Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L2{L}^2 of the Circle

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L^2-sense on the unit circle, to functions f, where f is a sum of a Cauchy transform of a complex measure \mu supported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of \mu.Comment: 28 page

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Pade interpolants, from which convergence follows.Comment: 42 pages, 3 figure

Pseudo-holomorphic functions at the critical exponent

We study Hardy classes on the disk associated to the equation \bar\d w=\alpha\bar w for $\alpha\in L^r$ with $2\leq r<\infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.Comment: 43 pages; to appear in the Journal of the European Mathematical Societ

Constrained extremal problems in the Hardy space H2 and Carleman's formulas

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the bounded extremal problem

Constrained optimization in classes of analytic functions with prescribed pointwise values

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart \& Leblond, 1998]. In the present work, we extend such a formulation to the case where the additional interpolation conditions are imposed. We also obtain some new results that can be applied to the original problem: we carry out stability analysis and propose a novel method of evaluation of the approximation and blow-up rates of the solution in terms of a Lagrange parameter leading to a highly-efficient computational algorithm for solving the problem

Uniqueness results for inverse Robin problems with bounded coefficient

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain \Omega\subset\RR^n, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, r\textgreater{}n. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context

Estimates in the Hardy-Sobolev space of the annulus and stability result

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k, \infty}; k \in {\mathbb{N}}^*$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner: On the recovery of functions from pointwise boundary values in a Hardy-sobolev class of the disk. J.Comput.Apll.Math 46(1993), 255-69 and by S. Chaabane and I. Feki: Logarithmic stability estimates in Hardy-Sobolev spaces $H^{k,\infty}$. C.R. Acad. Sci. Paris, Ser. I 347(2009), 1001-1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.Comment: 14 pages. To be published in Czechoslovak Mathematical Journa

Hardy spaces of the conjugate Beltrami equation

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation ${div}(\sigma\nabla u)=0$ with $L^p$-boundary data