Search results for "Hilbert space"

Frame-related Sequences in Chains and Scales of Hilbert Spaces

2022

Frames for Hilbert spaces are interesting for mathematicians but also important for applications in, e.g., signal analysis and physics. In both mathematics and physics, it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties of a certain sequence in one of the spaces, such as completeness or the property of being a (semi-) frame, propagate to the other ones in a scale of Hilbert spaces. We link that to the properties of the respective frame-related operators, such as analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism, the properties of frame se…

Frames and weak frames for unbounded operators

2020

In 2012 G\u{a}vru\c{t}a introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$, in order to decompose its range $\mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:\mathcal{D}(A)\to\mathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the norm of $\mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the graph norm of $A$.

Partial $\ast$-algebras of distributions

2005

The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of C ∞ -vectors of a self-adjoint operator, is considered in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an

Beyond frames: Semi-frames and reproducing pairs

2017

Frames are nowadays a standard tool in many areas of mathematics, physics, and engineering. However, there are situations where it is difficult, even impossible, to design an appropriate frame. Thus there is room for generalizations, obtained by relaxing the constraints. A first case is that of semi-frames, in which one frame bound only is satisfied. Accordingly, one has to distinguish between upper and lower semi-frames. We will summarize this construction. Even more, one may get rid of both bounds, but then one needs two basic functions and one is led to the notion of reproducing pair. It turns out that every reproducing pair generates two Hilbert spaces, conjugate dual of each other. We …

PIP-Spaces and Signal Processing

2009

Contemporary signal processing makes an extensive use of function spaces, always with the aim of getting a precise control on smoothness and decay properties of functions. In this chapter, we will discuss several classes of such function spaces that have found interesting applications, namely, mixed-norm spaces, amalgam spaces, modulation spaces, or Besov spaces. It turns out that all those spaces come in families indexed by one or more parameters, that specify, for instance, the local behavior or the asymptotic properties. In general, a single space, taken alone, does not have an intrinsic meaning, it is the family as a whole that does, which brings us to the very topic of this volume. In …

Tensor product characterizations of mixed intersections of non quasianalytic classes and kernel theorems

2009

Mixed intersections of non quasi-analytic classes have been studied in [12]. Here we obtain tensor product representations of these spaces that lead to kernel theorems as well as to tensor product representations of intersections of non quasi-analytic classes on product of open or of compact sets (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Examples of Indexed PIP-Spaces

2009

This chapter is devoted to a detailed analysis of various concrete examples of pip-spaces. We will explore sequence spaces, spaces of measurable functions, and spaces of analytic functions. Some cases have already been presented in Chapters 1 and 2. We will of course not repeat these discussions, except very briefly. In addition, various functional spaces are of great interest in signal processing (amalgam spaces, modulation spaces, Besov spaces, coorbit spaces). These will be studied systematically in a separate chapter (Chapter 8).

Invertibility in tensor products of Q-algebras

2002

Spectrum and Pseudo-Spectrum

2019

In this book all Hilbert spaces will be assumed to separable for simplicity. In this section we review some basic definitions and properties; we refer to Kato (Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer, New York, 1966), Reed and Simon (Methods of modern mathematical physics. I. Functional analysis, 2nd edn. Academic, New York, 1980; Methods of modern mathematical physics. II. Fourier analysis, self adjointness. Academic, New York, 1975; Methods of modern mathematical physics. IV. Analysis of operators. Academic, New York, 1978), Riesz and Sz.-Nagy (Lecons d’analyse fonctionnelle, Quatrieme edition. Academie des Sciences d…

Current Algebras as Hilbert Space Operator Cocycles

1994

Aspects of a generalized representation theory of current algebras in 3 + 1 dimensions axe discussed. Rules for a systematic computation of vacuum expectation values of products of currents are described. Their relation to gauge group actions in bundles of fermionic Fock spaces and to the sesquilinear form approach of Langmann and Ruijsenaars is explained. The regularization for a construction of an operator cocycle representation of the current algebra is explained. An alternative formula for the Schwinger terms defining gauge group extensions is written in terms of Wodzicki residue and Dixmier trace.