493 research outputs found
Numerical hyperinterpolation over nonstandard planar regions
We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes
Constructing Cubature Formulas of Degree 5 with Few Points
This paper will devote to construct a family of fifth degree cubature
formulae for -cube with symmetric measure and -dimensional spherically
symmetrical region. The formula for -cube contains at most points
and for -dimensional spherically symmetrical region contains only
points. Moreover, the numbers can be reduced to and if
respectively, the later of which is minimal.Comment: 13 page
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 -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
Bivariate Lagrange interpolation at the Padua points: the ideal theory approach
Padua points is a family of points on the square given by explicit
formulas that admits unique Lagrange interpolation by bivariate polynomials.
The interpolation polynomials and cubature formulas based on the Padua points
are studied from an ideal theoretic point of view, which leads to the discovery
of a compact formula for the interpolation polynomials. The convergence
of the interpolation polynomials is also studied.Comment: 11 page
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