4,002 research outputs found
On the Beurling dimension of exponential frames
We study Fourier frames of exponentials on fractal measures associated with a
class of affine iterated function systems. We prove that, under a mild
technical condition, the Beurling dimension of a Fourier frame coincides with
the Hausdorff dimension of the fractal
Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms
Let F be a characteristic zero differential field with an algebraically
closed field of constants, E be a no-new-constant extension of F by
antiderivatives of F and let y1, ..., yn be antiderivatives of E. The
antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the
derivatives of yi in E satisfies certain conditions. We will discuss a new
proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower
of extensions by J-I-E antiderivatives and use this generalized version of the
theorem to classify the finitely differentially generated subfields of this
tower. In the process, we will show that the J-I-E antiderivatives are
algebraically independent over the ground differential field. An example of a
J-I-E tower is extensions by iterated logarithms. We will discuss the normality
of extensions by iterated logarithms and produce an algorithm to compute its
finitely differentially generated subfields.Comment: 66 pages, 1 figur
Fourier duality for fractal measures with affine scales
For a family of fractal measures, we find an explicit Fourier duality. The
measures in the pair have compact support in \br^d, and they both have the
same matrix scaling. But the two use different translation vectors, one by a
subset in \br^d, and the other by a related subset . Among other
things, we show that there is then a pair of infinite discrete sets
and in \br^d such that the -Fourier exponentials are
orthogonal in , and the -Fourier exponentials are
orthogonal in . These sets of orthogonal "frequencies" are
typically lacunary, and they will be obtained by scaling in the large. The
nature of our duality is explored below both in higher dimensions and for
examples on the real line.
Our duality pairs do not always yield orthonormal Fourier bases in the
respective -Hilbert spaces, but depending on the geometry of certain
finite orbits, we show that they do in some cases. We further show that there
are new and surprising scaling symmetries of relevance for the ergodic theory
of these affine fractal measures.Comment: v
Isospectral measures
In recent papers a number of authors have considered Borel probability
measures in \br^d such that the Hilbert space has a Fourier
basis (orthogonal) of complex exponentials. If satisfies this property,
the set of frequencies in this set are called a spectrum for . Here we fix
a spectrum, say , and we study the possibilities for measures
having as spectrum.Comment: v
Meromorphic Szego functions and asymptotic series for Verblunsky coefficients
We prove that the Szeg\H{o} function, , of a measure on the unit circle
is entire meromorphic if and only if the Verblunsky coefficients have an
asymptotic expansion in exponentials. We relate the positions of the poles of
to the exponential rates in the asymptotic expansion. Basically,
either set is contained in the sets generated from the other by considering
products of the form, with
in the set. The proofs use nothing more than iterated Szeg\H{o} recursion
at and
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