Abstract: Four. Functions of One Complex Variable. Special Part.- Five. The Location of Zeros.- Six. Polynomials and Trigonometric Polynomials.- Seven. Determinants and Quadratic Forms.- Eight. Number Theory.- Nine. Geometric Problems.- x 1 Additional Problems to Part One.- New Problems in English Edition.- Author Index.- Topics.

Abstract: : This book contains the proceedings of a Symposium on Approximation Theory which was held in Austin on January 18-21, 1976 The program of the symposium included seven one-hour invited expository lectures, more than forty one-half-hour invited research talks, and a large number of contributed fifteen minute talks The expository papers describe recent developments in some important special fields For example, D Braess discusses nonlinear approximation, C de Boor surveys the central role of B-splines in the general theory of splines, R DeVore discusses the degree of approximation by polynomials, trigonometric polynomials, and splines, obtaining the results from a relation between the moduli of smoothness and Peetre's K-functionals, A G Vituskin has a new point of view upon entropy-he indicates its applications to coding, G P Nevai describes recent results (by himself, G Freud and others) in the theory of convergence of interpolation formulas and of orthogonal polynomials, C K Chui surveys Pade approximations of arbitrary analytic functions, and L L Schumaker discusses fitting surfaces

Abstract: The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA.** Some machine examples with timing are included.

Abstract: The potential distribution and the wave propagation in a horizontally stratified earth is considered and the analogy of the mathematical expression for seismic transfer function, electromagnetic and electric kernel functions, and magnetotelluric input impedance is discussed. Although these specific functions are conveniently treated by a separate expression in each method, it is indicated that the function for seismic and electromagnetic methods is mathematically the same with a change in the physical meaning of the variables from one method to the other. Similarly, the identity of the mathematical expressions of the resistivity kernel function and magnetotelluric input impedance is noticed. In each method a specific geophysical function depends on the thickness and the physical properties of the various layers. Every specific function involves two interdependent fundamental functions, that is Pn and Qn, or Pn and P*n, having different physical meaning for different methods. Specific functions are expressible as a ratio Pn/Qn or P*n/Pn. Fundamental functions may be reduced to polynomials. The fundamental polynomials Q*n and P*n describing the horizontally stratified media are a system of polynomials orthogonal on the unit circle, of first and second order, respectively. The interpretation of geophysical problems corresponds to the identification of the parameters of a system of fundamental orthogonal polynomials. The theorems of orthogonal polynomials are applied to the solution of identification problems. A formula for calculating theoretical curves and direct resistivity interpretation is proposed for the case of arbitrary resistivity of the substratum. The basic equation for synthetic seismograms is reformulated in appendix A. In appendix B a method is indicated for the conversion of the seismic transfer function from arbitrary to perfectly reflective substratum.

Abstract: The Q-matrix technique of bilinear transformation of a single-variable polynomial is extended to multivariable polynomials. A computer program for the transformation is included in the Appendix.

Abstract: A method of constructing 28-point, 26-point and 25-point cubature formulas with polynomial precision 11 is given for planar regions and weight functions, which are symmetric in each variable. The nodes are computed as common zeros of a set of linearly independent orthogonal polynomials.

Abstract: The oldest distribution is that defined by the Bernoulli polynomials, although of course their classical recurrence property was not called by that name.

Abstract: (1960). Some Inequalities for Polynomials. The American Mathematical Monthly: Vol. 67, No. 9, pp. 847-851.

Abstract: Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lame differential equation. The problem of determining the location of the zeros of such polynomials has been studied under quite general conditions by Marden. He has obtained (see Trans. Amer. Math. Soc. 33 (1931), 934944) varied generalizations of certain results proved earlier by Stieltjes, Van Vleck, B6cher, Klein, and P6lya. Our object in this paper is to study certain aspects of the corresponding problem in relation to yet another form of the generalized Lame differential equation. Furthermore, applications of our theorems to the standard form of the generalized Lame differential equation immediately furnish the corresponding results due to Stieltjes, Van Vleck, and Marden (cf. the paper cited above).

Abstract: By the use of a classical result of Strassman an estimate is obtained for the number of zeros in Z p of p -adic exponential polynomials. The p -adic method employed improves on estimates obtained by p -adic analogues of the method applicable in the complex case.

Abstract: In 1972 Shanks conjectured that the least squares inverse of a two-dimensional polynomial is stable, and verified the conjecture numerically for certain low-degree two-dimensional polynomials. Recently the conjecture was proved false. However, in this note we prove the conjecture for all polynomials of a restricted and low degree. The key to the verification lies in utilizing the centrosymmetric properties of the Toeplitz matrix which arises in an equation yielding the coefficients of the approximate inverse.

Abstract: Publisher Summary This chapter focuses on Legendre's polynomials. It discusses Kodaira's identity, Weyl's theory, Green's formula, symmetric boundary conditions, T-positive theory, S-positive theory, and other theorems.

Abstract: Interpolation series (divided difference, Gregory-Newton, Gauss, Stirling, Bessel) are converted into Chebyshev (or Jacobi) series by applying a previously derived general five-term recurrence formula [31. It employs the coefficients in three-term linear recurrence formulas (same kind as for orthogonal polynomials) which have been found for the mth degree nonorthogonal polynomial coefficients of the differences used in the interpolation series. In the Gauss, Stirling and Bessel series, the coefficients in the recurrence formulas vary with the parity of m. The basic five-term recurrence formula is applicable also to: (1) interand intraconversion of power series in ax + b, divided difference and equal-interval interpolation series (including subtabulation), and Chebyshev series, (2) obtaining Chebyshev series for solutions of difference equations, (3) the derivation of formulas for Chebyshev coefficients in terms of differences, and (4) the conversion of interpolation series into Chebyshev series, for more than one variable. Introduction. In a previous note on converting from one orthogonal polynomial series 12 0amqm(x) into another orthogonal polynomial series 1 = oAmQm(x) [3], the basic five-term recurrence formula [3, (22)] is deduced just from these three-term recurrence formulas for qm qm(x) and Qm Qm(x) (la) q-1 = 0, qm + 1 + (a(m) + b(m)x)qm + c(m)qm -1 = 0, (lb) Q-1 = 0, Qm+1 + (A(m) ?B(m)X)Qm + C(m)Qmi = 0 m 0(l)n -1. Since the orthogonality of qm or Qm is not necessary for (la) or (lb) respectively, it happens that [3, (22)] has many applications when either of, or both, qm and Qm are not orthogonal. We give here some useful applications to converting a number of different interpolation series into Chebyshev series. These seven interpolation series, namely, Newton's divided difference formula, the Gregory-Newton formulas with forward and backward differences, Gauss's forward and backward formulas, Stirling's formula and Bessel's formula [2], through the nth degree terms,1 are each expressible as n2=0amqm where qm satisfies a three-term recurrence relation of the form (la). This Received June 16, 1975. AMS (MOS) subject classifications (1970). Primary 65D05, 33A65, 40-04; Secondary 41A05, 41A10, 41A30.

Abstract: In this paper we determine the order of the best one-sided approximation by polynomials and splines of minimal defect of the classes WrLp in the Lp-metric.

Abstract: In this paper various transformations of trigonometrical polynomials are introduced; these are then used to deduce asymptotic formulas for the behavior of sequences of integrals of the moduli of the polynomials. As a consequence a definitive form is found for a relationship (in both directions) between the summability and absolute convergence of trigonometric Fourier series which has been noted earlier by the author.Bibliography: 26 titles.

Abstract: This paper is concerned with the efficient computation of the ubiquitous electron repulsion integral in molecular quantum mechanics. Differences and similarities in organization of existing Gaussian integral programs are discussed, and a new strategy is developed. An analysis based on the theory of orthogonal polynomials yields a general formula for basis functions of arbitrarily high angular momentum. (ηiηj∥ηkηl) = Σα=1,nIx(uα) Iy(uα) I*z(uα) By computing a large block of integrals concurrently, the same I factors may be used for many different integrals. This method is computationally simple and numerically well behaved. It has been incorporated into a new molecular SCF program HONDO. Preliminary tests indicate that it is competitive with existing methods especially for highly angularly dependent functions.