Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCohen class of time-frequency representations and operators: boundedness and uncertainty principles
97
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper presents a proof of an uncertainty principle of Donoho-Stark type involving $varepsilon$-concentration of localization operators. More general operators associated with time-frequency representations in the Cohen class are then considered. For these operators, which include all usual quantizations, we prove a boundedness result in the $L^p$ functional setting and a form of uncertainty principle analogous to that for localization operators.
rate research
Read More
We obtain sufficient conditions for a densely defined operator on the Fock space to be bounded or compact. Under the boundedness condition we then characterize the compactness of the operator in terms of its Berezin transform.
Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although the study of self-adjoint operators goes back a few decades, the non self-adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like $mathcal A({Omega})$, where $Omega subseteq mathbb C^m$ is a bounded domain, consisting of complex valued functions which are holomorphic on some open set $U$ containing $overline{Omega}$, the closure of $Omega$. The book, Hilbert Modules over function algebra, R. G. Douglas and V. I. Paulsen showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The book, Spectral decomposition of analytic sheaves, J. Eschmeier and M. Putinar and the book, Analytic Hilbert modules, X. Chen and K. Guo, provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.
We characterize the (essentially) decreasing sequences of positive numbers $beta$ = ($beta$ n) for which all composition operators on H 2 ($beta$) are bounded, where H 2 ($beta$) is the space of analytic functions f in the unit disk such that $infty$ n=0 |c n | 2 $beta$ n < $infty$ if f (z) = $infty$ n=0 c n z n. We also give conditions for the boundedness when $beta$ is not assumed essentially decreasing.
A class of doubly stochastic graph shift operators (GSO) is proposed, which is shown to exhibit: (i) lower and upper $L_{2}$-boundedness for locally stationary random graph signals; (ii) $L_{2}$-isometry for textit{i.i.d.} random graph signals with the asymptotic increase in the incoming neighbourhood size of vertices; and (iii) preservation of the mean of any graph signal. These properties are obtained through a statistical consistency analysis of the graph shift, and by exploiting the dual role of the doubly stochastic GSO as a Markov (diffusion) matrix and as an unbiased expectation operator. Practical utility of the class of doubly stochastic GSOs is demonstrated in a real-world multi-sensor signal filtering setting.
In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatricks family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brondsted-Rockafellar property. In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons. We will also prove that an auxiliary condition used by the authors to prove the restricted Brondsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
