MA4229 Lectures 13, 14 Week 12 Nov 1, Chapter 13 Approximation to periodic functions

Periodic Functions We will let denote the real vector space represent continuous periodic functions on of continuous real valued functions Functions in that satisfy that have for period This means that they satisfy the condition Examples has as its set of periods the subgroup therefore so

Trigonometric Polynomials denote the real vector space of functions Question Why is Clearly where the coefficients Consider the function for every However for the function satisfies therefore can be approximated with arbitrary accuracy by trigonometric polynomials.

Trigonometric Polynomials This space satisfies some properties similar to those Property 2. Property 1. They are both real vector spaces. Property 3. They are both closed under translation Question Show that Question Describe satisfied by the space of algebraic polynomials. using Laurent polynomials. Question How many distinct roots can have? Define the space of (all) trig. polynomials

Approximation Theorem Theorem 13.1 For every there exist and such that Proof (pages ) uses the fact that continuous functions can be approximated to arbitrary accuracy by algebraic polynomials are dense. First express with even and odd Next define by Thm 6.1  there exists an algebraic poly.such that

Approximation Theorem Now the inequality so by defining the trigonometric polynomialby we obtain the approximation Since bothand are even functions we have

Approximation Theorem Now we need to construct that satisfies because then Approximating the odd functionis trickier than satisfies approximating the even function so we apply some magic. First observe that Choose the largestso that and choose the smallestso that

Approximation Theorem Now define the EVEN function to by and extending it on Now apply to to make it an even function. the method that we applied to obtain an even trigonometric polynomial such that and define the odd by Question Why is this odd?

Approximation Theorem We must examine three cases: Question Derive this bound for the case Therefore This concludes the proof of Theorem 13.1

The Fourier Series Operator Question Show that the set of functions is an orthonormal basis for the real Hilbert space with scalar product Define Fourier series operators

Tutorial 7 Due Thursday 4 November 1. Use the Bernstein approximation with methods 2. Compute developed in these notes to approximate defined by by defined above. 3. Derive (without using the book) the formula for the function 4. Do exercise 13.1 on page Do exercise 13.2 on page 161.