MA5241 Lecture 1

1. TO BE COMPLETED Background Convexity, inequalities and norms MA5241 Lecture 1 1.2 Some tools of the trade 1.3 Fourier series: Lp-theory Dirichlet and Fejer kernels 1.1 Characters 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

2. 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

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

4. 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

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

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

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

8. 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

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

10. 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

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

12. Spaces Lebesgue or 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

13. 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

14. 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

15. Geometry of a Norm Given a normed space centred at is the closed ball of radius 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

16. Geometry of a Norm Consider the measure space wi centred at is the closed ball of radius 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

17. Geometry of Best Approximation To be completed