Extensions of Szegö's theory of orthogonal polynomials,II

Let {? _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior of? _n (dμ) when logμ'∈L ¹. In what follows we will discuss the asymptotic behavior of the ratio φ_n(dμ ₁)/φ_n(dμ ₂) off the unit circle in casedμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=g dμ ₁ whereg≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0, for the monic polynomials Φ _n . The consequences for orthogonal polynomials on the real line are also discussed.     

Szegő’s theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg?’s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.     

Approximation with Bernstein–Szegö polynomials

We present approximation kernels for orthogonal expansions with respect to Bernstein–Szegö polynomials. Theconstruction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein–Szegö polynomials.     

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Given {P_n}_n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., 

     

Rate of Convergence in Szegö's Asymptotic Formula for Toeplitz Determinants

Rate of convergence in Szegö's asymptotic formula for Toeplitz determinants is given in terms of L₂-moduli of continuity of functions.     

PC-fractions and Szegö polynomials associated with starlike univalent functions

Each classS , 0<1, functions starlike of order , can be associated with a Carathéodory function mapping the unit disk onto a subset of the right halfplane. This Carathéodory function determines a certain continued fraction (PC-fraction) and a family of polynomials orthogonal on the unit circle (Szegö polynomials). We compute the PC-fraction and Szegö polynomials corresponding to eachS and do some investigations on these PC-fractions and Szegö polynomials.     

Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $A$ and $B$ , if $0\le B \le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $r(A)$ is a pole of the resolvent of $A$ . In this paper, we prove that the same result holds if $B$ is irreducible and $r(B)$ is a pole of the resolvent for $B$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.     

Companion theorems to G. Szegö’s inequality for pairs of coefficients of bounded polynomials

We consider real univariate polynomials $P_n$ of degree $ \le n $ from class $$\begin{aligned} \mathbf {C}_n = \{P_n:|P_n \left( \cos \displaystyle \frac{(n -i)\pi }{n}\right) |\le 1 \; \text{ for }\; 0\le i\le n \} \end{aligned}$$ which encompasses the unit ball of polynomials with respect to the uniform norm on $[- 1, 1]$ . For pairs of consecutive coefficients of $ P_n(x) = \sum \nolimits _{k=0}^{n}a_kx^k$ there holds the inequality 1 $$\begin{aligned} |a_{k-1}|+|a_k|\le |t_{n,k}|, \quad \text{ if }\; k\equiv n\; \text{ mod }\; 2, \end{aligned}$$ where $T_n(x)=\sum \nolimits _{k=0}^{n} t_{n,k}x^k$ is the $n$ -th Chebyshev polynomial of the first kind. (1) implies Markov’s classical coefficient inequality of 1892 (Math. Ann. 77:213–258, 1916, p. 248) and goes back to Szegö, but was made public by P. Erdös (Bull. Am. Math. Soc. 53:1169–1176, 1947, p. 1176) in 1947. We ask here: will the (nonzero) coefficients of $T_n$ likewise majorize complementary pairs $|a_k| + |a_{k+1}|$ ? More generally: does there hold 2 $$\begin{aligned} |a_k| + |a_j| \le |t_{n,k}| \quad \text{ for } \text{ all }\; P_n \in \mathbf {C_n}, \end{aligned}$$ $\text{ where }\; k < j \;\text{ and }\; k\equiv n\mod 2\;\text{ but }\; j\not \equiv n\mod 2 ?$ We treat the marginal cases $n < 12$ separately, and for $n \ge 12$ we provide answers to this question with the aid of the explicitly determined optimal bound $K \sim \lceil \frac{n}{\sqrt{2}}\rceil $ which incorporates the height and the length of $ \frac{T'_n(x)}{n}$ . Theorem 2.1: (2) holds, provided $K \le k < j$ ; in particular, provided $\frac{n}{\sqrt{2}}<k<j$ . As a corollary we reveal new extremal properties of the leading coefficients of $\pm T_n$ . Theorem 2.4: (2) does not hold if $k < j < K$ . Theorem 2.5: If $k < K < j$ , then (2) holds for certain, but not for all, $k$ and $j$ . If we keep fixed $j = n-1> K$ , then (2) holds for all $k$ with $k_{*}\le k < j$ , where the bound $k_{*} < K$ is explicitly determined, and is optimal for $n\le 43$ . In Theorem 2.6 we return to G. Szegö’s original inequality (1) and constructively prove the non - uniqueness of its extremizer $\pm T_n$ .     

Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product

Discrete orthogonal polynomials – polynomial modification of a classical functional

Polynomial modifications of a classical discrete linear functional are examined in detail, in particular when the new linear functional remains classical. New addition formulas are deduced for Charlier, Meixner and Hahn polynomials from the Christoffei representation and results are also given for a particular generalized Meixner family.     

A note on the zeros of Freud–Sobolev orthogonal polynomials

We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e^-x4${\mathrm{e}}^{-x^4}$ on R$\mathbb{R}$ are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e^-x4${\mathrm{e}}^{-x^4}$. Some numerical examples are shown.     

A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.Three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.Three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.     

Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature

The connection between orthogonal polynomials, Padé approximants and Gaussian quadrature is well known and will be repeated in section 1. In the past, several generalizations to the multivariate case have been suggested for all three concepts [4,6,9,...], however without reestablishing a fundamental and clear link. In sections 2 and 3 we will elaborate definitions for multivariate Padé and Padé-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new m-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree 2m−1, in any number of variables. A numerical application of the new integration rules can be found in sections 4 and 5. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.