195 research outputs found
On Fourier frame of absolutely continuous measures
Let be a compactly supported absolutely continuous probability measure
on , we show that 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 admits Fourier frame, then the
measure must be a characteristic function of self-similar tile. In particular,
this shows for almost everywhere , the -Bernoulli
convolutions cannot admit Fourier frames
Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution
Let be a fundamental domain of some full-rank lattice in and
let and be two positive Borel measures on such that
the convolution is a multiple of . We consider the problem
as to whether or not both measures must be spectral (i.e. each of their
respective associated space admits an orthogonal basis of exponentials)
and we show that this is the case when . 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 and we show that it implies the classical Fuglede's conjecture
on
Some reductions of the spectral set conjecture to integers
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 , there are many results in the literature that
discuss the relations among various forms of the Fuglede conjecture on
, and 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 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 satisfy the Coven-Meyerowitz property, then both
sides of the Fuglede conjecture on are true
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