Extremal Polynomials

Extremal polynomials that are polynomials which deviate least from zero on a given compact set among all polynomials of degree n with leading coefficient one as orthogonal - or Chebyshev polynomials appear in many areas of applied mathematics, as in the fields numerical analysis, signal theory , probability theory etc.. In recent years it turned out that they are of importance also in fields as random matrix theory, combinatorics, integrable systems. But in contrast to the classical fields of applications where extremal polynomials which live on a single interval are mainly of foremost interest now extremal polynomials on nonconnected sets, as on several intervals, Cantor sets, Julia sets, ... play the crucial role. One of the main goals of this project is to derive asymptotic representations for extremal polynomials on nonconnected sets with respect to the L_q-norms, including the maximum norm, and to obtain information about the behaviour of their zeros. In contrast to the single interval case where precise asymptotics and very detailed informations on the zeros are available for the nonconnected case, even for the several interval case, the situation is opposite. Apart from the L_2 case, i.e., the case of orthogonal polynomials, for which in the sixties Achieser and Widom derived asymptotics for the several interval and arc case not much is known about these problems. For instance concerning the zeros such simple questions as how many zeros has the extremal polynomial in each of the components, outside of the components, what is the spacing of the zeros,...(question of particular interest in random matrix theory, combinatorics and integrable systems) are still open, though they were already mentioned and studied by old masters like Markoff, Stieltjes, Heine etc.. An understanding of the behaviour of the zeros of extremal polynomials would also enable us to understand better the dynamical behaviour of the solutions of certain integrable systems, as of the particles of the Toda lattice. Many of the above problems are closely related to classical problems of analysis on Riemann surfaces, as to the Jacobi- inversion problem or the representation and the determination of the number of zeros of functions analytic on a Riemann-surface.

Extremal polynomials that are polynomials which deviate least from zero on a given compact set among all polynomials of degree n with leading coefficient one as orthogonal - or Chebyshev polynomials appear in many areas of applied mathematics, as in the fields numerical analysis, signal theory , probability theory etc.. In recent years it turned out that they are of importance also in fields as random matrix theory, combinatorics, integrable systems. But in contrast to the classical fields of applications where extremal polynomials which live on a single interval are mainly of foremost interest now extremal polynomials on nonconnected sets, as on several intervals, Cantor sets, Julia sets, ... play the crucial role. One of the main goals of this project is to derive asymptotic representations for extremal polynomials on nonconnected sets with respect to the L_q-norms, including the maximum norm, and to obtain information about the behaviour of their zeros. In contrast to the single interval case where precise asymptotics and very detailed informations on the zeros are available for the nonconnected case, even for the several interval case, the situation is opposite. Apart from the L_2 case, i.e., the case of orthogonal polynomials, for which in the sixties Achieser and Widom derived asymptotics for the several interval and arc case not much is known about these problems. For instance concerning the zeros such simple questions as how many zeros has the extremal polynomial in each of the components, outside of the components, what is the spacing of the zeros,...(question of particular interest in random matrix theory, combinatorics and integrable systems) are still open, though they were already mentioned and studied by old masters like Markoff, Stieltjes, Heine etc.. An understanding of the behaviour of the zeros of extremal polynomials would also enable us to understand better the dynamical behaviour of the solutions of certain integrable systems, as of the particles of the Toda lattice. Many of the above problems are closely related to classical problems of analysis on Riemann surfaces, as to the Jacobi- inversion problem or the representation and the determination of the number of zeros of functions analytic on a Riemann-surface.