94 research outputs found
On Optimal Frame Conditioners
A (unit norm) frame is scalable if its vectors can be rescaled so as to
result into a tight frame. Tight frames can be considered optimally conditioned
because the condition number of their frame operators is unity. In this paper
we reformulate the scalability problem as a convex optimization question. In
particular, we present examples of various formulations of the problem along
with numerical results obtained by using our methods on randomly generated
frames.Comment: 11 page
Invertibility of graph translation and support of Laplacian Fiedler vectors
The graph Laplacian operator is widely studied in spectral graph theory
largely due to its importance in modern data analysis. Recently, the Fourier
transform and other time-frequency operators have been defined on graphs using
Laplacian eigenvalues and eigenvectors. We extend these results and prove that
the translation operator to the 'th node is invertible if and only if all
eigenvectors are nonzero on the 'th node. Because of this dependency on the
support of eigenvectors we study the characteristic set of Laplacian
eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish
on large neighborhoods and then explicitly construct a family of non-planar
graphs that do exhibit this property.Comment: 21 pages, 7 figure
Local well-posedness of nonlinear dispersive equations on modulation spaces
By using tools of time-frequency analysis, we obtain some improved local
well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in
modulation spaces .Comment: 11 page
Scalable Frames and Convex Geometry
The recently introduced and characterized scalable frames can be considered
as those frames which allow for perfect preconditioning in the sense that the
frame vectors can be rescaled to yield a tight frame. In this paper we define
-scalability, a refinement of scalability based on the number of non-zero
weights used in the rescaling process, and study the connection between this
notion and elements from convex geometry. Finally, we provide results on the
topology of scalable frames. In particular, we prove that the set of scalable
frames with "small" redundancy is nowhere dense in the set of frames.Comment: 14 pages, to appear in Contemporary Mat
A Beurling-Helson type theorem for modulation spaces
We prove a Beurling-Helson type theorem on modulation spaces. More precisely,
we show that the only changes of variables that leave
invariant the modulation spaces \M{p,q}(\rd) are affine functions on \rd. A
special case of our result involving the Sj\"ostrand algebra was considered
earlier by A. Boulkhemair
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