341 research outputs found
Weighted projections and Riesz frames
Let be a (separable) Hilbert space and a
fixed orthonormal basis of . Motivated by many papers on scaled
projections, angles of subspaces and oblique projections, we define and study
the notion of compatibility between a subspace and the abelian algebra of
diagonal operators in the given basis. This is used to refine previous work on
scaled projections, and to obtain a new characterization of Riesz frames.Comment: 23 pages, to appear in Linear Algebra and its Application
Linear combinations of generators in multiplicatively invariant spaces
Multiplicatively invariant (MI) spaces are closed subspaces of
that are invariant under multiplications of (some)
functions in . In this paper we work with MI spaces that
are finitely generated. We prove that almost every linear combination of the
generators of a finitely generated MI space produces a new set on generators
for the same space and we give necessary and sufficient conditions on the
linear combinations to preserve frame properties. We then apply what we prove
for MI spaces to system of translates in the context of locally compact abelian
groups and we obtain results that extend those previously proven for systems of
integer translates in .Comment: 13 pages. Minor changes have been made. To appear in Studia
Mathematic
Invariances of Frame Sequences under Perturbations
This paper determines the exact relationships that hold among the major Paley–Wiener perturbation theorems for frame sequences. It is shown that major properties of a frame sequence such as excess, deficit, and rank remain invariant under Paley–Wiener perturbations, but need not be preserved by compact perturbations. For localized frames, which are frames with additional structure, it is shown that the frame measure function is also preserved by Paley–Wiener perturbations
Continuous K-g-fusion frames in Hilbert spaces
This paper aims at introducing the concept of c-K-g-fusion frames, which are generalizations of K-g-fusion frames, proving some new results on c-K-g-fusion frames in Hilbert spaces, defining duality of c-K-g-fusion frames and characterizing the kinds of the duals, and discussing the perturbation of c-K-g-fusion frames.Publisher's Versio
Operator-Valued Frames for the Heisenberg Group
A classical result of Duffin and Schaeffer gives conditions under which a
discrete collection of characters on , restricted to , forms a Hilbert-space frame for . For the case of characters
with period one, this is just the Poisson Summation Formula. Duffin and
Schaeffer show that perturbations preserve the frame condition in this case.
This paper gives analogous results for the real Heisenberg group , where
frames are replaced by operator-valued frames. The Selberg Trace Formula is
used to show that perturbations of the orthogonal case continue to behave as
operator-valued frames. This technique enables the construction of
decompositions of elements of for suitable subsets of in
terms of representations of
Multipliers for p-Bessel sequences in Banach spaces
Multipliers have been recently introduced as operators for Bessel sequences
and frames in Hilbert spaces. These operators are defined by a fixed
multiplication pattern (the symbol) which is inserted between the analysis and
synthesis operators. In this paper, we will generalize the concept of Bessel
multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be
shown that bounded symbols lead to bounded operators. Symbols converging to
zero induce compact operators. Furthermore, we will give sufficient conditions
for multipliers to be nuclear operators. Finally, we will show the continuous
dependency of the multipliers on their parameters.Comment: 17 page
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