197 research outputs found

    On Fourier frame of absolutely continuous measures

    Get PDF
    Let μ\mu be a compactly supported absolutely continuous probability measure on Rn{\Bbb R}^n, we show that μ\mu admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support. As a consequence, we prove that if an equal weight absolutely continuous self-similar measure on R1{\Bbb R}^1 admits Fourier frame, then the measure must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2<λ<11/2<\lambda<1, the λ\lambda-Bernoulli convolutions cannot admit Fourier frames

    Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution

    Full text link
    Let QQ be a fundamental domain of some full-rank lattice in Rd{\Bbb R}^d and let μ\mu and ν\nu be two positive Borel measures on Rd{\Bbb R}^d such that the convolution μ∗ν\mu\ast\nu is a multiple of χQ\chi_Q. We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated L2L^2 space admits an orthogonal basis of exponentials) and we show that this is the case when Q=[0,1]dQ = [0,1]^d. This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede's conjecture for spectral measures on R1{\Bbb R}^1 and we show that it implies the classical Fuglede's conjecture on R1{\Bbb R}^1

    Some reductions of the spectral set conjecture to integers

    Full text link
    The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on R1{\mathbb R}^1, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on Zn{\mathbb Z}_n, Z{\mathbb Z} and R1{\mathbb R}^1 and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on R1{\mathbb R}^1 is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven-Meyerowitz property for finite sets of integers, introduced in \cite{CoMe99}, and we show that if the spectral sets and the tiles in Z{\mathbb Z} satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on R1{\mathbb R}^1 are true
    • …
    corecore