Estimation For The Cardinality Of Finite Orthogonal Exponentials Under Certain Self-affine Measures

The self-affine measure ?M,D associated with an affine iterated function system{?_d?x?= M^-1{x + d)}d?D is uniquely determined.There are many open problems on the self-affine measures.All these researches focus mainly on the conditions under which ?M,D is a spectral measure or a non-spectral measure.Based on the previous research we will study the non-spectrality of self-affine measures,which is to estimate the number of orthogonal exponentials in L²??M,D?and to find them.We have the following results:In the first part,we study the spectrality of self-affine measure ?M,D corre-sponding to an expanding matrix M = diag[p₁,p₂,p₃]?p_j?Z\{0,±1},j =1,2,3?and digit set D={0,e₁,e₂,e₃,e₁+e₂,e₁+e₃,e₂ + e₃,e₁+e₂ + e₃},where e₁,e₂,e₃ are the standard basis of unit column vectors.In this part,by characterizing the zero set Z??M,D?of Fourier transform ?M,D,we prove when p_j??2Z + 1?\{0,±1}?j =1,2,3?,?M,D is non-spectral of self-affine measures,there exist at most 8 mutually orthogonal exponential functions in L²??M,D?,where the number 8 is the best.In the second part,we study the question of ?M,D-orthogonal exponentials for the special four-element digit sets in the plane.By using the residue class modulo 8,we can write the expanding matrix expanding matrix M*as M*= 8??? + M?,?,?,when det?M?? 2Z+ 1,the number of orthogonal exponentials in L²??M,D?is given.