Sequences of orthogonal Laurent polynomials,bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.     

Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out. Received February 29, 2000 / Published online August 17, 2001     

On chebyshev quadrature and variance of quadrature formulas

The purpose of this note is to give an example which demonstrates that one can achieve much higher algebraic precision with a quadrature rule with small but not minimal variance than with a Chebyshev rule with minimal variance.     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Optimal stochastic quadrature formulas for convex functions

We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods, we prove that adaptive Monte Carlo methods are much better.Supported by a Heisenberg scholarship of the DFG.     

Construction of asymptotically best quadrature formulas

Summary. We show that, for integrals with arbitrary integrable weight functions, asymptotically best quadrature formulas with equidistant nodes can be obtained by applying a certain scheme of piecewise polynomial interpolation to the function to be integrated, and then integrating this interpolant. Received August 7, 1991     

Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circle

Summary A method is proposed for the computation of the Riesz-Herglotz transform. Numerical experiments show the effectiveness of this method. We study its application to the computation of integrals over the unit circle in the complex plane of analytic functions. This approach leads us to the integration by Taylor polynomials. On the other hand, with the goal of minimizing the quadrature error bound for analytic functions, in the set of quadrature formulas of Hermite interpolatory type, we found that this minimum is attained by the quadrature formula based on the integration of the Taylor polynomial. These two different approaches suggest the effectiveness of this formula. Numerical experiments comparing with other quadrature methods with the same domain of validity, or even greater such as Szeg? formulas, (traditionally considered as the counterpart of the Gauss formulas for integrals on the unit circle) confirm the superiority of the numerical estimations. This work was supported by the ministry of education and culture of Spain under contract PB96-1029.     

Rational quadrature formulae on the unit circle with arbitrary poles

Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature. The work of the first author was supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and by UPM and Comunidad de Madrid under grant CCG06-UPM/MTM-539. The work of the second author was partially supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grant MTM2005-08571.     

Convergence and computation of simultaneous rational quadrature formulas

We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.     

Quadrature formulas for Fourier coefficients

We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the Fourier-Tchebycheff coefficients given by Micchelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives.     

Gauss-Radau and Gauss-Lobatto interval quadrature rules for Jacobi weight function

In this paper we prove the existence and uniqueness of the Gauss-Lobatto and Gauss-Radau interval quadrature formulae for the Jacobi weight function. An algorithm for numerical construction is also investigated and some suitable solutions are proposed. For the special case of the Chebyshev weight of the first kind and a special set of lengths we give an analytic solution. The authors were supported in parts by the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320–111079 ``New Methods for Quadrature') and the Serbian Ministry of Science and Environmental Protection. Serbian Ministry of Science and Environmental Protection.     

Gaussian rational quadrature formulas for ill-scaled integrands

A flexible treatment of Gaussian quadrature formulas based on rational functions is given to evaluate the integral , when f is meromorphic in a neighborhood V of the interval I and W(x) is an ill-scaled weight function. Some numerical tests illustrate the power of this approach in comparison with Gautschi’s method.     

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szeg?-Radau and Szeg?-Lobatto quadrature formulas on the unit circle. These are (n+m)-point formulas for which m nodes are fixed in advance, with m=1 and m=2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. This means that the free parameters (free nodes and positive weights) are chosen such that the quadrature formula is exact for all powers z^j, −p≤j≤p, with p=p(n,m) as large as possible.     

Ramanujan continued fractions via orthogonal polynomials

Some Ramanujan continued fractions are evaluated using asymptotics of polynomials orthogonal with respect to measures with absolutely continuous components.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Some extremal properties of orthogonal polynomials

Using a particular way of normalizing the orthogonal polynomials, which is most commonly encountered in the synthesis of filtering networks in communication and electronic engineering, two theorems concerning the extremal properties of orthogonal polynomials are first proved. The results are then applied to find the minimum value and the minimizing function for various definite integrals involving weight functions of classical orthogonal polynomials.     

Error estimates of anti-Gaussian quadrature formulae

Anti-Gauss quadrature formulae associated with four classical Chebyshev weight functions are considered. Complex-variable methods are used to obtain expansions of the error in anti-Gaussian quadrature formulae over the interval [−1,1]$[-1,1]$. The kernel of the remainder term in anti-Gaussian quadrature formulae is analyzed. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L^∞$L^{\infty }$-error bounds of anti-Gauss quadratures. Moreover, the effective L¹$L^1$-error estimates are also derived. The results obtained here are an analogue of some results of Gautschi and Varga (1983) [11], Gautschi et al. (1990) [9] and Hunter (1995) [10] concerning Gaussian quadratures.     

Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(u)=u. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.