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In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev type case is studied as an illustrative example. Finally, the bispectrality of such matrix orthogonal polynomials is pointed out. [ABSTRACT FROM AUTHOR]

The constrained mock-Chebyshev least squares approximation (CMCLS-approximation) is a method that has been recently introduced. It operates on a grid of equidistant points, aiming to eliminate the Runge phenomenon. The implementation of the idea behind this approximation method involves interpolating the function exclusively on the subset of nodes closer to the set of Chebyshev–Lobatto nodes of a suitable order and using the remaining nodes to enhance the accuracy of the approximation through a simultaneous regression. The main goal of this article is to extend the CMCLS-approximation through the interpolation on zeros of orthogonal polynomials, leveraging their inherent favorable properties. [ABSTRACT FROM AUTHOR]

We consider a varying discrete Sobolev inner product such as ( f , g ) S = ∫ f ( x ) g ( x ) d μ + M n f ( j ) ( c ) g ( j ) ( c ) , where μ is a finite positive Borel measure supported on an infinite subset of the real line, c is adequately located on the real axis, j ≥0, and { M n } n ≥0 is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler-Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c , just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler–Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones. [ABSTRACT FROM AUTHOR]

By using the Szegő’s transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle. Moreover, we study the relation between the corresponding S -functions and C -functions. [ABSTRACT FROM AUTHOR]

By using the explicit form of the eigenvectors of the finite Jacobi matrix associated to a family of orthogonal polynomials and some asymptotic expressions, we obtain quadrature formulas for the integral transforms arising from linear generating functions of the classical orthogonal polynomials. As a bypass product, we obtain simple and accurate Riemann–Steklov quadrature formulas and as an application of this quadrature formalism, we obtain the relationship between the fractional Fourier transform and the canonical coherent states. [ABSTRACT FROM AUTHOR]

Let us consider a Hermitian linear functional defined on the linear space of Laurent polynomials with complex coefficients. In the literature, canonical spectral transformations of this functional are studied. The aim of this research is focused on perturbations of Hermitian linear functionals associated with a positive Borel measure supported on the unit circle. Some algebraic properties of the perturbed measure are pointed out in a constructive way. We discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated Verblunsky coefficients. Then, the structure of the Θ matrices of the perturbed linear functionals, which is the main tool for the comparison of their corresponding CMV matrices, is deeply analyzed. From the comparison between different CMV matrices, other families of perturbed Verblunsky coefficients will be considered. We introduce a new matrix, named Fundamental matrix, that is a tridiagonal symmetric unitary matrix, containing basic information about the family of orthogonal polynomials. However, we show that it is connected to another family of orthogonal polynomials through the Takagi decomposition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider sequences of polynomials orthogonal with respect to the discrete Sobolev inner product where f and g are polynomials with real coefficients, and the vectors are with and the mass point c is located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of classical Laguerre polynomials and we analyze the behavior of the coefficients of the corresponding five-term recurrence relation when the degree of the polynomials is large enough. Also, the outer relative asymptotics of the Laguerre–Sobolev type with respect to the Laguerre polynomials is analyzed. [Copyright &y& Elsevier]

Abstract: A pair of regular linear functionals in the linear space of polynomials with complex coefficients is said to be an -coherent pair of order m if their corresponding sequences of monic orthogonal polynomials and satisfy a structure relation where , and m are non-negative integers, , and , are sequences of complex numbers such that if if , and if . When is called an -coherent pair. In this work, we give a matrix interpretation of -coherent pairs of linear functionals. Indeed, an algebraic relation between the corresponding monic tridiagonal (Jacobi) matrices associated with such linear functionals is stated. As a particular situation, we analyze the case when one of the linear functionals is classical. Finally, the relation between the Jacobi matrices associated with -coherent pairs of linear functionals of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the Sobolev inner product defined by the pair is deduced. [Copyright &y& Elsevier]

Abstract: We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler–Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give us an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other special functions. We generalize some results appeared very recently in the literature for both the varying and non-varying cases. [Copyright &y& Elsevier]

Abstract: In this paper we deal with Hankel determinants of the form where r is a non-negative integer, and is a sequence complex numbers. When and the Hankel determinants associated with the sequence are not identically zero, we show that is a sequence of polynomials satisfying a three term recurrence relation. We illustrate our result by evaluating the Hankel determinant associated with the sequence for and . [Copyright &y& Elsevier]

Abstract: In this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the Uvarov perturbation of a positive Borel measure . When the measure is semiclassical, then its electrostatic interpretation is given. [Copyright &y& Elsevier]

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential 〈 p , q 〉 = ∫ R p ( x ) q ( x ) e − x 4 + 2 t x 2 d x + M p ( 0 ) q ( 0 ) . We analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on t is given. [ABSTRACT FROM AUTHOR]

We present a new structure relation for the sequence of orthogonal polynomials associated with a D ν -semiclassical linear functional of class s , and then we use it to obtain a matrix characterization of the D ν -semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthonormal polynomials, and the nonsingular lower triangular matrix that represents the orthogonal polynomials with respect to some bases of polynomials. We also provide a matrix characterization of D ν -coherent pairs of linear functionals. [ABSTRACT FROM AUTHOR]

We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we show how to compute in steps the Fourier coefficients associated with the Gelfand–Levitan approach for discrete Sobolev orthogonal polynomials on the unit circle when the support of the discrete component involving derivatives is located outside the closed unit disk. As a consequence, we deduce the outer relative asymptotics of these polynomials in terms of those associated with the original orthogonality measure. Moreover, we show how to recover the discrete part of our Sobolev inner product. [Copyright &y& Elsevier]

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form Iμ(f)=(1/2π)∫lower limit 0, upper limit 2π f(eiθ) dμ(θ). For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szego¨ polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Pade´ approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szego¨ quadrature formulae is presented through some illustrative numerical examples. [Copyright &y& Elsevier]

Abstract: Let be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional and a sequence of polynomials defined by with for . We obtain a new characterization of the orthogonality of the sequence with respect to a linear functional , in terms of the coefficients of a quadratic polynomial such that . We also study some cases in which the parameters and can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with and is presented. [Copyright &y& Elsevier]