Showing papers by "Albrecht Böttcher published in 2011"

Spectral theory of large Wiener-Hopf operators with complex-symmetric kernels and rational symbols

Abstract: This paper is devoted to the asymptotic behaviour of individual eigenvalues of truncated Wiener‐Hopf integral operators over increasing intervals. The kernel of the operators is complex-symmetric and has a rational Fourier transform. Under additional hypotheses, the main result describes the location of the eigenvalues and provides their asymptotic expansions in terms of the reciprocal of the length of the truncation interval. Also determined is the structure of the eigenfunctions.

Inequalities of the Markov type for partial derivatives of polynomials in several variables

Abstract: Markov-type inequalities give upper bounds for the derivatives of an algebraic polynomial by the polynomial itself. To be more precise, they provide a constant C such that ‖Df‖ ≤ C‖f‖ for all polynomials of degree at most n, where D is the operator of differentiation. The constant C depends on n, on the order ν of the derivative, and on the norm ‖ · ‖. We here consider the case where ‖ · ‖ is one of the classical L norms and study the problem of extending such inequalities to the situation when f is a polynomial of several variables and D is replaced by a partial differential operator. Let Pn be the linear space of all polynomials f(t) = ∑n j=0 fjt j of degree at most n with complex coefficients fj. We equip Pn with one of the classical Hermite, Laguerre, or Gegenbauer norms. These are defined by

On the best constants in Markov-type inequalities involving Gegenbauer norms with different weights

Abstract: The paper is concerned with best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial is taken in L2 with the Gegenbauer weight corresponding to a parameter α , while the derivative is measured in L2 with the Gegenbauer weight for a parameter β . Under the assumption that β −α is an integer, we determine the first order asymptotics of the best constants as the degree of the polynomial goes to infinity. Mathematics subject classification (2010): Primary 41A44; Secondary 15A18, 26D10, 45D05, 47B35.