Showing papers on "Discrete orthogonal polynomials published in 1974"
139 citations
TL;DR: In this article, it is demonstrated that there is a close parallel between the theory of a class of orthogonal polynomials and scattering theory, and that by appropriate limiting procedures one can pass from a result about orthogonality to one in scattering theory.
Abstract: It is demonstrated that there is a close parallel between the theory of a class of orthogonal polynomials and scattering theory. In both cases a fundamental role is played by a particular solution of the basic difference (differential) equation which we call the Jost function. Under rather general conditions this function has simple analytic properties. It determines and is largely determined by either the asymptotic phases or the continuous part of the weight (spectral) function. Indeed this is more than an analogy. By appropriate limiting procedures one can pass from a result about orthogonal polynomials to one in scattering theory. Conversely, scattering theory throws considerable light on theorems about orthogonal polynomials.
99 citations
TL;DR: The discrete orthogonal polynomials (DLOPs) as mentioned in this paper have been used for approximation purposes in engineering applications of weighted residual methods, and their properties have been studied extensively.
Abstract: The discrete (Legendre) orthogonal polynomials, (DLOP's) are useful for approximation purposes. This set of mth degree polynomials {Pm(K, N)} are orthogonal with unity weight over a uniform discrete interval and are completely determined by the normalization Pm(O, N) = 1. The authors are employing these polynomials as assumed modes in engineering applications of weighted residual methods. Since extensive material on these discrete orthogonal polynomials, and their properties, is not readily available, this paper is designed to unify and summarize the presently available information on the DLOP's and related polynomials. In so doing, many new properties have been derived. These properties, along with sketches of their derivation, are included. Also presented are a representation of the DLOP's as a product of vectors and matrices, and an efficient computational scheme for generating these polynomials.
79 citations
TL;DR: In this article, it was shown that Ritt's original proof of Theorem 3 does not make any essential use of the topological manifold structure of the Riemann surface; it consists of combinatorial arguments about extensions of primes to a composite of fields, and depends on the fact that the completion of a field k,(t), at each of its prime spots, is quasifinite when k, is algebraically closed of characteristic 0.
72 citations
69 citations
TL;DR: In this article, the authors further extend these results to obtain many results on when projection formulas of the above types hold for the discrete orthogonal polynomials of Hahn, Krawtchouk, Meixner, and Charlier.
45 citations
34 citations
TL;DR: In this article, Ullman defined the restricted support of a weight w(x) is defined as the set of points for which w < 0, and defined the support of w(X) can be characterized as the sets of points e for which W(x ) > 0.
Abstract: In this case the points 0k,, = are sin xkn are equidistributed in Weyl's sense . A non-negative measure da for which the array xkn (da) has the distribution function fl o(t) will be called an arc-sine measure . If du(x) = w(x) dx is absolutely continuous, we apply, replacing dca by w, the notations pn(W,x), yn (w), xkn(w) and call a non-negative w(x) an arc-sine weight if da(x) = w(x) dx is an arc-sine measure. A fairly complete treatise of are-sine weights with compact support is given in [9] by Ullman . The restricted support of a weight w(x) is defined as the set {x : w(x) > 0} . The support of w(x) can be characterized as the set of points e for which
33 citations
TL;DR: An algorithm for computing the coefficients of a polynomial equation resulting from a bilinear transformation of its variable is described, which compares favorably with the matrix method proposed by Power and improved by Jury and Chan.
Abstract: An algorithm for computing the coefficients of a polynomial equation resulting from a bilinear transformation of its variable is described, which compares favorably with the matrix method proposed by Power and improved by Jury and Chan, and which is simpler and easier to program.
TL;DR: In this paper, the authors introduce the method of symmetrization of the measure algebra of a compact hypergroup, and derive a theorem about nonnegative expansions of one family of Krawtchouk polynomials in terms of another family.
Abstract: This paper introduces the method of symmetrization of the measure algebra of a compact $P_ * $-hypergroup. This method is used to form a measure algebra whose characters are Krawtchouk polynomials (these are the finite sets of polynomials orthogonal with respect to the binomial distribution on $\{ 0,1, \cdots ,N\} $. As a further application, one derives a theorem about nonnegative expansions of one family of Krawtchouk polynomials in terms of another family.
01 Sep 1974
TL;DR: In this paper, a table of irreducible polynomials and their exponents for certain small nonprime Galois fields is presented, up to and including degree 5 for GF(4), degree 3 for GF (8) and GF(9), and degree 2 for GF 16.
Abstract: Tables of irreducible polynomials and their exponents are listed for certain small nonprime Galois fields. These include all such polynomials up to and including degree 5 for GF(4), degree 3 for GF(8) and GF(9), and degree 2 for GF(16). In addition, a single primitive polynomial is given for each degree up to and including degree 11 for GF(4), degree 7 for GF(8) and GF(9), and degree 5 for GF(16). A brief summary is given of several areas where these results may prove useful in providing an alternative to the more conventional approach.
TL;DR: The first of a series of papers which will give simple proofs of a number of recent formulas for Jacobi polynomials was given in this article, where one of Gegenbauer's proofs for his integral representation of ultraspherical polynomial is given, and then a fractional integration gives Koornwinder's integral representation.
Abstract: This is the first of a series of papers which will give simple proofs of a number of recent formulas for Jacobi polynomials. In this paper one of Gegenbauer’s proofs for his integral representation of ultraspherical polynomials is given, and then a fractional integration gives Koornwinder’s integral representation for Jacobi polynomials. This is then combined with Koornwinder’s product formula to give a new proof of a bilinear sum of Bateman.
TL;DR: In this article, it was proved that there exists a set of polynomials orthogonal on (-1, 1 ) with respect to the weight function, which is a polynomial orthogonality on the weighted weight function.
Abstract: It is proved that there exists a set of polynomials orthogonal on (-1, 1 ) with respect to the weight function
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01 Jan 1974
Abstract: We investigate in this chapter real functions of the real variable x. In particular we assume that the coefficients a 0, a 1, a 2,... of the polynomials a 0 + a 1 x +a 2 x 2 + ... + a n x n and of the power series a 0 + a 1 x + a 2 x 2 + ... which we shall be considering are real. We assume further, unless the contrary is stated, that all functions are analytic in the corresponding intervals. The theorems, however, are changed only slightly or not at all if we introduce more general assumptions, e.g. the existence of derivatives up to some order. The zeros in the following are always to be counted according to their multiplicity.
TL;DR: In this paper, the first primitive polynomial of the third kind of degree n over GF(pd) was given for each p, d, n satisfying p < 102, pd < 10 pdn < 106.
Abstract: This paper gives the first primitive polynomial of the third kind of degree n over GF(pd) for each p, d, n satisfying p < 102, pd < 10 pdn < 106. In the preceding paper [1, Section 3] Beard introduced an exponential representation for GF(pd) which allows full use of its multiplicative structure and permits direct rational calculations in GF(pd). As indicated in [1, Section 4] , such representations are easily and quickly obtained once primitive polynomials of the third kind of degree d Received May 14, 1973: AMS (MOS) subject classifications (1 970). Primary 12-04, 12C05, 12C 1 5.
IBM1
TL;DR: An exact order of growth of Szego polynomial kernels and moduli of polynomials orthogonal in the unit circle is obtained in the zeros of a weight function of special form.
Abstract: An exact order of growth of Szego polynomial kernels and moduli of polynomials orthogonal in the unit circle is obtained in the zeros of a weight function of special form. Other questions are also examined.
01 Nov 1974
TL;DR: In this paper, the zero distribution of real aperiodic polynomials with real and complex Hurwitz polynomial has been analyzed and the results are applied to the relation between real and Hurwitz aperiodics.
Abstract: Results are obtained connecting the zero distribution of a real polynomial and a complex polynomial of approximately half the degree of the real polynomial. The results are applied to the relating of real aperiodic polynomials with real and complex Hurwitz polynomials. "Unit circle" results are also outlined.
TL;DR: In this article, Brugia's work on the noniterative computation of high order derivatives of rational functions with application to multiple-pole fraction expansion is simplified by the use of operators.
Abstract: Brugia's work on the noniterative computation of high order derivatives of rational functions with application to multiple-pole fraction expansion is simplified by the use of operators. The formulas for the derivatives are given in a recursive scheme and finally the Laurent expansion is developed and applied to the partial fraction expansion of improper rational functions.
01 Jan 1974
TL;DR: For a given matrix with dominant diagonal, a number of convenient upper and lower bounds for the corresponding determinant are available in the literature as discussed by the authors, and these inequalities are used here to derive bounds for a general system of orthogonal polynomials for a specified range of values of the argument.
TL;DR: The Ratios of Integer-Valued Polynomials Over Any Algebraic Number Field as mentioned in this paper is an example of such an approach, and it can be seen as a form of algebraic number fields.
Abstract: (1974). On the Ratios of Integer-Valued Polynomials Over Any Algebraic Number Field. The American Mathematical Monthly: Vol. 81, No. 9, pp. 997-999.