On the -orthogonal exponentials

✍ Scribed by Jian-Lin Li

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 360 KB
• Volume: 73
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.04.017

✦ Synopsis

Let µ M,D be a self-affine measure corresponding to a given affine iterated function system {φ d (x) = M -1 (x + d)} d∈D . In the present paper we will study the problem of how to determine the L 2 (µ M,D )-space has finite or infinite orthogonal exponentials. Such research is motivated by a conjecture on the non-spectrality of µ M,D . We first obtain a nonspectral criterion which extends the result of Dutkay and Jorgensen. In opposition to the condition of this criterion, we then obtain some conditions which imply the infinite orthogonal systems in L 2 (µ M,D ). These are necessary for further investigation on the spectrality of µ M,D . As an application, we completely settle the corresponding problem for the generalized planar Sierpinski family.