Spectral self-affine measures on the planar Sierpinski family

The present research will concentrate on the topic of Fourier analysis on fractals. It mainly deals with the problem of determining spectral self-affine measures on the typical fractals: the planar Sierpinski family. The previous researches on this subject have led to the problem within the possible fifteen cases. We shall show that among the fifteen cases, the nine cases correspond to the spectral measures, and reduce the remnant six cases to the three cases. Thus, for a large class of such measures, their spectrality and non-spectrality are clear. Moreover, an explicit formula for the existent spectrum of a spectral measure is obtained. We also give a concluding remark on the remnant three cases.