Orthogonal polynomials for modified Gegenbauer weight and corresponding quadratures (original) (raw)
Orthogonal polynomials for the oscillatory-Gegenbauer weight
Publications de l'Institut Math?matique (Belgrade), 2008
This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional L : P?C, where L = ?1 -1 p(x) d?(x), d?(x) = (1-x?)?-1/2 exp(i?x) dx, and P is a linear space of all algebraic polynomials. Here, we prove an extension of our previous existence theorem for rational ? ? (-1/2,0], give some hypothesis on three-term recurrence coefficients, and derive some differential relations for our orthogonal polynomials, including the second order differential equation.
Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions
Journal of Computational and Applied Mathematics, 2005
In this paper we consider polynomials orthogonal with respect to an oscillatory weight function w(x) = xe im x on [-1, 1], where m is an integer. The existence of such polynomials as well as several of their properties (threeterm recurrence relation, differential equation, etc.) are proved. We also consider related quadrature rules and give applications of such quadrature rules to some classes of integrals involving highly oscillatory integrands.
Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type
In the first part of this survey paper we present a short account on some important properties of orthogonal polynomials on the real line, including computational methods for constructing coefficients in the fundamental three-term recurrence relation for orthogonal polynomials, and mention some basic facts on Gaussian quadrature rules. In the second part we discuss our Mathematica package OrthogonalPolynomials (see [2]) and show some applications to problems with strong nonclassical weights on (0, +∞), including a conjecture for an oscillatory weight on [−1, 1]. Finally, we give some new results on orthogonal polynomials on radial rays in the complex plane.
Orthogonal polynomials related to the oscillatory: Chebyshev weight function
Bulletin: Classe des sciences mathematiques et natturalles, 2005
In this paper we discuss the existence question for polynomials orthogonal with respect to the moment functional Since the weight function alternates in sign in the interval of orthogonality, the existence of orthogonal polynomials is not assured. A nonconstructive prof of the existence is given. The three-term recurrence relation for such polynomials is investigated and the asymptotic formulae for recursion coefficients are derived.
Generalized Gegenbauer orthogonal polynomials
Journal of Computational and Applied Mathematics, 2001
In this paper we explore a speciÿc semi-classical orthogonal sequence, namely the generalized Gegenbauer orthogonal polynomials (GG) which appear in many applications such as the weighted L p mean convergence of Hermite-Fejà er interpolation or the chain of harmonic oscillators in the absence of externally applied forces. First we trace back the genesis of GG underlining its links with the Jacobi orthogonal polynomials. Second we establish a di erential-di erence relation and the second-order di erential equation satisÿed by this sequence. We end by giving the fourth-order di erential equation satisÿed by the association (of arbitrary order) of the GG.
Journal of Computational and Applied Mathematics, 2014
In this work, we derive a family of symmetric numerical quadrature formulas for finite-range integrals dx, where w(x) is a symmetric weight function. In particular, we will treat the commonly occurring case of w( ] p , p being a nonnegative integer. These formulas are derived by applying a modification of the Levin L transformation to some suitable asymptotic expansion of the function H(z) = 1 -1 w(x)/(zx) dx as z → ∞, and they turn out to be interpolatory. The abscissas of these formulas have some rather interesting properties: (i) they are the same for all α, (ii) they are real and in [-1, 1], and (iii) they are related to the zeros of some known polynomials that are biorthogonal to certain powers of log(1 -x 2 ) -1 . We provide tables and numerical examples that illustrate the effectiveness of our numerical quadrature formulas.
Quadrature formulae connected to σ-orthogonal polynomials
Journal of Computational and Applied Mathematics, 2002
Let d (t) be a given nonnegative measure on the real line R, with compact or inÿnite support, for which all moments k = R t k d (t); k = 0; 1; : : : ; exist and are ÿnite, and 0 ¿ 0. Quadrature formulas of Chakalov-Popoviciu type with multiple nodes where = n = (s1; s2; : : : ; sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness dmax = 2 n =1 s + 2n -1 if and only if The proof of the uniqueness of the extremal nodes 1; 2; : : : ; n was given ÿrst by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1-15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an in uence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes ; = 1; 2; : : : ; n, which are the zeros of the corresponding -orthogonal polynomial, is presented. Finally, in order to show a numerical e ciency of the proposed procedure, a few numerical examples are included.
In this letter we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A n (z) and B n (z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w 0 and a standard jump function, then A n (z) and B n (z) have apparent simple poles at these jumps. We exemplify the approach by taking w 0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painlevé IV.