129 research outputs found
Frames, semi-frames, and Hilbert scales
Given a total sequence in a Hilbert space, we speak of an upper (resp. lower)
semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently,
for an upper semi-frame, the frame operator is bounded, but has an unbounded
inverse, whereas a lower semi-frame has an unbounded frame operator, with
bounded inverse. For upper semi-frames, in the discrete and the continuous
case, we build two natural Hilbert scales which may yield a novel
characterization of certain function spaces of interest in signal processing.
We present some examples and, in addition, some results concerning the duality
between lower and upper semi-frames, as well as some generalizations, including
fusion semi-frames and Banach semi-frames.Comment: 27 pages; Numerical Functional Analysis and Optimization, 33 (2012)
in press. arXiv admin note: substantial text overlap with arXiv:1101.285
Unconditional convergence and invertibility of multipliers
In the present paper the unconditional convergence and the invertibility of
multipliers is investigated. Multipliers are operators created by (frame-like)
analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or
necessary conditions for unconditional convergence and invertibility are
determined depending on the properties of the analysis and synthesis sequences,
as well as the symbol. Examples which show that the given assertions cover
different classes of multipliers are given. If a multiplier is invertible, a
formula for the inverse operator is determined. The case when one of the
sequences is a Riesz basis is completely characterized.Comment: 31 pages; changes to previous version: 1.) the results from the
previous version are extended to the case of complex symbols m. 2.) new
statements about the unconditional convergence and boundedness are added
(3.1,3.2 and 3.3). 3.) the proof of a preliminary result (Prop. 2.2) was
moved to a conference proceedings [29]. 4.) Theorem 4.10. became more
detaile
Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing
For the solution of operator equations, Stevenson introduced a definition of
frames, where a Hilbert space and its dual are {\em not} identified. This means
that the Riesz isomorphism is not used as an identification, which, for
example, does not make sense for the Sobolev spaces and
. In this article, we are going to revisit the concept of
Stevenson frames and introduce it for Banach spaces. This is equivalent to
-Banach frames. It is known that, if such a system exists, by defining
a new inner product and using the Riesz isomorphism, the Banach space is
isomorphic to a Hilbert space. In this article, we deal with the contrasting
setting, where and are not identified, and
equivalent norms are distinguished, and show that in this setting the
investigation of -Banach frames make sense.Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis
and Optimization
Weighted frames, weighted lower semi frames and unconditionally convergent multipliers
In this paper we ask when it is possible to transform a given sequence into a
frame or a lower semi frame by multiplying the elements by numbers. In other
words, we ask when a given sequence is a weighted frame or a weighted lower
semi frame and for each case we formulate a conjecture. We determine several
conditions under which these conjectures are true. Finally, we prove an
equivalence between two older conjectures, the first one being that any
unconditionally convergent multiplier can be written as a multiplier of Bessel
sequences by shifting of weights, and the second one that every unconditionally
convergent multiplier which is invertible can be written as a multiplier of
frames by shifting of weights. We also show that these conjectures are also
related to one of the newly posed conjectures.Comment: 14 page
-dual Frames in Hilbert -module Spaces
In this paper, we introduce the concept of -dual frames for Hilbert -modules, and then the properties and stability results of -dual frames are given. A characterization of -dual frames, approximately dual frames and dual frames of a given frame is established. We also give some examples to show that the characterization of -dual frames for Riesz bases in Hilbert spaces is not satisfied in general Hilbert -modules
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