# Fourier Analysis

Mind Map by lucio_milanese, updated more than 1 year ago
 Created by lucio_milanese almost 6 years ago
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### Description

Mind Map on Fourier Analysis, created by lucio_milanese on 03/31/2014.

## Resource summary

Fourier Analysis
1 Fourier Series
1.1 A Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials).
1.1.1 It is very useful in Physics because it allows us to detect and analyse periodicities in apparently random functions
1.1.2 Useful for solving linear differential equations - superposition
1.2 Every function can be represented as a sum of even and odd functions
1.3 The Euler-Fourier formulae (TO REMEBER!) help up to compute the coefficients an and bn
1.3.1 If f(x) is odd, no an coefficients will be present
1.3.2 If f(x) is even, no bn coefficients will be present
1.3.3 Dirichlet conditions, sufficient but not necessary
1.3.3.1 1. f(x) must be periodic 2. f(x) must be single valued with a finite number of discontinuities in one period 3.f(x) must have a finite number of discontinuities in one period 4.It has to be possible to compute the integral of the absolute value of f(x) over a period
1.4 Convergence of Fourier series
1.4.1 IMPORTANT: Gibbs Phenomenon
1.4.1.1 Fourier series overshoot at a jump and this overshooting is not eliminated even with a very high number of elements in the sum
1.5 Fourier/Frequency Space
1.5.1 The frequency components, spread across the frequency spectrum, are represented as peaks in the frequency domain. (See wikipedia's excellent animation on the "frequency domain" page)
1.6 Parseval's theorem
1.6.1 The average value of the square of a function is equal to the sum of the average values of the square of the Fourier compontents
2 Fourier Transforms
2.1 Provide a Fourier representation of non-periodic functions
2.1.1 It is employed to transform signals between time (or spatial) domain and frequency domain. It is reversible, being able to transform from either domain to the other.
2.2 It could be useful to remember some of the most important fourier transform pairs
2.2.1 sine/cosine - is a combination of dirac delta functions
2.2.2 delta function - is a constant (if we are absolutely sure about the position in x or t domain, then we will have an infinite spread in the frequency domain - very important in quantum mechanics)
2.2.3 gaussian - is a gaussian!
2.2.4 The Fourier transform of a Fourier transform is the original function over 2Pi
2.3 Theorems
2.3.1 1. Linearity: F[af(x)+bg(x)] = aF(f(x))+bF(g(x)) 2.Shift theorem F(f(x+a))=e^iaω*g(ω) 3.Scaling F(f(ax))=1/|a|*g(ω/a) 4.Exponential moltiplication F(e^ax*f(x))=g(ω+ia) 5.Convolution Theorem F[f(x) convoluted with g(x)] = 2PiF[f(x)]F[g(x)]
3 Convolution
3.1 Convolution Theorem F[f(x) convoluted with g(x)] = 2PiF(x)Fg(x)
3.2 Be careful with extrema of integration

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