Asymptotic behavior of Müntz orthogonal polynomials

SIAM Student Seminar
Friday, October 30, 2009 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 255
Ulfar Stefansson – School of Mathematics, Georgia Tech
Linwei Xin
After a brief introduction of the theory of orthogonal polynomials, where we touch on some history and applications, we present results on Müntz orthogonal polynomials. Müntz polynomials arise from consideration of the Müntz Theorem, which is a beautiful generalization of the Weierstrass Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials which holds on the interval of orthogonality, and in particular we get new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. We also look at the asymptotic behavior outside the interval, where we apply the method of stationary phase.