132 research outputs found
Minimal sequences and the Kadison-Singer problem
The Kadison-Singer problem asks: does every pure state on the diagonal
sublgebra of the C*-algebra of bounded operators on a separable infinite
dimensional Hilbert space admit a unique extension? A yes answer is equivalent
to several open conjectures including Feichtinger's: every bounded frame is a
finite union of Riesz sequences. We consider the special case: Feichtinger's
conjecture for exponentials and prove that the set of projections onto a
measurable subset of the circle group of the set of exponential functions
equals a union of a finite number of Reisz sequences if and only if there
exists a Reisz subsequence corresponding to integers whose characteristic
function is a nonzero minimal sequence.Comment: 10 pages, Theorem 1.1 was announced during conferences in St.
Petersburg, Russia, June 14-20, 2009, and in Kuala Lumpur, Malaysia, June
22-26, 200
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
Measure Functions for Frames
This paper addresses the natural question: ``How should frames be compared?''
We answer this question by quantifying the overcompleteness of all frames with
the same index set. We introduce the concept of a frame measure function: a
function which maps each frame to a continuous function. The comparison of
these functions induces an equivalence and partial order that allows for a
meaningful comparison of frames indexed by the same set. We define the
ultrafilter measure function, an explicit frame measure function that we show
is contained both algebraically and topologically inside all frame measure
functions. We explore additional properties of frame measure functions, showing
that they are additive on a large class of supersets-- those that come from so
called non-expansive frames. We apply our results to the Gabor setting,
computing the frame measure function of Gabor frames and establishing a new
result about supersets of Gabor frames.Comment: 54 pages, 1 figure; fixed typos, reformatted reference
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