How fast do radial basis function interpolants of analytic functions converge?

The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analyzed. The starting point is a complex variable contour integral formula for the remainder in RBF interpolation. We study a generalized Runge phenomenon and explore how the location of centers and affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip $|Im(z)| < (1/2\epsilon)$, where $\epsilon$ is the shape parameter, converge exponentially

Using global interpolation to evaluate the Biot-Savart integral for deformable elliptical Gaussian vortex elements

This paper introduces a new method for approximating the Biot-Savart integral for elliptical Gaussian functions using high-order interpolation and compares it to an existing method based on small aspect ratio asymptotics. The new evaluation technique uses polynomials to approximate the kernel corresponding to the integral representation of the streamfunction. We determine the polynomial coefficients by interpolating precomputed values from look-up tables over a wide range of aspect ratios. When implemented in a full nonlinear vortex method, we find that the new technique is almost three times faster and unlike the asymptotic method, provides uniform accuracy over the full range of aspect ratios. As a proof-of-concept for large scale computations, we use the new technique to calculate inviscid axisymmetrization and filamentation of a two-dimensional elliptical fluid vortex. We compare our results with those from a pseudo-spectral computation and from electron vortex experiments, and find good agreement between the three approaches

Piecewise smooth chebfuns

Algorithms are described that make it possible to manipulate piecewise-smooth functions on real intervals numerically with close to machine precision. Breakpoints are introduced in some such calculations at points determined by numerical rootfinding, and in others by recursive subdivision or automatic edge detection. Functions are represented on each smooth subinterval by Chebyshev series or interpolants. The algorithms are implemented in object-oriented MATLAB in an extension of the chebfun system, which was previously limited to smooth functions on [-1, 1]

Freedom of Association and State Regulation of Delegate Selection: Potential for Conflict at the 1984 Democratic National Convention

This Note begins with a discussion of the history of the regulation of state parties by state law and national party rules. The Note then traces the development of case law concerning state regulation of party delegate selection procedures. Finally, the Note explores the potential for credentials disputes and litigation on the primacy of state party rules over contrary state laws if both the party rules and the state regulations comply with the Delegate Selection Rules for the 1984 Democratic National Convention. The Note concludes that the first amendment right of freedom of association guarantees that a state party may select delegates to the national convention in any manner consistent with the national party rules

Recent Development

While few commentators question the international status of the Community in relation to the Member States and those countries with which it has negotiated treaties, the question of whether the Common Market possesses a universally recognizable personality remains open. In determining the international status of the United Nations, the ICJ in the Reparations Case stated that fifty states, representing the vast majority of the members of the international community, had the power in conformity with international law to bring into being an entity possessing an objective international personality and not only personality recognized by them alone. If this recognition standard determines when an entity has international legal personality, then the increasing membership in the Common Market may indicate that it has objective international personality. Furthermore, over fifty states have implicitly recognized the EEC by entering into commercial negotiations with it or sending diplomatic missions to it

Recovering piecewise smooth functions from nonuniform Fourier measurements

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree