The central idea in approximation of functions can be illustrated by the question: Given a set of functions A and an element u in A, if we select a subset B of A, can we choose an element U in B so that U approximates u in some way? The course focuses on this question in the context of functions when the way we measure 'goodness' of approximation is either with an integral least square norm or with an infinity norm of the difference u-U. The choice of measure leads to further questions: is there a best approximation; if a best approximation exists, is it unique, how accurate is a best approximation and can we develop algorithms to generate good approximations? This course aims to give a grounding in the advanced theory of such ideas, the analytic methods used and important theorems for real functions. As well as being a beautiful subject in its own right, approximation theory is the foundation for many of the algorithms of computational mathematics and numerical analysis.
Introduction to approximation. Approximation in L2. Approximation in L∞: Oscillation Theorem, Exchange Algorithm. Approximation with splines. Rational approximation. Approximation of perodic functions.
