Franz Peherstorfer

Asymptotic behavior of polynomials orthonormal on a homogeneous set

LetE be a homogeneous compact set, for instance a Cantor set of positive length. Further, let σ be a positive measure with 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 orthonomal with respect to σ. For the special case thatE consists of a finite number of intervals and that σ has no singular component, this is a well-known result of Widom. IfE=[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 ofE which are supposed to accumulate only onE 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. In other words, the Jacobi matrices associated with the above discussed orthonomal polynomials are compact perturbations of a onesided 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.

Orthogonal and extremal polynomials on several intervals

Abstract Let a1 1 ρ , where ρ is positive on El, are characterised by orthogonality conditions and some new results are proved. Then the connection with polynomials orthogonal with respect to |H| |ρ| , where ρ has an odd number of zeros in each interval[a2j, a2j+1], j = 1,…,l − 1, is shown and a full description of orthogonal polynomials having periodic respectively asymptotic periodic recurrence coefficients is given. Finally the asymptotic behaviour of polynomials orthogonal on several intervals is discussed.

On Bernstein-Szego¨ orthogonal polynomials on several intervals. II.: Orthogonal polynomials with periodic recurrence coefficients

Abstract Geronimus has shown that a sequence of orthogonal polynomials (pn) with periodic recurrence coefficients for n ⩾ n0 is orthogonal on a set of disjoint intervals el = Uj = 1l = [a2j − 1, a2j] with respect to a distribution of the form dψ(x)= − ∏ j=1 2l (x−a j |p v (x)|dx+dμ(x) where ρv,(x) = Πj = 1v (x − Wj) with sgn νv(x) = (−1)l + 1 − j on (a2j − 1, a2j) for j = 1, …, l, v ⩾ l − 1, and where μ is a certain point measure with supp(μ) ⊂{w1, …, wv}. In this paper we show (in fact a more general result is presented) that a sequence of polynomials (pn) orthogonal with respect to dψ has recurrence coefficients of period N, N ⩾ l, for n ⩾ n0, if and only if there exists a so-called Chebyshev polynomial T N of degree N on El, where a polynomial T N is called a Chebyshev polynomial on El if ¦ T N| attains its maximum value on El at N + l points from El. Furthermore it is demonstrated how to get in a simple way a (nonlinear) recurrence relation for the recurrence coefficients of the orthogonal polynomials. Results on Chebyshev polynomials on several intervals are also given.

On Bernstein-Szego¨ orthogonal polynomials on several intervals

Let

Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points

l \in \mathbb{N}

Zeros of polynomials orthogonal on several intervals

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On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on the Gauss-Kronrod quadrature formulas

a_1 < a_2 < \cdots < a_{2l}

Linear combinations of orthogonal polynomials generating positive quadrature formulas

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ON GENERALIZED SUM RULES FOR JACOBI MATRICES

E_l = \bigcup _{k = 1}^l [a_{2k - 1} ,a_{2k} ]

Strong Asymptotics of Orthonormal Polynomials with the Aid of Green's Function

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