ∈ ( n + 1 ) N is necessary and sufficient for μ R , D to admit a typical spectrum if n + 1 is prime. And some necessary conditions for μ R , D to admit a typical spectrum are provided when n + 1 is not a prime number. Furthermore, under the condition on real Hadamard matrix, we prove that μ R , D admits a quasi-typical spectrum if and only if ϱ ∈ 2 N . These results show that the spectral properties of the Sierpinski measure μ R , D are really different between n + 1 is prime and non-prime. As a corollary, we prove that ϱ ∈ 2 N are the only integers such that μ R , D becomes a spectral measure when n = 3 .