Analysis Seminar: Müntz-Szasz Approximation Theorems

Abstract: Polynomial approximation theory was initiated by Weierstrass in the late 1800s. Bernstein conjectured what would eventually become the C[0,1] Müntz-Szasz theorem (Müntz gave a proof of the continuous version and Szasz gave an L^2 formulation). Although purely real in statement, the theorem proves particularly amenable to complex analytic techniques. An asymptotic form of the theorem was later derived by Agler and McCarthy; who also introduced a convenient formalism for discussing limits of monomial spaces. The Sarason transform with also be discussed for its utility in transforming real problems into Hardy space language. As time allows-new questions and ideas related to limits of monomial spaces will be discussed.