The Study Of The Cardinality Of Orthogonal Exponential Functions Of Planar Self-affine Measures

Let A(?)Rn be a countable set,εΛ＝{e2πi<λ,x>:λ ∈ Λ},and let μ be a Borel probability measure with compact support on Rn.We call μ a spectral measure and Λ a spectrum of μ if εΛ is an orthogonal basis for L2(μ),we also say that(μ,Λ)is a spectral pair.In recent years,the spectral and non-spectral problems of self-affine measures have received a lot of attention.In this thesis,we will mainly study the cardinality of orthogonal exponential functions of planar self-affine measure μM,D This thesis consists of three chapters.In Chapter 1,we summarize the research background,research significance and latest research results of spectral and non-spectral problems of self-affine measures,and we also introduce the main results of this thesis.In Chapter 2,we mainly introduce some related knowledge of self-affine measures,and give some lemmas and propositions to prove the main theorem of this paper.In Chapter 3,we mainly study the cardinality of orthogonal exponential functions of planar self-affine measure μM,D generated by an expanding integer matrix M and a three-element integer digit set D={(0 0),(α1 α2),(β1 β2)}with α1β2-α2β1 ≠0.We show that if det(M)(?)3Z,then the mutually or-thogonal exponential functions in L2(μM,D)is finite,and the exact maximal cardinality of mutually orthogonal exponential functions in L2(μM,D)is given.Lastly,we give some examples and related conclusions.