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Analysis of μM, D‐orthogonal exponentials for the planar four‐element digit sets

The self‐affine measure μM,D is a unique probability measure satisfying the self‐affine identity with equal weight. It only depends upon an expanding matrix M and a finite digit set D. In this paper we study the question of when the L2(μM,D) ‐space has infinite families of orthogonal exponentials. Such research is necessary to further understanding the spectrality of μM,D . For a class of planar four‐element digit sets, we present several methods to deal with this question. The application of each method is also given, which extends the known results in a simple manner.