12) Fourier Series Flashcards
Fourier series as infinite trigonometric series
The sum is from n=1 to ∞.
S(x) = a_0 /2 + Σ (a_ncosnx + b_nsinnx)
Where constants a_i and b_i are called FOURIER COEFFICIENTS defined…
Fourier vs Taylor
Taylor shows local behaviour of stationary points and coefficients found by differentiation. Used in grad and hessian.
Fourier series uses integration and dirichlet conditions, even and odd functions.
Dirichlet conditions
On x between -pi and pi. Conditions are:
1) f(x) must be single valued and bounded
2) f(x) continuous on interval except for finite discontinuities
3) f(x) finite number of maxima and minima in the interval
If there are finite discontinuities then we use improper Integrals. Often if there are we evaluate these parts separately to continuous part.
Fourier coefficients
Defined as Integrals:
a_0 = (1/π) integral_-π_to_π( f(x).dx)
a_n = (1/π) integral_-π_to_π( f(x)cos(nx) .dx)
For n=1,2,3,…
b_n = (1/π) integral_-π_to_π( f(x)sin(nx).dx)
For n=1,2,3,…
They are found from orthogonality relations Integrals of cosnxsinmx or cosnxcosmx etc for the interval =0 pi times kronecker delta function
Integrand is a product so we often use BY PARTS
Periodicity in the Fourier series
Fundamental period 2π:
S(x+2kπ) =S(x) for all x in the reals and k in Integers.
For all x outside the interval f(x +2kπ) =f(x)
The dirichlet conditions ensure that for values of x at which f(x) is continuous Fourier series converges to value f(x).
At finite discontinuities in Fourier series
At finite discontinuities x=a Fourier series converges the the mean of the two limiting values if f(x) as x tends to a from both sides.
Hence it converges to:
½lim_h->0{ (f(a-h) + f(a+h)}
Even or odd functions in Fourier series
EVEN: f(-x)=f(x). Hence if f(x) is even then f(x)cosmx is even and f(x)sinmx is odd. Then Fourier series becomes: COSINE SERIES
f(x) = (a_0/2) + Σ (a_n*cosnx )
As b_n=0
ODD: f(-x)=-f(x). Hence if f(x) is odd then f(x)cosmx is odd and f(x)sinmx is even.hen Fourier series becomes: SINE SERIES
f(x) = (a_0/2) + Σ (b_n*sinnx )
As a_n=0
And we have, for any odd function g(x): integral_-pi_to_pi(g(x).dx) =0
Sum from 1 to infinity.
(Hence Fourier coeffs)
Hence check if even or odd function first. Then usually use by parts to find coefficients for the series. Often the periodic extension has discontinuities. Where Fourier series gives mean of left right .
Fourier series over other intervals
For series over general interval (-L,L) for some positive constant L. The fundamental periodicity is 2L. Must satisfy the dirichlet conditions on this interval but Fourier series given by:
f(x) = (a_0/2) + Σ (a_ncos( nπx/L) + b_nsin( nπx/L))
Sum is from 1 to infinity.
Where coeffiecients are also different.
Fourier series over different interval coefficients.
Coefficients:
Defined as Integrals:
a_0 = (1/L) integral_-L_to_L( f(x).dx)
a_n = (1/L) integral_-L_to_L( f(x)cos(nπx/L) .dx)
For n=1,2,3,…
b_n = (1/π) integral_-L_to_L( f(x)sin(nπx/L).dx)
For n=1,2,3,…
REPLACING WITH L.
Similarly even and odd functions represented by cosine and sine functions.
Integrand is a product so we often use BY PARTS
Fourier series over general interval THAT IS NOT SYMMETRICAL ABOUT x=0
E.g. For (a,b] then the function g(x) =f(x-x_0) is defined on the symmetric interval (-L,L]. Where
x_0 = (a+b)/2 and L= (b-a)/2
Fourier coefficients extra detail
The magnitude of the Fourier coefficients decreases as N increases
and the partial sums of the Fourier series provide more accurate approximations to the function
