2024/07/18 by Deng, Guotai, Fu, Yan-Song, Kang, Qingcan
#28A80 #42A65 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42A10
paper · doi:10.48550/arxiv.2407.13075
In this work we investigate the question of constructions of the possible Fourier bases E(Λ)=\e2πi λx:λ∈Λ\ for the Hilbert space L2(μ4), where μ4 is the standard middle-fourth Cantor measure and Λ is a countable discrete set. We show that the set \mathop \bigcapp∈ 2\Z+1\Λ⊂ \R: E(Λ) and E(pΛ) are Fourier bases for L2(μ4)\ has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space L2(μ4) from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set Λ such that E(Λ) and its all odd scaling sets E(Λ),p∈2\Z+1, are still Fourier bases for L2(μ4).