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  • 1.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    A generalization of the Heine-Stieltjes theorem2011In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 34, no 1, p. 135-148Article in journal (Refereed)
    Abstract [en]

    The Heine-Stieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.

  • 2.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ottergren, Elin
    A Characterization of Multiplier Sequences for Generalized Laguerre Bases2014In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 39, no 3, p. 585-596Article in journal (Refereed)
    Abstract [en]

    We give a complete characterization of multiplier sequences for generalized Laguerre bases. We also apply our methods to give a short proof of the characterization of Hermite multiplier sequences achieved by Piotrowski.

  • 3. Daubechies, I.
    et al.
    Runborg, Olof
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Sweldens, W.
    Normal multiresolution approximation of curves2004In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 20, no 3, p. 399-463Article in journal (Refereed)
    Abstract [en]

    A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular, we show that these properties critically depend on the underlying subdivision scheme and that, in general, the convergence of normal multiresolution approximations equals the convergence of the underlying subdivision scheme.

  • 4.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Painlevé Kernels in Hermitian Matrix Models2014In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 39, no 1, p. 173-196Article in journal (Refereed)
    Abstract [en]

    After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4x4 Riemann-Hilbert problem related to the Painlev, II equation. Also an open problem will be presented.

  • 5. Hallnäs, Martin
    et al.
    Langmann, Edwin
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero-Sutherland Type2010In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 31, no 3, p. 309-342Article in journal (Refereed)
    Abstract [en]

    In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero-Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.

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