43 research outputs found
Growth estimates in the Hardy–Sobolev space of an annular domain with applications
AbstractWe give an explicit estimate on the growth of functions in the Hardy–Sobolev space Hk,2(Gs) of an annulus. We apply this result, first, to find an upper bound on the rate of convergence of a recovery interpolation scheme in H1,2(Gs) with points located on the outer boundary of Gs. We also apply this result for the study of a geometric inverse problem, namely we derive an explicit upper bound on the area of an unknown cavity in a bounded planar domain from the difference of two electrostatic potentials measured on the boundary, when the cavity is present and when it is not
Boundary value problems and Heisenberg uniqueness pairs
We describe a general method for constructing Heisenberg uniqueness pairs
in the euclidean space based on the study
of boundary value problems for partial differential equations. As a result, we
show, for instance, that any pair made of the boundary of a bounded
convex set and a sphere is an Heisenberg uniqueness pair if
and only if the square of the radius of is not an eigenvalue of the
Laplacian on . The main ingredients for the proofs are the Paley-Wiener
theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial
boundary value problem, the continuity of single layer potentials, and some
complex analysis in . Denjoy's theorem on topological conjugacy
of circle diffeomorphisms with irrational rotation numbers is also useful
Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach
We investigate the asymptotic behavior of the polynomials p, q, r of degrees
n in type I Hermite-Pade approximation to the exponential function, defined by
p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are
characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We
use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to
obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz),
and r(3nz) in every domain in the complex plane. An important role is played by
a three-sheeted Riemann surface and certain measures and functions derived from
it. Our work complements recent results of Herbert Stahl.Comment: 60 pages, 13 figure
An extremal problem for the Bergman kernel of orthogonal polynomials
Let be a curve of class . For
in the unbounded component of , and for
, let be a probability measure with supp which minimizes the Bergman function
at among all
probability measures on (here,
are an orthonormal basis in for
the holomorphic polynomials of degree at most ). We show that
tends weak-* to , the balayage of the point
mass at onto , by relating this to an optimization problem for
probability measures on the unit circle. Our proof makes use of estimates for
Faber polynomials associated to .Comment: To appear in Constructive Approximatio