RII type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle

Let (P ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ωυ) the monic associated polynomials of (P v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (APυλ+BΩυλ)/Aλ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP ν and of the corresponding orthogonality measure and Szego function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.

We characterize the sequences of orthogonal polynomials on the unit circle whose derivatives are also orthogonal polynomials on the unit circle. Some relations for the sequences of derivatives of orthogonal polynomials are provided. Finally, we pose some problems about orthogonality-preserving maps and differential equations for orthogonal polynomials on the unit circle.

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.

Semiorthogonal functions and orthogonal polynomials on the unit circle

ABSTRACTA stochastic representation is derived for a vector random field that is stationary, isotropic, and mean square continuous on a sphere or unit circle. The established stochastic representation is an infinite series involving the sequence of ultraspherical or Gegenbauer's polynomials, looks like a mimic of the series representation of the covariance matrix function of the isotropic vector random field, but differs from the spectral representation in terms of the ordinary spherical harmonics. It is also shown in this paper that some isotropic and continuous covariance matrix functions on the real line or , if they are compactly supported, can be adopted as covariance matrix functions on the unit circle or .

About a system of anti-periodic trigonometric functions

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

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

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

Markov–Bernstein inequalities for generalized Gegenbauer weight

Matrix methods for quadrature formulas on the unit circle. A survey

We describe an algorithm for complex discrete least squares approximation, which turns out to be very efficient when function values are prescribed in points on the real axis or on the unit circle. In the case of polynomial approximation, this reduces to algorithms proposed by Rutishauser, Gragg, Harrod, Reichel, Ammar, and others. The underlying reason for efficiency is the existence of a recurrence relation for orthogonal polynomials, which are used to represent the solution. We show how these ideas can be generalized to least squares approximation problems of a more general nature.

Orthogonal polynomials and some q-beta integrals of Ramanujan

The location of the zeros of discrete systems characteristic polynomials with respect to the unit circle is investigated. A new sequence of symmetric polynomials of descending degrees is defined for the characteristic polynomial and the number of zeros inside and outside the unit circle is shown to be related to a certain sign variation pattern of the polynomials in the sequence. A stability table based on this sequence is presented to obtain the sought distribution of zeros. The new table is close in appearance, size, and number of arithmetic operations to the Routh table used for continuous-time system polynomials. By comparison with the table of Jury, based on the theory of Marden, Cohn, and Schur, the new table involves about half the number of entries and a corresponding significant saving in computations. The study includes a detailed consideration of all possible cases of singular conditions so that the complete information on the number of zeros inside, outside, and on the unit circle is always obtained (including some additional information on possible reciprocal pairs of zeros). Necessary and sufficient conditions for a polynomial to have all its zeros inside the unit circle are obtained as a special outcome. Other additional necessary conditions, that are useful to shorten procedures when the table is used only for testing the stability of the system, are also given.