Abstract: Publisher Summary This chapter discusses the approximation of multiple integrals by using interpolatory cubature formula. It presents a survey of the theory of interpolatory cubature formulae that has been developed after 1970. In the chapter the following are considered: lower bounds for the number of knots of cubature formula, which are exact for polynomials of some fixed degree, the connection between orthogonal polynomials and cubatureformulae, the method of reproducing kernels, and invariant formula. The extension of cubature formulae of Gaussian type for the multivariate case is important in the theory of interpolatory cubature formula. If two polynomials of degree k orthogonal with respect to Ω and weight-function p (x, y) have exactly k2 roots in common, finite and distinct, then these roots can be taken as knots of a cubature formula for an integral on Ω with the weight-function p (x, y). This formula is exact for all polynomials of degree not higher than 2k - 1.

Abstract: Results in the approximation of functions by polynomials with coefficients which are integers have been appearing since that of Pal in 1914. The body of results has grown to an extent which seems to justify this book. The intention here is to make these results as accessible as possible. The book addresses essentially two questions. The first is the question of what functions can be approximated by polynomials whose coefficients are integers and the second question is how well are they approximated (Jackson type theorems). For example, a continuous function $f$ on the interval $-1,1$ can be uniformly approximated by polynomials with integral coefficients if and only if it takes on integral values at $-1,0$ and $+1$ and the quantity $f(1)+f(0)$ is divisible by $2$.The results regarding the second question are very similar to the corresponding results regarding approximation by polynomials with arbitrary coefficients. In particular, nonuniform estimates in terms of the modules of continuity of the approximated function are obtained. Aside from the intrinsic interest to the pure mathematician, there is the likelihood of important applications to other areas of mathematics; for example, in the simulation of transcendental functions on computers.In most computers, fixed point arithmetic is faster than floating point arithmetic and it may be possible to take advantage of this fact in the evaluation of integral polynomials to create more efficient simulations. Another promising area for applications of this research is in the design of digital filters. A central step in the design procedure is the approximation of a desired system function by a polynomial or rational function. Since only finitely many binary digits of accuracy actually can be realized for the coefficients of these functions in any real filter the problem amounts (to within a scale factor) to approximation by polynomials or rational functions with integral coefficients.

Abstract: Using an integral inequality, contributions are made towards the solutions of two long open problems. The first one concerns the determination of the best constant C(p) in Bernstein's inequality [H~P dt < C(p) nP IH.(t) JPdt 0 < p <1 where Ho is a trigonometric polynomial of degree n. We prove C(p) < 11, while previously C(p) < 8/p was known. The second one concerns orthogonal polynomials p"(da) corresponding to positive measures da defined on [-1, 11. We prove that

Abstract: This paper constructs an example of a weight function on the interval such that 0$ SRC=http://ej.iop.org/images/0025-5734/36/4/A07/tex_sm_1864_img3.gif/>, , whereas the corresponding sequence of orthonormal polynomials is unbounded at . Bibliography: 6 titles.

Abstract: Let be the system of polynomials orthonormal on the unit circumference with respect to the measure . By way of generalizing and strengthening a number of previous results, we show that if , continuous and positive on , and , then the polynomials converge uniformly in , inside , to the Szego function. The result so formulated is shown to be definitive. Bibligraphy: 16 titles.

Abstract: For each infinite family of Chevalley groups over a finite field an Erdos–Ko–Rado theorem is given. The technique uses orthogonal polynomials to find upper bounds for the independence number of specific graphs. In all but one case the bound is realizable.

Abstract: It is shown that the Kronrod extension to the n-point Gauss integration rule, with respect to the weight function (1 x2)M-1/2, 0 < ,u < 2, ,u $ 1, is of exact precision 3n + 1 for n even and 3n + 2 for n odd. Similarly, for the (n+1)-point Lobatto rule, with -1/2 < , 6 1, ,u # 0, the exact precision is 3n for n odd and 3n + 1 for n even.

Abstract: The object of this paper is to present a systematic introduction to and several interesting applications of a general method of obtaining bilinear, bilateral or mixed multilateral generating functions for a fairly wide variety of special functions in one, two and more variables. The main results, contained in Theorems 2 and 3 below, are shown to apply not only to the Bessel polynomials, the classical orthogonal polynomials including, for example, Hermite, Jacobi (and, of course, Gegenbauer, Legendre, and Tchebycheff), and Laguerre polynomials, and to their various generalizations studied in recent years, but indeed also to such other special functions as the Bessel functions, a class of generalized hypergeometric functions, the Lauricella polynomials in several variables, and the familiar Lagrange polynomials which arise in certain problems in statistics. It is also indicated how these general results are related to a number of known results scattered in the literature.

Abstract: A counterexample is given to an assertion by K. M. Case that if the coefficients in the three-term recurrence formula for orthogonal polynomials converge as fast as $n^{ - 2} $, the corresponding distribution function has only finitely many discrete points in its spectrum. Some positive results concerning this situation are also given, and continuity of the distribution function is investigated.

Abstract: In this paper we examine the existence of polynomials orthogonal with respect to weight functions which are given by the product of a nonnegative (on the interval of orthogonality) function and a polynomial. In particular, we give some results when the polynomial factor is orthogonal with respect to the remaining (nonnegative) part of the weight function.

Abstract: Work on orthogonal polynomials by Tatian has been incorporated into a computer program for interferogram analysis. For obscured-aperture optical systems, the data reduction is far more accurate than with programs based only on Zernike polynomials. Results are shown for spherical aberration, coma, and astigmatism.

Abstract: Publisher Summary This chapter discusses the interpolatory cubature formulae and real ideals. It is a follow-up to I. P. Mysovskikh's survey on interpolatory cubature formulae. Real ideals satisfying some additional properties are necessary and sufficient for the existence of interpolatory cubature formulae. As a consequence, properties known from the one-dimensional case can be regained in several dimensions. For instance, the knots of an interpolatory cubature formula are the pairwise distinct common zeros of certain orthogonal polynomials. These polynomials have only common simple real zeros. In the two-dimensional case the number of knots attains the lower bounds.

Abstract: First the notion of a digital filter is briefly introduced. Next the part of the design process for such filters where approximation by polynomials with integral coefficients enters is outlined. Finally a result is stated and proved, which reduces the determination of the error involved in approximating a function by a polynomial whose coefficients are integers to a finite number of calculations.

Abstract: Product formulas for general q-Hahn polynomials are derived from counting arguments involving subspaces of a finite vector space.

Abstract: Model reduction of linear, time-invariant, single-input, single-output systems over desired frequency intervals (low-pass, band-pass and high-pass) is considered in this paper. Order reduction is effected by manipulating two orthogonal polynomial series, one representing the high-order system and the other representing the approximating low-order model. The method is a generalization of the classical Pade approximations, however, by a special transformation it becomes the simple (classical) Pade problem, thus retaining the computational attractiveness of the latter.