There are eight-element orthogonal exponentials on the spatial Sierpinski gasket

摘要

The self-affine measure mu(M,D) corresponding to an expanding matrix M = diag[p(1), p(2), p(3)] and the digit set D = {0, e(1), e(2), e(3)} in the space R-3 is supported on the spatial Sierpinski gasket, where e(1), e(2), e(3) are the standard basis of unit column vectors in R-3 and p(1), p(2), p(3) is an element of Z {0,+/- 1}. In the case p(1) is an element of 2Z and p(2), p(3) is an element of 2Z + 1, it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space L-2(mu(M,D)) is at most "4", where the number 4 is the best upper bound. That is, all the four-element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five-element orthogonal exponentials in L-2(mu(M,D)). In the present paper, we construct a class of the eight-element orthogonal exponentials in the corresponding Hilbert space L-2(mu(M,D)) to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.