39,281 research outputs found
A new Hilbert-type integral inequality and the equivalent form
We give a new Hilbert-type integral inequality with
the best constant factor by estimating the weight function. And
the equivalent form is considered
The Dirichlet space: A Survey
In this paper we survey many results on the Dirichlet space of analytic
functions. Our focus is more on the classical Dirichlet space on the disc and
not the potential generalizations to other domains or several variables.
Additionally, we focus mainly on certain function theoretic properties of the
Dirichlet space and omit covering the interesting connections between this
space and operator theory. The results discussed in this survey show what is
known about the Dirichlet space and compares it with the related results for
the Hardy space.Comment: 35 pages, typoes corrected, some open problems adde
Convexity and the Euclidean metric of space-time
We address the question about the reasons why the "Wick-rotated",
positive-definite, space-time metric obeys the Pythagorean theorem. An answer
is proposed based on the convexity and smoothness properties of the functional
spaces purporting to provide the kinematic framework of approaches to quantum
gravity. We employ moduli of convexity and smoothness which are eventually
extremized by Hilbert spaces. We point out the potential physical significance
that functional analytical dualities play in this framework. Following the
spirit of the variational principles employed in classical and quantum Physics,
such Hilbert spaces dominate in a generalized functional integral approach. The
metric of space-time is induced by the inner product of such Hilbert spaces.Comment: 41 pages. No figures. Standard LaTeX2e. Change of affiliation of the
author and mostly superficial changes in this version. Accepted for
publication by "Universe" in a Special Issue with title: "100 years of
Chronogeometrodynamics: the Status of Einstein's theory of Gravitation in its
Centennial Year
Local properties of Hilbert spaces of Dirichlet series
We show that the asymptotic behavior of the partial sums of a sequence of
positive numbers determine the local behavior of the Hilbert space of Dirichlet
series defined using these as weights. This extends results recently obtained
describing the local behavior of Dirichlet series with square summable
coefficients in terms of local integrability, boundary behavior, Carleson
measures and interpolating sequences. As these spaces can be identified with
functions spaces on the infinite-dimensional polydisk, this gives new results
on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as
well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on
the infinite dimensional unit ball. We use both techniques from the theory of
sampling in Paley-Wiener spaces, and classical results from analytic number
theory.Comment: 27 pages, 1 figur
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