Introduction and motivation Full range Fourier series Completeness and convergence theorems Fourier series of odd and even functions Arbitrary range Fourier series Integral transform: a primer of Fourier transform CALCULUS III CHAPTER 6: Short introduction to Fourier series

t f(t) 24  Introduction and motivation Suppose you want to digitalize music (MP3) You need a way to describe signals as a function of time, that is, functions f(t) !!!  Harmonic analysis will do that for you: this branch of maths is concerned with the representation of functions or signals as the superposition (linear combination) of basic (simple) waves  Fourier analysis does that, the basic waves are trigonometric functions and the superposition outcome is called a Fourier series.  For instance, a pure tone is f(t) = A sen(2 π f t), where the frequency f describes the pitch of the tone. Suppose you want to study problems modeled by partial differential equations such as the problem of heat propagation  Solutions can typically be expanded as a linear combination of ‘blocks’

Introduction and motivation  (algebraically) the function f(x) is ‘decomposed’ in a linear combination of more basic functions  (geometrically) the function f(x) is ‘projected’ in a basis, where each ‘direction’ is given by a sine/cosine Such basis can be indeed understood as an orthogonal curvilinear coordinate system defined in an infinite dimension Hilbert space

Full range Fourier series Orthogonal basis

Full range Fourier series: calculating the Fourier coefficients The case m=0 actually works in the same way (that’s why the ½ is plugged in a 0 )

Full range Fourier series: calculating the Fourier coefficients summarizing

Full range Fourier series: calculating the Fourier coefficients

 Recall that f(x) is periodic f(x+T)=f(x), therefore S(x) is periodic as well S(x+T)=S(x).  Graphically, this is equivalent to just making a copy of the resulting S(x) infinitely many times.  The Fourier series is an infinite one, each of the terms is called an harmonic.  The function f(x) and its expansion in Fourier series S(x) only coincide exactly for the inifinite series, if we truncate the series up to harmonic p, the expansion will only be an approximation, called a partial sum

Gibbs phenomenon  If the periodic function to be expanded in Fourier series has jump discontinuities, a number of oscillations appear close to the discontinuities in the representation of the Fourier partial sums  These oscillations do not disappear in the limit of taking an infinite amount of harmonics, but rather approach a bounded, finite limit (Wilbrehem-Gibbs constant)  In the limit of taking the whole Fourier expansion, these oscillations are concentrated in a point that tends to be at the discontinuity, so that f(x)=S(x) at every point except at the discontinuity, where S(x) will tend towards the average of the functions on either side of the jump.  This is an example of pointwise convergence rather than uniform convergence.

Completeness and convergence theorem

Parseval theorem  Parseval theorem provides us a recipe to quantify how good are we doing if we approximate f(x) by a partial sum of S(x) instead of the infinite series.

Parseval theorem  Parseval theorem provides us a recipe to quantify how good are we doing if we approximate f(x) by a partial sum of S(x) instead of the infinite series.  Define the function how is the Fourier series of this new function? The Fourier series of that function has null coefficients up to n=N, and the same coefficients of f(x) for n>N Applying Parseval’s theorem to that new function, it turns out that where a n, b n are the coefficients associated to f(x)

Odd and even functions Parity of function multiplication Parity of trigonometric functions Parity of functions Fourier coefficients of f(x) Even/odd functions integrated over symmetric ranges [-x,x]

Fourier series for a half range Any function φ(x) can always be written as φ(x)= ½ [φ(x)+ φ(-x)] + ½ [φ(x)- φ(-x)] f(x) g(x)

Arbitrary range Fourier series: rescaling

A primer on Fourier Transforms