Exact Reconstruction using Beurling Minimal Extrapolation

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc). Generalized minimal extrapolation shares related geometric properties with basis pursuit of Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points,can be exactly recovered from only 2s + 1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.Comment: 27 pages, 3 figures version 2 : minor changes and new titl

Efficient algorithm for solving semi-infinite programming problems and their applications to nonuniform filter bank designs

An efficient algorithm for solving semi-infinite programming problems is proposed in this paper. The index set is constructed by adding only one of the most violated points in a refined set of grid points. By applying this algorithm for solving the optimum nonuniform symmetric/antisymmetric linear phase finite-impulse-response (FIR) filter bank design problems, the time required to obtain a globally optimal solution is much reduced compared with that of the previous proposed algorith

Spectrahedral cones generated by rank 1 matrices

[no abstract available

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

AbstractThe main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz–Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the problem of finding the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to undesirably conservative results in many applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the $n^\text{th}$-degree polynomial curve with largest convex hull enclosed in a given $n$-simplex. Then, we present a formulation that is \emph{independent} of the $n$-simplex or $n^{\text{th}}$-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any $n\in\mathbb{N}$ and prove numerical global optimality for $n=1,2,3$. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-degree polynomial curve, the MINVO basis is able to obtain an enclosing simplex whose volume is $2.36$ and $254.9$ times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When $n=7$, these ratios increase to $902.7$ and $2.997\cdot10^{21}$, respectively.Comment: 25 pages, 16 figure

On Nudel'man's Problem and Indefinite Interpolation in the Generalized Schur and Nevanlinna Classes

This work is a revised and corrected version of the authors' joint paper with T. Constantinescu (Trans Am Math Soc 355:813-836, 2003). Some of the theorems from the original paper that are withdrawn are recast as new open problems for indefinite interpolation. Partial results are obtained by other methods, including Kronecker's theorem for Hankel operators

Contractions with rank one defect operators and truncated CMV matrices

AbstractThe main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some Hilbert space of dimension N⩽∞. These models complement nicely the well-known models of Livšic and Sz.-Nagy–Foias. We show that each such operator acting on some finite-dimensional (respectively, separable infinite-dimensional Hilbert space) is unitarily equivalent to some finite (respectively semi-infinite) truncated CMV matrix obtained from the “full” CMV matrix by deleting the first row and the first column, and acting in CN (respectively ℓ2(N)). This result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to Cantero, Moral and Velázquez, as well as an analog for contraction operators of the result from [Yu. Arlinskiĭ, E. Tsekanovskiĭ, Non-self-adjoint Jacobi matrices with a rank-one imaginary part, J. Funct. Anal. 241 (2006) 383–438] concerning dissipative non-self-adjoint operators with a rank one imaginary part. It is shown that another functional model for contractions with rank one defect operators takes the form of the compression f(ζ)→PK(ζf(ζ)) on the Hilbert space L2(T,dμ) with a probability measure μ onto the subspace K=L2(T,dμ)⊖C. The relationship between characteristic functions of sub-matrices of the truncated CMV matrix with rank one defect operators and the corresponding Schur iterates is established. We develop direct and inverse spectral analysis for finite and semi-infinite truncated CMV matrices. In particular, we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving Schur parameters. It is pointed out that if the mixed spectral data contains zero eigenvalue, then no solution, unique solution or infinitely many solutions may occur in the inverse problem for truncated CMV matrices. The uniqueness theorem for recovered truncated CMV matrix from the given mixed spectral data is established. In this part the paper is closely related to the results of Hochstadt and Gesztesy–Simon obtained for finite self-adjoint Jacobi matrices

The Gauss-Green theorem in stratified groups

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.Comment: 69 page