INFINITE ORTHOGONAL EXPONENTIALS OF A CLASS OF SELF-AFFINE MEASURES

In this paper, we study infinite families of orthogonal exponentials of some self-affine measures. The digit set D = {(0 0), (1 0), (0 2)} and any 2 x 2 expanding integer matrix M is an element of M-2(Z) can generate a self-affine measure mu(M,D). Let epsilon(7) = (1/3, 1/3)(t) and M* := 3 (M) over tilde + M-alpha be the transposed conjugate of M, where (M) over tilde is an element of M-2(Z) and the elements of M-alpha come from {0, 1, 2}. In this paper, we prove the following results. For M-alpha is an element of{M-alpha : M-alpha epsilon(7) is an element of Z(2), det(M-alpha) is an element of 3Z}, mu(M,D) is a spectral measure. For M-alpha is an element of{M-alpha : M-alpha(2)epsilon(7) is an element of Z(2), M-alpha epsilon(7) is not an element of Z(2), det(M-alpha) is an element of 3Z}, there are infinite families of orthogonal exponentials, but none of them forms an orthogonal basis in L-2(mu(M,D)).