Sum Of Dirac Measures Research Articles

Prony’s Method Under an Almost Sharp Multivariate Ingham Inequality

The parameter reconstruction problem in a sum of Dirac measures from its low frequency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham inequality which improves the previously best known dimension-dependent constant from square-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham inequality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function.

Nontopological Condensates for the Self‐Dual Chern‐Simons‐Higgs Model

For the abelian self‐dual Chern‐Simons‐Higgs model we address existence issues of periodic vortex configurations—the so‐called condensates—of nontopological type as k → 0, where k > 0 is the Chern‐Simons parameter. We provide a positive answer to the longstanding problem on the existence of nontopological condensates with magnetic field concentrated at some of the vortex points (as a sum of Dirac measures) as k → 0, a question that is of definite physical interest. © 2015 Wiley Periodicals, Inc.

An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes

AbstractWe propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defect‐laden point set with noise and outliers. We introduce an optimal‐transport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex considered as a sum of uniform measures on 0‐ and 1‐simplices. A fine‐to‐coarse scheme is devised to construct the resulting simplicial complex through greedy decimation of a Delaunay triangulation of the input point set. Our method performs well on a variety of examples ranging from line drawings to grayscale images, with or without noise, features, and boundaries.

Duality in static hedging of barrier options

During the past 10 years static hedging strategies for barrier options have been the focus of various studies. But while the primal optimization problem is sufficiently well understood, it has been unclear so far what the dual of the static hedging problem looks like. This article closes this gap by exploiting a linear semi-infinite reformulation of the static super-replication problem in the Black–Scholes model. As it turns out the feasible set of the dual problem consists of all measures which are price-consistent with the calls in the hedge portfolio–a class which is much bigger than the equivalent martingale measures which form the feasible set of the dual of a dynamic super-replication problem. Moreover, we prove the existence of an optimal measure solving the dual problem which is the finite sum of Dirac measures and analyze the duality result from a financial point of view.