SPECTRALITY OF A CLASS OF MORAN MEASURES ON THE PLANE

Let {(R-k, D-k)}(k=1)(infinity) be a sequence of pairs, where D-k = {0, 1,..., q(k) - 1}(1, 1)(T) is an integer vector set and R-k is an integer diagonal matrix or upper triangular matrix, i.e., R-k = ( [GRAPHICS] ) or R-k = ( [GRAPHICS] ). Associated with the sequence {(R-k, D-k)}(k=1)(infinity), Moran measure mu({Rk},{Dk}) is defined by mu({Rk},{Dk}) = delta(R1-1 D1) * delta(R1-1R2-1 D2) * ... * delta(R1-1) (R2-1) ... Rk-1 (Dk) * ... . In this paper, we consider the spectrality of mu({Rk},{Dk}). We prove that mu({Rk},{Dk}) is a spectral measure under certain conditions in terms of (R-k, D-k), i.e., there exists a Fourier basis for L-2 (mu({Rk},{Dk})).

Automated summary

This paper investigates the spectral properties of the sequence {(R-k, D-k)}(k=1)(infinity) and its associated Moran measure mu({Rk},{Dk}).