Showing papers on "Biorthogonal system published in 1988"

q-Series and Orthogonal Polynomials Associated with Barnes’ First Lemma

Abstract: We exploit symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions. Similar methods have been used by Nikiforov, Suslov, and Uvarov. Here we apply these ideas to find new proofs of Barnes’ First Lemma and some of its q-analogues. We show that these integrals correspond to the weight functions determining the orthogonality relations for Hahn, q-Hahn, and big q-Jacobi polynomials. As another example of our method we introduce a biorthogonal system of rational functions whose weight function corresponds to the q-analogue of Kummer’s Theorem.

On biorthogonal and orthonormal Clebsch–Gordan coefficients of SU(3): Analytical and algebraic approaches

Abstract: Minimal biorthogonal systems of the Clebsch–Gordan (Wigner) coefficients of SU(3)⊇U(2) are discussed as well as the dual coupled bases. The closed system of analytical expressions for the dual isofactors (reduced Wigner coefficients) and the overlaps of coupled states is obtained with the help of analytical inversion symmetry. The Regge‐type symmetry of the overlaps and the boundary orthonormal isofactors (orthogonalization coefficients) is discovered. The polynomial structure of the alternative complete algebraic systems of the orthonormal SU(3) isofactors (characterized by the null spaces, symmetries, and additional selection rules and obtained by means of the Hecht or Gram–Schmidt process) is considered. The realizations of the external ‘‘missing label’’ operators of the third and the fourth orders in the minimal coupled bases, which lead to preferable algorithms to evaluate the orthonormal SU(3) coupling coefficients satisfying different symmetry properties, are presented. With the help of the 6j coef...

Some Extensions of Askey-Wilson's Q-Beta Integral and the Corresponding Orthogonal Systems

Abstract: A seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

Canonical forms for a class of distributed parameter control systems

Abstract: A class of linear distributed parameter control systems with scalar control is considered in which the uncontrolled system is governed by a holomorphic semigroup. A canonical form is constructed for these systems, and is used to help construct feedback controls solving an eigenvalue specification problem. Explicit formulas for the feedback element and the closed-loop eigenvectors are obtained in terms of the original eigenvectors and the basis which is biorthogonal to the eigenvectors. This will be applied to an eigenvalue specification problem for a structurally damped beam. These feedback controls are given by infinite series, which cannot be computed in practice, so the canonical form is used to analyze the effect of truncated series on the distributed system.The canonical form is a functional equation with an associated canonical state space equation that is equivalent to it. The development in this paper is primarily done in the frequency domain, and hinges on an infinite-dimensional generalization o...

Quadrature Errors and Biorthogonality

Abstract: For functions with possibly low continuity there are two main principles to derive quadrature error bounds. One method involves degrees of approximation (or upper bounds for them), the other one is based on series expansions (e. g. Chebyshev series). Using biorthogonal systems we recently presented an error estimate (called BOGS method) which contains the above mentioned methods as special cases. In the present work we first choose Chebyshev polynomials in the biorthogonal system and derive error bounds for Polya type quadratures. In the second part there are employed Legendre polynomials in order to treat Gauss rules and bilinear quadrature. We further show how to replace some of the Fourier coefficient functionate by point functional with lower norm.

A pair of biorthogonal polynomials for the szego-hermite weight function

Abstract: A pair of polynomial sequences {S(x'k)} and {T(x;k)} where S(x;k) is n m n of degree n in x k and T(x;k) is of degree m in x, is constructed. It is shown m that this pair is biorthogonal with respect to the Szeg-Hermite weight function

Autocorrelation and power spectrum of Hadamard signalling

Abstract: The paper presents results for the autocorrelation and power spectrum of digital modulation for M-ary orthogonal and biorthogonal signal sets based on binary Hadamard matrices. Such signal sets provide easily-mechanised orthogonal (and related) constructions, and are important in coded signalling for jammed and/or fading channels using large-alphabet codes.