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 $i$'th node is invertible if and only if all eigenvectors are nonzero on the $i$'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

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 $\mathcal{C}^{1}$ 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

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 $M{p, 1}_{0,s}$.Comment: 11 page

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

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 $m$-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