Fourier series/transform

1. Fourier series (−π, π)
   

   Fourier coefficient Fourier series Fm = 1 2π ⌠ ⌡ π −π  f(x) e−i m xdx f(x) = m=∞ ∑ m=−∞  Fm ei m x

   (1)
2. Fourier transform (−∞, ∞) Three definitions:
   
   

   Fourier transform Inverse Fourier transform F(ω) = 1 2π ⌠ ⌡ ∞ −∞  f(x) e−i ωxdx f(x) = ⌠ ⌡ ∞ −∞  F(ω) ei ωx dω F(ω) = ⌠ ⌡ ∞ −∞  f(x) e−i ωxdx f(x) = 1 2π ⌠ ⌡ ∞ −∞  F(ω) ei ωx dω F(ω) = 1 √ 2π ⌠ ⌡ ∞ −∞  f(x) e−i ωxdx f(x) = 1 √ 2π ⌠ ⌡ ∞ −∞  F(ω) ei ωx dω

   (2)

Properties of Fourier transform

(1) (Linearity)



F(α f + β g) = α F(f) + β F(g)

(5)

where α and β are constants. This property comes from the linearity of the integral operator.
(2)



F(f(n)(x)) = (iω)n F(f(x))

(6)

(Proof)
For n=1,



F(f ′(x)) = ⌠ ⌡ ∞ −∞  f  ′(x) e−iωx dx = ⎡ ⎣ f(x) e−iωx ⎤ ⎦ ∞ −∞  − ⌠ ⌡ ∞ −∞  f(x) (−iω) e−iωx dx = iω ⌠ ⌡ ∞ −∞  f(x) e−iωx dx = iωF(ω), (7)

where f(x)→ 0 as x→ ±∞ was used. This condition is required for the existence of Fourier transforms. Use mathematical induction for general n. It is seen that once transformed into the frequency domain, differentiation becomes multiplication by iω.
(Example):


u"" + u = w(x).

(8)

where w(x) is a given function defined over (−∞,∞). Fourier transforming both sides of Eq.(8) into the frequency domain yields



(iω)4 U (ω) + U(ω) = W(ω),

(9)

where U(ω) and W(ω) are the Fourier transforms of u(x) and w(x), respectively. Equation (9) can be solved as



U(ω) = W(ω) 1 + ω4 .

(10)

By using the inverse Fourier transform, one obtains



u(x) = 1 2π ⌠ ⌡ ∞ −∞  U(ω) eiωx dω = 1 2π ⌠ ⌡ ∞ −∞  W(ω) 1 + ω4 eiωxdω = 1 2π ⌠ ⌡ ∞ −∞  ⎛ ⎝ ⌠ ⌡ ∞ −∞  w(y) e−i y ω dy ⎞ ⎠ ei x ω 1 + ω4 dω = ⌠ ⌡ ∞ −∞  ⎛ ⎝ 1 2π ⌠ ⌡ ∞ −∞  eiω(x − y) 1 + ω4 dω ⎞ ⎠ w(y) dy = ⌠ ⌡ ∞ −∞  G(x − y) w(y) dy, (11)

where


G(x − y) ≡ 1 2π ⌠ ⌡ ∞ −∞  eiω(x − y) 1 + ω4 dω.

(12)

The function, G(x − y), is called the Green's function and is the inverse Fourier transform of [1/(1+ω⁴)].



(3) (Fourier convolution)
Fourier convolution between two functions, f(x) and g(x), is defined as


f * g ≡ ⌠ ⌡ ∞ −∞  f(x−y) g(y) dy.

(13)

Note that


f * g = g * f.

(Proof)


f * g = ⌠ ⌡ ∞ −∞  f(x−y) g(y) dy = ⌠ ⌡ −∞ ∞  f(z) g(x−z) (−dz) = ⌠ ⌡ ∞ −∞  g(x−y) f(y) dy = g * f. (14)

where x−y ≡ z was used in Eq.(14).

Enter the function to be inverse Fourier transformed. with respect to .