Showing papers on "Wilson polynomials published in 1972"

Weak-star density of polynomials.

Abstract: The purpose of this paper is to investigate the following approximation problem: For which finite positive measures μ in the complex plane are the polynomials weak-star dense in L (μ) ? By a polynomial we mean an analytic polynomial, that is, a function p of the form p (z) = CQ + ολζ + · · · + ν> where c0, . . ., cn are complex constants. For our question to make sense it is necessary that the polynomials belong to L (μ), in other words, that μ be a Borel measure of compact support. All measures considered in this paper will be finite Borel measures. We shall obtain here a necessary and sufficient condition on μ for the required density to hold. Although the condition cannot be called simple, it seems appropriate to the problem, for the following two reasons. (1) Together with its proof, it makes transparent, in case density fails, what it is that causes this failure. (2) The reasoning that leads to the condition yields, in addition, a description for arbitrary μ of the weak-star closure of the polynomials in LTM (μ). The Statement of the condition requires considerable preparation; it is therefore deferred until later. Our treatment of the approximation problem makes extensive use of known results on rational approximation and function algebras (especially Dirichlet algebras). Some of the needed results are set forth in the preliminary §§ 3 and 4. We assume the reader is famili r with the most basic terminology and results in the theory of function algebras; the book of Gamelin [7] is an excellent reference. Although our bibliographical references are mainly to the original sources, most of what we need can be found in Gamelin's book. § 2 is also preliminary; it contains a few simple but useful remarks on balayage. The approximation problem is attacked in §§ 5—7. In § 5 we study a notion of convexity with respect to families of bounded holomorphic functions. The main result here is the recognition of a class of Dirichlet algebras. § 6 deals with a related notion, that of the h ll of a measure with respect to a family of bounded holomorphic functions. The density criterion is obtained in § 7. Our approximation problem was suggested by, and is in fact equivalent to, a certain question about the invariant subspaces of normal operators on Hubert spaces. That question is discussed in § 8.

An inequality of Turán type for Jacobi polynomials

Abstract: For Jacobi polynomials P^^Hx), a, ß> —1, let RAX) = ^"„„/^ > A»M = KM Rn^(x)R„ , AX).

BEST APPROXIMATIONS OF FUNCTIONS IN THE Lp METRIC BY HAAR AND WALSH POLYNOMIALS

Abstract: In this work the modulus of continuity of functions in the Lp metric (1 ≤ p < ∞) is estimated through its best approximations in this metric by Haar and Walsh polynomials. Besides, estimates of best approximations of functions by Haar and Walsh polynomials in the Lq metric are obtained by the same approximations in the Lp metric (1 ≤ p ≤ ∞). In the last case, the results are analogous to those which were proved for approximations by trigonometric polynomials by P. L. Ul'janov and also by S. B. Steckin and A. A. Konjuskov. Bibliography: 26 items.

Condensing generalized polynomials

Abstract: An extension of geometric programming to includegeneralized polynomials, and not onlyPositive polynomials, is described, together with an algorithm and a numerical example.

Solution of the Difference Equations of Generalized Lucas Polynomials

Abstract: Barakat and Baumann have introduced polynomials U(N) (a1, a2, …, aN) termed the generalized Lucas polynomials satisfying a difference equation with a set of initial conditions. We show that these polynomials can be obtained directly from the symmetric functions hn, which are of basic importance in combinatorial analysis. Moreover, we extend the definition of V(a1, a2) to V(N) (a1, a2, …, aN) and establish that these polynomials too can be obtained from the symmetric functions Sn. Further, closed expressions for the U and V are obtained.

The location of the zeros of the derivatives of Faber polynomials

Abstract: We prove a property of Faber polynomials which supports a conjecture concerning the zeros of Faber polynomials

Generating functions for Jacobi and related polynomials

Abstract: In the present paper, we have established the following relation involving Kampe de Feriet's double hypergeometric function of superior order _ al,***, a. F I .b cx I (b)f F_Oci,, cq -y= nE J(b)=1 (X)(b)n F ?bX-; yn rr (bj). LI + b n; j=l which yields a number of interesting generating formulae for Jacobi and related polynomials. A large number of special cases have been also discussed.

A unified presentation of certain classical polynomials

Abstract: This paper attempts to present a unified treatment of the classical orthogonal polynomials, viz. Jacobi, Laguerre and Hermite polynomials, and their generalizations introduced from time to time. The results obtained here include a number of linear, bilinear and bilateral generating functions and operational formulas for the polynomials {Tn(a )(X, a, b, c, d, p, r) I n = 0, 1, 2, * }, defined by Eq. (3) below.*

On polynomials which commute with a given polynomial

Abstract: By extending a theorem of Jacobsthal, the following resuLlt is obtained: if g is a nonlinear polynomial, there is an integer J(g)> 1 such that for each m>0 there are either J(g) or zero distinct polynomials of degree mn which commute with, . A formula is given for computing J(g) from the coefficients of g. We say that a pair of polynomialsf, g commute if f (g(x))=g(f(x)) for all x. Fifty years ago, J. F. Ritt [4] proved that commuting polynomials must be, within a linear homeomorphism, either both powers of x, both iterates of the same polynomial, or both Tchebycheff polynomials. Yet Ritt's proof is so complicated and relies on such deep results from analysis that there is still interest in what can be proved about commuting polynomials without resorting to his methods (see [1], [2], [3]). In this paper we offer results extending the algebraic, nonanalytic approach to the problem. A principal result presented here concerns the number of polynomials of degree m>0 which commute with a given polynomial g of degree n> 1. We show that there is an integer J(g)> 1 such that for each m>O, either J(g) or zero distinct polynomials of degree m commute with g. In addition we show how J(g) may be computed from the coefficients of g. Our main tool is Theorem 1, which is an extension of Theorem 19 of Jacobsthal [3]. Although Jacobsthal claims for Theorem 19 only the result that J(g) 1 and m>0 respectively, g(x) = E b2x'", f(x) = E aixm-, aobo $ 0, i= () i=0 Presented to the Society, September 1, 1971; received by the editors August 25, 1971. AMS 1970 suibject classifications. Primary 12A20, 30A20; Secondary lOA30, 13F20, 20M20.

On a New Discrete Analogue of the Legendre Polynomials

Abstract: A new system of orthogonal polynomials has been introduced recently by the author. These polynomials exhibit many of the “nice” properties of the Legendre polynomials. The evidence, theoretical and computational, implies that, as a discrete analogue to the Legendre polynomials, these polynomials are “superior” to the classical Hahn polynomials. In this paper, proofs of the announced results are presented and further development and generalizations are indicated.

Partially-orthogonal polynomials

Abstract: This paper contains a discussion of partiallyorthogonal polynomials This is an extension of the concept of quasi-orthogonal polynomials Some relationships between various partially-orthogonal polynomials are obtained The concept of pseudo-polynomials is defined and used as an example of partially-orthogonal polynomials Polynomials obtained from the simple Laguerre polynomials are also used as an example The concept of quasi-orthogonal polynomials is discussed by Dickinson [2] and by Chihara [1] It is the purpose of this paper to discuss some generalizations of the concept of quasi-orthogonal polynomials and to obtain recurrence relations between the various polynomials Some examples will be given Also, the concept of polynomials will be generalized DEFINITION 1 Let {Qn(x, m)}l=0 be a set of polynomials, where each Qn(x, m) is of degree n The Qn(x, m) will be called partially-orthogonal of deficiency m if there exists an interval (a, b) such that fw(x)x'Qk(x,m)dx=O forO m, $AO forj>k-m,k>m, where w(x) is a nonnegative weight function If m=O the set of polynomials are fully orthogonal If m=1 the set of polynomials are quasiorthogonal The m=O index will be omitted in this paper For simplicity all the examples of polynomials here will have leading coefficient unity, and this is assumed throughout the paper DEFINITION 2 We will call two partially-orthogonal sets of polynomials related if the weight function and interval of integration are the same but the deficiencies are different DEFINITION 3 Two polynomials will be said to share the same zero if they are both annihilated by the same operation By operation is meant any linear functional F, and F annihilates the polynomial Q if F(Q)=O Received by the editors February 1, 1971 AMS 1970 subject classifications Primary 33A65