4,921 research outputs found
Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas
In the present paper we introduce a new concept of Hardy type space naturally
defined on the Klein-Dirac quadric. We study different properties of the
functions belonging to these spaces, in particular boundary value problems. We
apply these new spaces to polyharmonic interpolation and to interpolatory
cubature formulas.Comment: 32 page
Is Gauss quadrature better than Clenshaw-Curtis?
We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at . Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at is only half as high, but which is nevertheless equally accurate near
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
Discrete spherical means of directional derivatives and Veronese maps
We describe and study geometric properties of discrete circular and spherical
means of directional derivatives of functions, as well as discrete
approximations of higher order differential operators. For an arbitrary
dimension we present a general construction for obtaining discrete spherical
means of directional derivatives. The construction is based on using the
Minkowski's existence theorem and Veronese maps. Approximating the directional
derivatives by appropriate finite differences allows one to obtain finite
difference operators with good rotation invariance properties. In particular,
we use discrete circular and spherical means to derive discrete approximations
of various linear and nonlinear first- and second-order differential operators,
including discrete Laplacians. A practical potential of our approach is
demonstrated by considering applications to nonlinear filtering of digital
images and surface curvature estimation
q-Identities from Lagrange and Newton Interpolation
Combining Newton and Lagrange interpolation, we give -identities which
generalize results of Van Hamme, Uchimura, Dilcher and Prodinger
An -regularity result with mean curvature control for Willmore immersions and application to minimal bubbling
In this paper we prove a convergence result for sequences of Willmore
immersions with simple minimal bubbles. To this end we replace the total
curvature control in T. Rivi\`ere's proof of the -regularity for
Willmore immersions by a control of the local Willmore energy.Comment: 42 pages, 2 figure
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