R.B Barrar

On the existence of optimal integration formulas for analytic functions

SummaryIn this paper we prove the existence of quadrature formulas that are optimal with respect to a Hilbert space of analytic functions solving a problem unsuccessfully attacked so far. Although we allow the formulas to be of type (2) the optimal formulas will be of the form (1).

CONVERGENCE OF THE VON ZEIPEL PROCEDURE

In the last fifteen years notable progress has been made in celestial mechanics, centered around the contributions of Siegel, Kolmogorov, Moser and Arnold. Perhaps the key paper in a whole series of brilliant papers by these distinguished mathematicians was the four page announcement by Kolmogorov [5]. Moser and Arnold generalized Kolmogorovs theorems, and extended them in various fashion with appropriate proofs, but no proof has been given following Kolmogorovs original outline. The present paper is devoted to doing this. Moreover, it is felt that as an introduction to this chain of ideas, the Kolmogorov approach is the least complicated. Before proceeding, let us give a brief history of the problem under consideration. Given a Hamiltonian:

Oscillating Tchebycheff systems

Abstract As is well known the Tchebycheff polynomial of degree n minimizes the sup norm over all monic polynomials with n simple zeros in [−1,+1). B. D. Bojanov [J. Approx. Theory, 26 (1979) , 293–300] recently investigated the situation for polynomials with a full set of zeros of higher multiplicities. In this paper we generalize these results to extended complete Tchebycheff systems.

Generalized polynomials of minimal norm

We consider the problem of finding optimal generalized polynomials of minimal Lp norm (1 ⩽ p < ∞). As an application of our results we obtain Gaussian quadrature formulas for extended Tchebycheff systems.

A necessary condition for controlled approximation

A necessary condition is given on the Fourier transforms of a finite family of functions in Rsso that the finite family and their translates will approximate an arbitrary function within certain precision.

On the Non-Existence of Transformations to Normal form in Celestial Mechanics

Kolmogorov (1954) has shown that under rather general conditions a Hamiltonian of the form:

Multiple node splines with boundary conditions: The fundamental theorem of algebra for monosplines and Gaussian quadrature formulae for splines

H=\lambda_{1} p_{1}+\lambda_{2} p_{2}+A_{0}\left (q_{1},q_{2}\right)+A_{1}\left ( q_{1},q_{2} \right )p_{1}+A_{2}\left ( q_{1}, q_{2}\right)p_{2}+\sum_{i+j=2}^{\infty} A_{ij}\left ( q_{1},q_{2} \right )p_{1}^{i}p_{2}^{j}

On a nonlinear characterization problem for monosplines

can be reduced to the form: