MA5241 Lecture 1 TO BE COMPLETED Background Convexity, inequalities and norms 1.1 Characters 1.2 Some tools of the trade 1.3 Fourier series: Lp-theory Dirichlet and Fejer kernels Convergence in norm of Fejer sums 1.4 Fourier series: L2-theory Convergence in norm of Dirichlet sums 1.5 Fourier analysis of measures Herglotz theorem for positive-definite functions

Convexity Definition A subset K of a real vector space V is convex if Definition A function f : [a,b]  R is convex if the cord connecting any two points on its graph lies on or above the graph

Convexity Question Show that if f is convex then f is continuous

Jensen’s Inequality Question Derive Jensen’s Inequality for convex f Suggestion Use the answer to the previous result in combination with an induction argument http://en.wikipedia.org/wiki/Jensen's_inequality

Arithmetic-Geometric Inequality Question Show that for the Geometric Mean and Arithmetic Mean satisfy Suggestion Consider the function

Harmonic-Geometric Inequality Question Show that for the Harmonic Mean satisfies

Young’s Inequality About Products satisfy If then for all Proof Set Since is convex

Legendre Transform Definition Let Definition The Legendre transform of a convex function http://en.wikipedia.org/wiki/Legendre_transform Question Show that Question Show that and use this to derive Young’s Inequality

Function Spaces a measure space and Definition For define For such that For let denote the p-th root of this integral. Question Show that and that is a complex vector space

Hölder's Inequality Theorem Let satisfy and Then and Proof Assume (WLOG) that Young’s inequality implies whence the assertion follows by integration. http://en.wikipedia.org/wiki/H%C3%B6lder's_inequality

Minkowski's Inequality Theorem Let and Then Proof http://en.wikipedia.org/wiki/H%C3%B6lder's_inequality

Lebesgue or Spaces are Normed Spaces since they satisfy properties: Positivity Homogeneity Triangle Inequality hence they are metric spaces with distance function Furthermore, every Cauchy sequence converges so they are complete normed spaces or Banach Spaces http://en.wikipedia.org/wiki/Lp_space http://en.wikipedia.org/wiki/Banach_space

The Approximation Problem Given an element f and a subset A of a metric space B find an approximation a from A to f An approximation a* is BEST if d(a*,f) d(a,f) for every a from A Theorem 1.1 If A is a compact subset of a metric space then for every f in B there exists a best approximation a* from A to f. Proof pages 4-5 in Powell

Approximation in a Normed Space Theorem 1.2 If A is a finite dimensional subspace of a normed space B, then for every f in B there exists a best approximation a* from A to f. Proof page 6 in Powell Question Show that C([a,b]) with norm is a Banach space. Theorem 1.3 For all Proof pages 8-9 in Powell

Geometry of a Norm Given a normed space the closed ball of radius centred at is Question Show that all balls are mutually similar Question Show that they are closed (contain all of their limit points) and bounded Question Show that they are convex Question Define open balls

Geometry of a Norm Consider the measure space wi the closed ball of radius centred at is Question Show that all balls are mutually similar Question Show that they are closed (contain all of their limit points) and bounded Question Show that they are convex

Geometry of Best Approximation To be completed