Showing papers on "Discrete orthogonal polynomials published in 1969"

Approximate Solutions of Systems of Singular Integral Equations

Abstract: Using the properties of the related orthogonal polynomials, approximate solutions of systems of simultaneous singular integral equations are obtained, in which the essential features of the singularity of the unknown functions are preserved In the system of integral equations of the first kind, the fundamental solution is the weight function of the Chebyshev polynomials of first or second kind In the system of singular integral equations of the second kind with constant coefficients, the elements of the fundamental matrix are the weights of Jacobi polynomials A direct method is introduced to obtain the fundamental matrix of the system The approximate solution is then expressed as the fundamental function, representing the singular behavior of the unknown functions, multiplied by a series of proper orthogonal polynomials with unknown coefficients The techniques of deriving the system of algebraic equations to determine these coefficients are described In order to have an idea about the effectiveness

A differential equation approach to interpolation at extremal points

Abstract: : A differential equation method is used to derive results for interpolation at extremal points for polynomials, trigonometric polynomials, linear combinations of a Tchebycheff system, splines, and monosplines

Summation of multiplicative functions of polynomials

Abstract: A method is proposed for estimating sums of multiplicative functions of polynomials. This method is used to study the question of the uniform distribution of integer points on a three-dimensional sphere. Eight references are quoted.

An Irreducibility Criterion for Polynomials Over the Integers

Abstract: (1969). An Irreducibility Criterion for Polynomials Over the Integers. The American Mathematical Monthly: Vol. 76, No. 7, pp. 795-797.

Mth Power of an N × N Matrix and Its Connection with the Generalized Lucas Polynomials

Abstract: The Mth power of an N × N matrix is expressed via the Cayley‐Hamilton theorem as a linear combination of the lower powers of the matrix. The polynomial coefficients of the lower powers of the matrix are expressed in terms of polynomials in N variables, termed the generalized Lucas polynomials. The independent variables in the generalized Lucas polynomials are the traces of the lower powers of the matrix.

A note on recursively defined orthogonal polynomials

Abstract: Favard showed that the polynomials {Pτ(x)} are orthonormal with respect to a bounded increasing function γ defined on (-co, +oo). This note generalizes recent constructive results which deal with connections between the two sequences {aτ} and {bi} and the spectrum of f. (The spectrum of ψ is the set S(ψ) = {λ : f(λ + e)—f(λ — e)>0 for all e > 0}.) It is shown that if bi —> 0 then every limit point of the sequence {α } is in S(f).

Polynomials with irreducible factors of specified degree

Abstract: Let d be a positive integer and let p be a prime > d. Set q = pm, where m ≥ 1, and let I (q, d) denote the number of distinct primary irreducible polynomials of degree d over GF(q). It is a simple deduction from the well-known expression for I(q, d) that (1) where d* is the largest positive integer < d which divides d if d > 1, and d* is 0 if d = 1.

Moments of Gaussian processes in terms of generalized multidimensional Hermite polynomials (Corresp.)

Abstract: New properties are derived for the moments of real and of complex Ganssian processes. The present analysis shows that all the properties of the multidimensional generalized Hermite polynomials can readily be transcribed into properties of Gaussian processes. Our results are general and hold for processes that are not necessarily of zero mean or stationary; the covariance matrix may be symmetric or nonsymmetric.

Moment sequences and the Bernstein polynomials

Abstract: The Bernstein polynomials (1.1) and the Bernstein power series (1.2) have been the subject of much research (e. g. [1; 2; 3; 6; 7; 8]). It is the purpose of this paper to demonstrate the relationship between these linear operators and certain classes of moment sequences defined below.

Application of Laguerre Polynomials in the Analysis of Neutron-Wave Propagation

Abstract: An energy-dependent transport theory solution for the infinite medium neutron-wave propagation problem is obtained by applying a Laguerre polynomial expansion to represent the flux energy dependenc...

Non-classical orthogonal polynomials with even weight functions on symmetric intervals

Abstract: the requirements for an advanced degree from the Georgia Institute of Technology, I agree that the Library of the Institute shall make it available for inspection and circulation in accordance with its regulations governing materials of this type. I agree that permission to copy from, or to publish from, this dissertation may be granted by the professor under whose direction it was written, or, in his absence, by the Dean of the Graduate Division when such copying or publication is solely for scholarly purposes and does not involve potential financial gain. It is under­ stood that any copying from, or publication of, this dis­ sertation which involves potential financial gain will not be allowed without written permission.