Oved Shisha

Variations on the Chebyshev and Lq theories of best approximation

Abstract A new method of approximation is proposed which maintains many of the essentials of the classical theory of best uniform approximation, while also using an L q -type measure of approximation.

Improper Integrals, Simple Integrals, and Numerical Quadrature

x is defined in the simplest case as a limit of proper Riemann integrals, which are themselves limits of Riemann sums. In this paper we discuss the representation of the improper integral as a limit of Riemann sums. Our main result-Theorem 3-gives a condition on the integrand that is necessary and sufficient for such represen- tation of the integral, with the largest natural class of Riemann sums. Some of the motivation for this paper comes from the theory of numerical integration. Most formulas for numerical quadrature-Simpson’s rule, the trapezoid rule, and the Gauss-Legendre formulas, for example -approximate the integral by calculating carefully chosen Riemann sums.* Thus it is the Riemann concept of the integral that is most appropriate for numerical analysis. The quadrature rules mentioned converge to the integral whenever the function being integrated is properly Riemann-integrable; there seems to be no larger class of bounded functions for which quadrature rules converge. In the case of the improper Riemann integral, the connection with numeri- cal quadrature is obscured by the double limiting process involved. If we wish to use a sequence of quadrature formulas, or quadrature rule, (with

Numerical solution of a singular integral equation in random rough surface scattering theory

Abstract A one-dimensional singular integral equation which appeared in a previous paper on random rough surface scattering theory ( J. Math. Phys. 13 , 1903 (1972) is solved numerically using quadratic splines. Its solution yields an approximation to the coherent (specular) scattered intensity for plane wave incidence on the surface. This approximate scattered intensity is plotted versus the Rayleigh factor Σ = κ 0 σ cos θ i , where κ i is the wavenumber of the incident plane wave, σ is the surface root mean square height, and θ i is the angle of the incident plane wave. For values of Σ > 1 this approximation yields more coherent intensity than the Kirchhoff approximation.

The generalized Riemann, simple, dominated and improper integrals

A powerful generalization of the Riemann integral has been introduced by making an innocent-looking modification in the usual definition. This generalized Riemann integral was defined in 1957 by Kurzweil [6]. It was independently defined and extensively studied and generalized by Henstock [3-51 who called it the Riemann-complete integral. Although this integral has been popularized somewhat (cf. [7]), it still is not as well known as it deserves to be. Among the virtues of this powerful integral are the following:

Infinite Riemann Sums, the Simple Integral, and the Dominated Integral

Simple integrability of a function f (defined by Haber and Shisha in [2]) is shown to be equivalent to the convergence of the infinite Riemann sum

The genesis of the generalized Riemann integral

\sum\limits_{k = 1}^\infty {f\left( {{\xi _k}} \right)\left( {{x_k} - {x_{k - 1}}} \right)}

Invulnerable queens on an infinite chessboard

to the improper Riemann integral \( \int_0^\infty f \) f as the gauge of the partition \( \left( {{x_k}} \right)_{k = 0}^\infty \) of [0,∞)converges to O. An analogous result is obtained for dominant integrability (defined by Osgood and Shisha in [5]). Also certain results of Bromwich and Hardy [1] are recovered.

Derivative without limit

When k =: 1 (k = 2), Eq. (1) states that each fn is increasing (convex) on (a, b). If k > 2, then [ 1, p. 381](l) implies that for each n,f~-2’ exists and is continuous in (a, b). Theorem I naturally raises the question : Can (I), which expresses convexity of the fn with respect to (tj)F