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Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball
This work is a continuation of the recent study by the authors on
approximation theory over the sphere and the ball. The main results define new
Sobolev spaces on these domains and study polynomial approximations for
functions in these spaces, including simultaneous approximation by polynomials
and relation between best approximation to a function and to its derivatives.Comment: 16 page
On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations
We describe a method for calculating the roots of special functions
satisfying second order linear ordinary differential equations. It exploits the
recent observation that the solutions of a large class of such equations can be
represented via nonoscillatory phase functions, even in the high-frequency
regime. Our algorithm achieves near machine precision accuracy and the time
required to compute one root of a solution is independent of the frequency of
oscillations of that solution. Moreover, despite its great generality, our
approach is competitive with specialized, state-of-the-art methods for the
construction of Gaussian quadrature rules of large orders when it used in such
a capacity. The performance of the scheme is illustrated with several numerical
experiments and a Fortran implementation of our algorithm is available at the
author's website
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