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Operator representations of frames: boundedness, duality, and stability
The purpose of the paper is to analyze frames
having the form for some linear operator T:
\mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}.
A key result characterizes boundedness of the operator in terms of
shift-invariance of a certain sequence space. One of the consequences is a
characterization of the case where the representation can be achieved for an operator that has an
extension to a bounded bijective operator
In this case we also characterize all the dual frames that are representable in
terms of iterations of an operator in particular we prove that the only
possible operator is Finally, we consider stability
of the representation rather surprisingly, it
turns out that the possibility to represent a frame on this form is sensitive
towards some of the classical perturbation conditions in frame theory. Various
ways of avoiding this problem will be discussed. Throughout the paper the
results will be connected with the operators and function systems appearing in
applied harmonic analysis, as well as with general group representations
Construction of scaling partitions of unity
Partitions of unity in formed by (matrix) scales of a fixed
function appear in many parts of harmonic analysis, e.g., wavelet analysis and
the analysis of Triebel-Lizorkin spaces. We give a simple characterization of
the functions and matrices yielding such a partition of unity. For invertible
expanding matrices, the characterization leads to easy ways of constructing
appropriate functions with attractive properties like high regularity and small
support. We also discuss a class of integral transforms that map functions
having the partition of unity property to functions with the same property. The
one-dimensional version of the transform allows a direct definition of a class
of nonuniform splines with properties that are parallel to those of the
classical B-splines. The results are illustrated with the construction of dual
pairs of wavelet frames
Hilbert space frames containing a Riesz basis and Banach spaces which have no subspace isomorphic to
We prove that a Hilbert space frame \fti contains a Riesz basis if every
subfamily \ftj , J \subseteq I , is a frame for its closed span. Secondly we
give a new characterization of Banach spaces which do not have any subspace
isomorphic to . This result immediately leads to an improvement of a
recent theorem of Holub concerning frames consisting of a Riesz basis plus
finitely many elements
Weyl-Heisenberg frames for subspaces of L^2(R)
We give sufficient conditions for translates and modulates of a function g in
L^2(R) to be a frame for its closed linear span. Even in the case where this
family spans all of L^2(R), wou conditions are significantly weaker than the
previous known conditions.Comment: 13 page
Explicit constructions and properties of generalized shift-invariant systems in
Generalized shift-invariant (GSI) systems, originally introduced by
Hern\'andez, Labate & Weiss and Ron & Shen, provide a common frame work for
analysis of Gabor systems, wavelet systems, wave packet systems, and other
types of structured function systems. In this paper we analyze three important
aspects of such systems. First, in contrast to the known cases of Gabor frames
and wavelet frames, we show that for a GSI system forming a frame, the
Calder\'on sum is not necessarily bounded by the lower frame bound. We identify
a technical condition implying that the Calder\'on sum is bounded by the lower
frame bound and show that under a weak assumption the condition is equivalent
with the local integrability condition introduced by Hern\'andez et al. Second,
we provide explicit and general constructions of frames and dual pairs of
frames having the GSI-structure. In particular, the setup applies to wave
packet systems and in contrast to the constructions in the literature, these
constructions are not based on characteristic functions in the Fourier domain.
Third, our results provide insight into the local integrability condition
(LIC).Comment: Adv. Comput. Math., to appea
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