We study inverse problems for the Poisson equation with source term the divergence of an R 3 -valued measure; that is, the potential Φ satisfies ∆Φ = div µ, and µ is to be reconstructed knowing (a component of) the field grad Φ on a set disjoint from the support of µ. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering µ based on total variation regularization. We provide sufficient conditions for the unique recovery of µ, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable.Numerical examples are provided to illustrate the main theoretical results.
A sharp version of the Balian-Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators {f k } K k=1 ⊂ L 2 (R d ) are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V , and if V has extra invariance by a suitable finer lattice, then one of the generators f k must sat-Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space H d/2+ǫ (R d ); our results provide an absolutely sharp improvement with H 1/2 (R d ). Our results are sharp in the sense that H 1/2 (R d ) cannot be replaced by H s (R d ) for any s < 1/2. 2010 Mathematics Subject Classification. Primary 42C15.
This paper complements the recent investigation of [4] on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve L. We study the specific case of L = {z = w−1+(w−1) −1 , |w| = R}, for some R > 2, providing an example that exhibits the new features discovered in [4], and for which the asymptotic behavior of the orthogonal polynomials is established over the entire domain of orthogonality. Surprisingly, this variation of the classical example of the ellipse turns out to be quite sophisticated. After properly normalizing the corresponding orthonormal polynomials pn, n = 0, 1, . . ., and on certain critical subregion of the orthogonality domain, a subsequence {pn k } converges if and only if log µ 4 (n k ) converges modulo 1 (µ being an important quantity associated to L). As a consequence, the limiting points of the sequence {pn} form a one parameter family of functions, the parameter's range being the interval [0, 1). The polynomials pn are much influenced by a certain integrand function, the explained behavior being the result of this integrand having a nonisolated singularity that is a cluster point of poles. The nature of this singularity sparks purely from geometric considerations, as opposed to the more common situation where the critical singularities come from the orthogonality weight.1991 Mathematics Subject Classification. 42C05, 30E10, 30E15.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.