1. 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

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

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

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
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
For
let
denote the p-th root of this integral.
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.

11. Minkowski's Inequality
Theorem Let
and
Then
Proof

12. 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

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 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

16. 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

17. Geometry of Best Approximation
To be completed