Translational absolute continuity and Fourier frames on a sum of singular measures

摘要

A finite Borel measure mu in R-d is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for L-2(mu). It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures nu and lambda without atoms whose supports form a packing pair, then nu * lambda + delta(t) * nu is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure mu(4) + mu(16) does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping R-0 with theta not equal +/-pi/2, the two-dimensional measure rho(0)(.) := (mu(4) x delta(0))(.) + (delta(0) x mu(16))(R-theta(-1).), supported on the union of x-axis and y = (cot theta)x, always admit a Fourier frame. Furthermore, we can find {e(2 pi i lambda,x )}(lambda is an element of Lambda theta), such that it forms a Fourier frame for rho(theta) with frame bounds independent of theta. Nonetheless, rho(+/-pi/2) does not admit any Fourier frame. (C) 2017 Elsevier Inc. All rights reserved.