The Study Of The Spectral Property Of Planar Fractal Measures With Four-element Digit Set

Let ? be a Borel probability measure with compact support on Rn and A(?)Rn be a countable set with.We call ? a spectral measure and A a spectrum of ? if is an orthogonal basis for L2(?)space?we also say that(?,?)is a spectral pair.In recent years,the problems of spectral and non-spectral measure of fractal measures have attracted extensive attention of many scholars at home and abroad,and many excellent achievements have made.In this paper,we mainly study the spectral property of a class of planar fractal measures with four-element digit set.The main work of this thesis is spread in following four chapters:In Chapter 1,we summarize the research background,research significance and latest research results of fractal spectral measures,we introduce some re-lated knowledge and the main results of this paper.In Chapter 2,we mainly study the planar self-affine measure ?M,D gener-ated by an expanding real matrix(?)element digit set D=(?)pace admits an orthogonal set of exponential functions if and only if |?|=(q/2p)1/r,where gcd(2p,q)=1 and p,g,r ? N.In Chapter 3,we consider the problem about cardinality of orthogonal exponential functions in L2{?M,D)space when there not exists infinite orthog-onal set,and we give the exact maximal cardinality exponential functions in L2{?M,D)space under differeLt conditions.In Chapter 4,we mainly explore the spectral property of planar self-affine measure ?M,D and we show that ?M,D is a spectral measure if and only if|?|=1/2p,where p?N.