arXiv subjects

Franz Peherstorfer

Publications and source records attributed to Franz Peherstorfer.

8 recordsLinked to original sources

arXiv2010

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.

math.CA↗

arXiv2010

On some classical problems concerning $L_{\infty}$-extremal polynomials with constraints

First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 π].$ We show that the minimal polynomial is on $[0,2 π]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $τ_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linear functionals on the space of trigonometric polynomials is determined. As a consequence a conjecture by Clenshaw from the sixties on the behavior of the ratio of the truncated Fourier series and the minimum deviation is proved.

math.CA↗

arXiv2010

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 ${\boldmath$ω$}(\infty) =(ω_1(\infty),...,ω_{l-1}(\infty))$, where $ω_j(\infty)$ is the harmonic measure of $[a_{2 j - 1}, a_{2 j}]$ at infinity. Let $μ$ be a measure which is on $E$ absolutely continuous and satisfies Szegő'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 $dμ$, satisfying the recurrence relation $\sqrt{λ_{2 + n}} y_{1 + n} = (x - α_{1 + n}) y_n -\sqrt{λ_{1 + n}} y_{-1 + n}$. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence $(n {\boldmath$ω$}(\infty))_{n\in \mathbb N}$ modulo 1; More precisely, putting $({\boldmath$α$}^{l-1}_{1 + n}, {\boldmath$λ$}^{l-1}_{2 + n}) = $ $(α_{[\frac{l 1}{2}]+1+n},...,$ $α_{1+n},...,$ $α_{-[\frac{l-2}{2}]+1+n},$ $λ_{[\frac{l-2}{2}]+2+n},$ $...,λ_{2+n},$ $...,$ $λ_{-[\frac{l-1}{2}]+2+n})$ we prove that $({\boldmath$α$}^{l-1}_{1 + n_ν},$ ${\boldmath$λ$}^{l-1}_{2 + n_ν})_{ν\in \mathbb N}$ converges if and only if $(n_ν{\boldmath$ω$}(\infty))_{ν\in \mathbb N}$ converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of $({\boldmath$α$}^{l-1}_{1 + n}, {\boldmath$λ$}^{l-1}_{2 + n})$ and $(n{\boldmath$ω$}(\infty))$ modulo 1.

math.CA↗

arXiv2010

Degenerating behavior of Green's function

Let the unions of real intervals $I = \cup_{j = 1}^l [a_{2 j -1},a_{2j}],$ $a_1 < ... < a_{2 l},$ and $I_n = \cup_{k = 1}^m [B_{k,n}, C_{k,n}]$ be such that $\cap_{k = 1}^{\infty} [B_{k,n},C_{k,n}] = \{c_k \}$ for $k = 1,...,m$ and ${\rm dist}(E,I_n) \geq const > 0.$ We show how to express asymptotically the Green's function $ϕ(z,\infty,E \cup I_n)$ of $E \cup I_n$ at $z = \infty$ in terms of the Green's function $ϕ(z,\infty,E)$ and $ϕ(z,c_k,E).$ The formula yields immediately asymptotics for $ϕ^n(z,\infty,E \cup I_n)$ with respect to $n$ which are important in many problems of approximation theory. Another consequence is an asymptotic representation of $cap(E \cup I_n)$ in terms of $cap(E)$ and $ϕ(z,c_k,E)$ and of the harmonic measure $ω(\infty, E_j,E \cup I_n).$

math.CV↗

arXiv2010

Asymptotic representation of minimal polynomials on several intervals

Asymptotic representation of minimal polynomials on several intervals is given. The last modifications and corrections of this manuscript were done by the author in the two months preceding his passing away in November 2009. The manuscript remained unsubmitted and is not published elsewhere.

math.CA↗

arXiv2006

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 π(\frac{1-a^2}{2a})^{k+{1/2}} Γ(k+\frac 1 2). \end{equation}

math.CA↗

arXiv2002

Asymptotic behavior of polynomials orthonormal on a homogeneous set

Let $E$ be a homogeneous compact set, for instance a Cantor set of positive length. Further let $σ$ be a positive measure with $\text{supp}(σ)=E$. Under the condition that the absolutely continuous part of $σ$ satisfies a Szegö--type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to $σ$. For the special case that $E$ consists of a finite number of intervals and that $σ$ has no singular component this is a nowaday well known result of Widom. If $E=[a,b]$ it becomes a classical result due to Szegö and in case that there appears in addition a singular component, it is due to Kolmogorov--Krein. In fact the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass--points outside of $E$ which are supposed to accumulate on $E$ only and to satisfy (together with the zeros of the associated Stieltjes function) the free--interpolation Carleson--type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one--sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character--automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one--dimensional Schrödinger equations.

math.FA↗

arXiv2002

Zeros of polynomials orthogonal on several intervals

First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the zeros of the orthogonal polynomials have given accumulation points in the gaps. As a consequence it follows that every point from the gaps is an accumulation point of the zeros of the orthogonal polynomials, if the harmonic measures of the intervals are linearly independent over the rationals. If the harmonic measures are rational then there is a finite number of accumulation points only.

math-ph↗