Fractal L <sup> 2 </ Sup> (¦Ì <sub> M-d </ Sub>) Space For A Class Of Self-affine Measure The Non-spectral Nature

This thesis mainly investigates the Bernoulli measureμλ(λ∈(0,1)) which is a one parameter system of compact supported Broel probably measures and non-spectrality of a class of planar self-affine measure.The main goal is to study the maximal family and the Fourier bases in L2(μλ) for a given A.and to estimate the number of orthogonal exponential functions with certain digit set.The main results are as follows:(1)If E(Γ) is not an ONBs in L2(μλ), it may be the maximal systems.By ap-plying the properties ofΓand the zero set ofμλ, using the 8-expansions,we show that E(pΓ1/8)(p is an odd) be the maximal orthogonal exponentials in L2(μp/8),by the similar method we can improve the special conclusion to the general conclusion,we will show that E(pΓ1/(2n))(p is an odd) be the maximal orthogonal exponentials in L2(μp/(2n)). (2)We study the non-spectral problems with certain digit set,The self-affine measureμM,D corresponding to the expanding integer matrix and the digit set where p1,p2,p3∈2Z+1,|pi|>1(i= 1,2,3) is a non-spectral measure.In the present paper,we show that there exists at most 4 mutually orthogonal exponentials in L2(μM,D) and the number 4 is the best.The results of this thesis improve the related conclusions finished by D.E.Dutkay, P.E.T.Jorgensen,J.-L. Li,and play an important role in deeply studing the non-spectrality of self-affine measure.