305 research outputs found
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 . 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
with -boundary data
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 -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
Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in 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
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 with . The paper seems to
be the first to deal with the case . 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 boundary data for 2-D isotropic conductivity equations
whose coefficients have logarithm in . In particular these are not
strictly elliptic. Our results depend on a new multiplier theorem for
-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
Constrained -approximation by polynomials on subsets of the circle
We study best approximation to a given function, in the least square sense on
a subset of the unit circle, by polynomials of given degree which are pointwise
bounded on the complementary subset. We show that the solution to this problem,
as the degree goes large, converges to the solution of a bounded extremal
problem for analytic functions which is instrumental in system identification.
We provide a numerical example on real data from a hyperfrequency filter
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
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
. 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
Robust identification from band-limited data
Consider the problem of identifying a scalar bounded-input/bounded-output stable transfer function from pointwise measurements at frequencies within a bandwidth. We propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity, the noise level to zero, and asymptotically meeting some gauge constraint outside. Error bounds are derived, and the procedure is illustrated by numerical experiment
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