Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros

Let $E = \cup_{j = 1}^l [a_{2j-1},a_{2j}],$ $a_1 < a_2 < ... < a_{2l},$ $l \geq 2$ and set {\boldmath\omega}(\infty) =(\omega_1(\infty),...,\omega_{l-1}(\infty)), where $\omega_j(\infty)$ is the harmonic measure of $[a_{2 j - 1}, a_{2 j}]$ at infinity. Let $\mu$ be a measure which is on $E$ absolutely continuous and satisfies Szeg\H{o}'s-condition and has at most a finite number of point measures outside $E$, and denote by $(P_n)$ and $({\mathcal Q}_n)$ the orthonormal polynomials and their associated Weyl solutions with respect to $d\mu$, satisfying the recurrence relation $\sqrt{\lambda_{2 + n}} y_{1 + n} = (x - \alpha_{1 + n}) y_n -\sqrt{\lambda_{1 + n}} y_{-1 + n}$. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence (n {\boldmath\omega}(\infty))_{n\in \mathbb N} modulo 1; More precisely, putting ({\boldmath\alpha}^{l-1}_{1 + n}, {\boldmath\lambda}^{l-1}_{2 + n}) = $(\alpha_{[\frac{l 1}{2}]+1+n},...,$ $\alpha_{1+n},...,$ $\alpha_{-[\frac{l-2}{2}]+1+n},$ $\lambda_{[\frac{l-2}{2}]+2+n},$ $...,\lambda_{2+n},$ $...,$ $\lambda_{-[\frac{l-1}{2}]+2+n})$ we prove that ({\boldmath\alpha}^{l-1}_{1 + n_\nu}, {\boldmath\lambda}^{l-1}_{2 + n_\nu})_{\nu \in \mathbb N} converges if and only if (n_\nu {\boldmath\omega}(\infty))_{\nu \in \mathbb N} converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of ({\boldmath\alpha}^{l-1}_{1 + n}, {\boldmath\lambda}^{l-1}_{2 + n}) and (n{\boldmath\omega}(\infty)) modulo 1.Comment: The last modifications and corrections of this manuscript were done by the author in the two months preceding this passing away in November 2009. The manuscript is not published elsewher

Positive trigonometric Quadrature Formulas and quadrature on the unit circle

We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally, asymptotics for the quadrature weights are presented.Comment: Submitted by F. Peherstorfer to Math. Com

Uniform approximation of sgn(x) by rational functions with prescribed poles

For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function \sgn(x) on the two symmetric intervals $[-1,-a]\cup[a,1]$ by rational functions with the only possible poles of degree $2k-1$ at the origin and of $2m-1$ at infinity. Then the following limit exists \begin{equation} \lim_{m\to \infty}L^k_m(a)(\frac{1+a}{1-a})^{m-{1/2}} (2m-1)^{k+{1/2}}=\frac 2 \pi(\frac{1-a^2}{2a})^{k+{1/2}} \Gamma(k+\frac 1 2). \end{equation

Explicit min–max polynomials on the disc

AbstractDenote by Πn+m−12≔{∑0≤i+j≤n+m−1ci,jxiyj:ci,j∈R} the space of polynomials of two variables with real coefficients of total degree less than or equal to n+m−1. Let b0,b1,…,bl∈R be given. For n,m∈N,n≥l+1 we look for the polynomial b0xnym+b1xn−1ym+1+⋯+blxn−lym+l+q(x,y),q(x,y)∈Πn+m−12, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial 2Pn,m(x,y)=xGn−1,m(x,y)+yGn,m−1(x,y)=2xnym+q(x,y) and q(x,y)∈Πn+m−12, where Gn,m(x,y)≔1/2n+m(Un(x)Um(y)+Un−2(x)Um−2(y)), and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients bj,j=0,…,l,l fixed, such that for every n,m∈N,n≥l+1, the linear combination ∑ν=0lbνPn−ν,m+ν(x,y) is a min–max polynomial. In fact the more general case, when the coefficients bj and l are allowed to depend on n and m, is considered. So far, up to very special cases, min–max polynomials are known only for xnym,n,m∈N0