Minimal cubature formulas exact for Haar polynomials

AbstractThe cubature formulas we consider are exact for spaces of Haar polynomials in one or two variables. Among all cubature formulas, being exact for the same class of Haar polynomials, those with a minimal number of nodes are of special interest. We outline here the research and construction of such cubature formulas

On Error Estimates for Weighted Quadrature Formulas Exact for Haar Polynomials

On the spaces Sp, estimates are found for the norm of the error functional of weighted quadrature formulas. For quadrature formulas exact for constants a lower estimate of �δN � ∗ is proved, and for p quadrature formulas possessing the Haar d-property upper estimates of the �δN � ∗ are obtained. pДля весовых квадратурных формул получены оценки нормы функционала погрешности на пространствах Sp — нижняя оценка величины �δN � ∗ для формул, точных на константах, и верх p ние оценки �δN � ∗ для формул, обладающих d-свойством Хаара.

Cubature rules from Hall-Littlewood polynomials

Discrete orthogonality relations for Hall-Littlewood polynomials are employed, so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group $SU(n;\mathbb{C})$). By passing to Macdonald's hyperoctahedral Hall-Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group $Sp (n;\mathbb{H})$). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to $SU(3;\mathbb{C})$ and $Sp (2;\mathbb{H})$), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively.Comment: 30 pages, 7 table

Signal reconstruction from the magnitude of subspace components

We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of $p$-fusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.Comment: 14 pages, 2 figure

Phase retrieval using random cubatures and fusion frames of positive semidefinite matrices

As a generalization of the standard phase retrieval problem,we seek to reconstruct symmetric rank- 1 matrices from inner products with subclasses of positive semidefinite matrices. For such subclasses, we introduce random cubatures for spaces of multivariate polynomials based on moment conditions. The inner products with samples from sufficiently strong random cubatures allow the reconstruction of symmetric rank- 1 matrices with a decent probability by solving the feasibility problem of a semidefinite program