Fourier Analysis

orthogonality properties:

to prove orthogonality the inner product = 0
the inner product (f(x),g(x)) on the interval [a,b]=∫abf(x)⋅g(x)dx
the norm of a f(x) ∣∣f(x)∣∣= square root of the inner product of the function and itself

Problem 1. prove that the system {1,x,23x2−21} is orthogonal over the interval [-1,1]. what is the value of ∣∣x∣∣2 ?

for the orthogonality of the system each two functions must be orthogonal

(1,x)=∫−11xdx=0

(1,23x2−21)=∫−1123x2−21dx=2∫0123x2−21dx=0

(x,23x2−21)=∫−1123x3−21xdx=0

∣∣x∣∣2=(x,x)=2∫01x2dx=2×31

Fourier series for functions of general period:

f(x)=2a0+n=1∑∞(ancoslnπx+bnsinlnπx)

a0=l1∫−llf(x)dx an=l1∫−llf(x)coslnπxdx bn=l1∫−llf(x)sinlnπxdx

l = half the period so if l=π→π/l=1
2a0= average value of f(x) over one period
if the period is not [−l,l]⟹f(x) can't be odd or even
If f(x) is neither odd nor even calculate all coefficients
If f(x) is odd only calculate bn (coeff of odd func: sin) as a0&an=0
If f(x) is even only calculate a0&an (coeff of even func: 1 and cos) as bn=0

Half-range Fourier series:

Fourier Sine series: complete the definition as odd function
Fourier Cosine series: complete the definition as even function
Full Fourier series: l = half the given period

Example (5): Find a Fourier series representation of f(x)=x20<x<1

(a) As a sine series
(b) As a cosine series
(c) As full Fourier series

(a) f(x) is odd so a0=an=0

bn=2∫01x2sin1nπxdx=2[x2×−nπ1cosnπx+2x×n2π21sinnπx+2×n3π31cosnπx]01

=2[(−nπ1×(−1)n+0+n3π32×(−1)n)−(n3π32)]=Bn

∴f(x)=n=1∑∞Bn(x)sinnπx

dxd	∫dx
x2	sinnπx
2x	−nπ1cosnπx
2	−n2π21sinnπx
0	n3π31cosnπx

(b) f(x) is even so bn=0

a0=2∫01x2dx=32

an=2∫01x2cosnπxdx=2[x2×nπ1sinnπx+2x×n2π21cosnπx−2×n3π31sinnπ]01

=2(2×n2π2(−1n)−0)=4n2π2(−1n)

∴f(x)=31+n=1∑∞4n2π2(−1n)cosnπx

a0=2∫01x2dx=32

an=2∫01x2cos2nπxdx

bn=2∫01x2sin2nπxdx

f(x)=2a0+n=1∑∞[ancos2πx+bnsin2nπx]

Fourier Integral:

f(x)=π1∫0∞A(ω)cosωx+B(ω)sinωxdω

A(ω)=∫−∞∞f(x)cosωxdx

B(ω)=∫−∞∞f(x)cos(ω)xdx

A(ω)=0 if f(x) is odd over the period [−∞,∞]

B(ω)=0 if f(x) is even over the period [−∞,∞]

Example (1):

Find the Fourier integral representation of the function

f(x)={10∣x∣<1∣x∣>1

f(x) is even so B(ω)=0

A(ω)=∫−∞∞f(x)cosωxdx=∫−11cosωxdx=2∫01cosωxdx=ω2sinωx∣∣∣x=01=ω2sinω

f(−1)=21+0=21=f(1)

f(x)=⎩⎪⎨⎪⎧1210∣x∣<1∣x∣=1∣x∣>1=π1∫0∞ω2sinω×cosωxdω

∫0∞ωsinω×cosωxdω=⎩⎪⎨⎪⎧π/2π/40∣x∣<1∣x∣=1∣x∣>1

(18)* Using the Fourier integral representation, verify the identities

4.∫0∞ω2+k2ωsinxωdω=2πe−kxk>0,x>0

let f(x)=2e−kx=π1∫0∞B(ω)sinωxdω

A(ω)=0 because it is a Fourier sine integral

B(ω)=2∫0∞21e−kxsinωxdx=k2+ω2ω

f(x)=π1∫0∞k2+ω2ωsinωxdω=2e−kx×π⟹#