85 research outputs found
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
Symmetric Contours and Convergent Interpolation
The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as
applied to the multipoint Pad\'e approximants is the fact that given a germ of
an algebraic function and a sequence of rational interpolants with free poles
of the germ, if there exists a contour that is "symmetric" with respect to the
interpolation scheme, does not separate the plane, and in the complement of
which the germ has a single-valued continuation with non-identically zero jump
across the contour, then the interpolants converge to that continuation in
logarithmic capacity in the complement of the contour. The existence of such a
contour is not guaranteed. In this work we do construct a class of pairs
interpolation scheme/symmetric contour with the help of hyperelliptic Riemann
surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author.
We consider rational interpolants with free poles of Cauchy transforms of
non-vanishing complex densities on such contours under mild smoothness
assumptions on the density. We utilize -extension of the
Riemann-Hilbert technique to obtain formulae of strong asymptotics for the
error of interpolation
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
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