53 research outputs found
A characterization of Schauder frames which are near-Schauder bases
A basic problem of interest in connection with the study of Schauder frames
in Banach spaces is that of characterizing those Schauder frames which can
essentially be regarded as Schauder bases. In this paper, we give a solution to
this problem using the notion of the minimal-associated sequence spaces and the
minimal-associated reconstruction operators for Schauder frames. We prove that
a Schauder frame is a near-Schauder basis if and only if the kernel of the
minimal-associated reconstruction operator contains no copy of . In
particular, a Schauder frame of a Banach space with no copy of is a
near-Schauder basis if and only if the minimal-associated sequence space
contains no copy of . In these cases, the minimal-associated
reconstruction operator has a finite dimensional kernel and the dimension of
the kernel is exactly the excess of the near-Schauder basis. Using these
results, we make related applications on Besselian frames and near-Riesz bases.Comment: 12 page
The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis
The Paulsen problem is a basic open problem in operator theory: Given vectors
that are -nearly satisfying the
Parseval's condition and the equal norm condition, is it close to a set of
vectors that exactly satisfy the Parseval's
condition and the equal norm condition? Given , the squared
distance (to the set of exact solutions) is defined as where the infimum is over the set of exact solutions.
Previous results show that the squared distance of any -nearly
solution is at most and there are
-nearly solutions with squared distance at least .
The fundamental open question is whether the squared distance can be
independent of the number of vectors .
We answer this question affirmatively by proving that the squared distance of
any -nearly solution is . Our approach is based
on a continuous version of the operator scaling algorithm and consists of two
parts. First, we define a dynamical system based on operator scaling and use it
to prove that the squared distance of any -nearly solution is . Then, we show that by randomly perturbing the input vectors, the
dynamical system will converge faster and the squared distance of an
-nearly solution is when is large enough
and is small enough. To analyze the convergence of the dynamical
system, we develop some new techniques in lower bounding the operator capacity,
a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor
changes in various place
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
A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators
This chapter offers a detailed survey on intrinsically localized frames and
the corresponding matrix representation of operators. We re-investigate the
properties of localized frames and the associated Banach spaces in full detail.
We investigate the representation of operators using localized frames in a
Galerkin-type scheme. We show how the boundedness and the invertibility of
matrices and operators are linked and give some sufficient and necessary
conditions for the boundedness of operators between the associated Banach
spaces.Comment: 32 page
Frame Theory for Signal Processing in Psychoacoustics
This review chapter aims to strengthen the link between frame theory and
signal processing tasks in psychoacoustics. On the one side, the basic concepts
of frame theory are presented and some proofs are provided to explain those
concepts in some detail. The goal is to reveal to hearing scientists how this
mathematical theory could be relevant for their research. In particular, we
focus on frame theory in a filter bank approach, which is probably the most
relevant view-point for audio signal processing. On the other side, basic
psychoacoustic concepts are presented to stimulate mathematicians to apply
their knowledge in this field
Probabilistic frames: An overview
Finite frames can be viewed as mass points distributed in -dimensional
Euclidean space. As such they form a subclass of a larger and rich class of
probability measures that we call probabilistic frames. We derive the basic
properties of probabilistic frames, and we characterize one of their subclasses
in terms of minimizers of some appropriate potential function. In addition, we
survey a range of areas where probabilistic frames, albeit, under different
names, appear. These areas include directional statistics, the geometry of
convex bodies, and the theory of t-designs
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