Function approximation: Fourier, Chebyshev, Lagrange
¾ Orthogonal functions
¾ Fourier Series
¾ Discrete Fourier Series
¾ Fourier Transform: properties
¾ Chebyshev polynomials
¾ Convolution
¾ DFT and FFT
Scope: Understanding where the Fourier Transform comes from. Moving from the continuous to the discrete world. The concepts are the basis for pseudospectral methods and the spectral element approach.
Fourier Series: one way to derive them
The Problem
we are trying to approximate a function f(x) by another function gn (x) which consists of a sum over N orthogonal functions Φ(x) weighted by some coefficients an .
) ( )
( )
(
0
x a
x g
x f
N
i
i i
N
∑
=
Φ
=
≈
... and we are looking for optimal functions in a least squares (l2 ) sense ...
... a good choice for the basis functions Φ(x) are orthogonal functions. What are orthogonal functions? Two functions f and g are said to be
orthogonal in the interval [a,b] if
∫
b =a
dx x
g x
f ( ) ( ) 0
How is this related to the more conceivable concept of orthogonal vectors? Let us look at the original definition of integrals:
The Problem
{
( ) ( )}
Min!) ( )
(
2 / 1 2
2 ⎥ =
⎦
⎢ ⎤
⎣
⎡ −
=
−
∫
ba
N
N x f x g x dx
g x
f
Orthogonal Functions
... where x0 =a and xN =b, and xi -xi-1 =Δx ...
If we interpret f(xi ) and g(xi ) as the ith components of an N component vector, then this sum corresponds directly to a scalar product of vectors.
The vanishing of the scalar product is the condition for orthogonality of vectors (or functions).
⎟⎠
⎜ ⎞
⎝
⎛ Δ
=
∑
∫
→∞ =N
i
i i
b
a f x g x dx N f x g x x
1
) ( ) ( lim
) ( ) (
fi gi • =
∑
i = 0i i i
i g f g
f
Periodic functions
-15 -10 -5 0 5 10 15 20
0 10 20 30 40
Let us assume we have a piecewise continuous function of the form
) ( )
2
(x f x
f + π =
) 2
( )
2
(x f x x
f + π = =
... we want to approximate this function with a linear combination of 2π periodic functions:
) sin(
), cos(
),..., 2
sin(
), 2 cos(
), sin(
), cos(
,
1 x x x x nx nx
{ }
∑
++
=
≈
⇒ f x gN x 1 a0 N ak cos(kx) bk sin(kx) )
( )
(
Orthogonality
... are these functions orthogonal ?
0 ,
0 0
) sin(
) cos(
0 0 ,
, 0
) sin(
) sin(
0 0 2
0 )
cos(
) cos(
>
≥
=
>
=
>
≠
⎪⎩
⎪⎨
⎧
=
>
=
=
=
≠
⎪⎩
⎪⎨
⎧
=
∫
∫
∫
−
−
−
k j
dx kx jx
k j
k j k j
dx kx jx
k j
k j
k j
dx kx jx
π π π π π
π
π π
π
... YES, and these relations are valid for any interval of length 2π.
Now we know that this is an orthogonal basis, but how can we obtain the coefficients for the basis functions?
from minimising f(x)-g(x)
Fourier coefficients
optimal functions g(x) are given if
{
( ) ( )}
0! Min )
( )
(x − f x 2 = or ∂∂ g x − f x 2 =
gn a n
k
leading to
... with the definition of g(x) we get ...
{ }
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎥⎦
⎢ ⎤
⎣
⎡ + + −
∂
= ∂
∂ −
∂ ∫ ∑
− =
dx x
f kx
b kx
a a a
x f x
a g
N
k
k k
k n
k
2
1 0
2 cos( ) sin( ) ( )
2 ) 1
( )
(
2
π π
{ }
∫
∫
∑
−
−
=
=
=
=
=
+ +
=
π π π
π
π π
N k
dx kx x
f b
N k
dx kx x
f a
kx b
kx a
a x
g
k k
N
k
k k
N
,..., 2 , 1 ,
) sin(
) 1 (
,..., 1 , 0 ,
) cos(
) 1 (
with )
sin(
) 2 cos(
) 1 (
1 0
Fourier approximation of |x|
... Example ...
.. and for n<4 g(x) looks like leads to the Fourier Serie
⎭⎬
⎫
⎩⎨
⎧ + + +
−
= ...
5 ) 5 cos(
3 ) 3 cos(
1 ) cos(
4 2
) 1
( 2 x 2 x 2 x
x
g π π
π π ≤ ≤
−
= x x
x
f ( ) ,
-20 -15 -10 -5 0 5 10
0 1 2 3 4
Fourier approximation of x
2... another Example ...
π 2 0
, )
(x = x2 < x <
f
.. and for N<11, g(x) looks like leads to the Fourier Serie
∑= ⎭⎬⎫
⎩⎨
⎧ −
+
= N
k
N kx
kx k x k
g
1
2 2
) 4 sin(
) 4 cos(
3 ) 4
( π π
-1 0 -5 0 5 1 0
-1 0 0 1 0 2 0 3 0 4 0
Fourier - discrete functions
N i xi 2π
=
.. the so-defined Fourier polynomial is the unique interpolating function to the function f(xj ) with N=2m
it turns out that in this particular case the coefficients are given by
,...
3 , 2 , 1 ,
) sin(
) 2 (
,...
2 , 1 , 0 ,
) cos(
) 2 (
1
*
1
*
=
=
=
=
∑
∑
=
=
k kx
x N f
b
k kx
x N f
a
N
j
j j
N
j
j j
k k
{ } cos( )
2 ) 1 sin(
) 2 cos(
) 1
( *
1
1
*
*
*
*
0 a kx b kx a kx
a x
g m
m
k
m = + ∑− k + k +
=
... what happens if we know our function f(x) only at the points
) ( )
* (
i i
m x f x
g =
Fourier - collocation points
... with the important property that ...
... in our previous examples ...
-10 -5 0 5
0 0.5 1 1.5 2 2.5 3 3.5
f(x)=|x| => f(x) - blue ; g(x) - red; xi - ‘+’
Fourier series - convergence
f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
Fourier series - convergence
f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
Gibb’s phenomenon
f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
0 0.5 1 1.5
-6 -4 -2 0 2 4 6
N = 32
0 0.5 1 1.5
-6 -4 -2 0 2 4 6
N = 16
0 0.5
-6 -4 -2 0 2 4 6
0 0.5 1 1.5
-6 -4 -2 0 2 4 6
N = 128
0 0.5 1 1.5
-6 -4 -2 0 2 4 6
N = 256
The overshoot for equi- spaced Fourier
interpolations is ≈14% of the step height.
Chebyshev polynomials
We have seen that Fourier series are excellent for interpolating (and differentiating) periodic functions defined on a regularly spaced grid. In many circumstances physical phenomena which are not periodic (in space) and occur in a limited area. This quest leads to the use of Chebyshev polynomials.
We depart by observing that cos(nϕ) can be expressed by a polynomial in cos(ϕ):
1 cos
8 cos
8 )
4 cos(
cos 3
cos 4
) 3 cos(
1 cos
2 )
2 cos(
2 4
3 2
+
−
=
−
=
−
=
ϕ ϕ
ϕ
ϕ ϕ
ϕ
ϕ ϕ
... which leads us to the definition:
Chebyshev polynomials - definition
N n
x x
x T T
n ) = n(cos( )) = n( ), = cos( ), ∈[−1,1], ∈
cos( ϕ ϕ ϕ
... for the Chebyshev polynomials Tn (x). Note that because of x=cos(ϕ) they are defined in the interval [-1,1] (which - however - can be extended to ℜ). The first polynomials are
0 2
4 4
3 3
2 2
1 0
and ]
1 , 1 [ for
1 )
(
where 1
8 8
) (
3 4
) (
1 2
) (
) (
1 ) (
N n
x x
T
x x
x T
x x
x T
x x
T
x x
T x T
n ≤ ∈ − ∈
+
−
=
−
=
−
=
=
=
Chebyshev polynomials - Graphical
The first ten polynomials look like [0, -1]
The n-th polynomial has extrema with values 1 or -1 at
0 0.2 0.4 0.6 0.8
-1 -0.5 0 0.5 1
x
T_n(x)
n k k
xk(ext ) = cos( π ), = 0,1,2,3,...,
Chebyshev collocation points
These extrema are not equidistant (like the Fourier extrema)
n n k
xk(ext ) = cos( kπ ), = 0,1,2,3,...,
k
x(k)
Chebyshev polynomials - orthogonality
... are the Chebyshev polynomials orthogonal?
2 0 1
1
, ,
0 0 2
/ 0 1
) ( )
( k j N
j k
for
j k
for
j k
for x
x dx T
x
Tk j ∈
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=
=
>
=
≠
− =
−
∫
ππ
Chebyshev polynomials are an orthogonal set of functions in the interval [-1,1] with respect to the weight function
such that
1 2
/
1 −x
... this can be easily verified noting that
) cos(
) ( ),
cos(
) (
sin ,
cos
ϕ ϕ
ϕ ϕ ϕ
j x
T k
x T
d dx
x
=
=
−
=
=
Chebyshev polynomials - interpolation
... we are now faced with the same problem as with the Fourier series. We want to approximate a function f(x), this time not a periodical function but a function which is defined between [-1,1].
We are looking for gn (x)
) ( )
2 ( ) 1
( )
(
1 0
0T x c T x
c x
g x
f
n
k
k k
n
∑
=
+
=
≈
... and we are faced with the problem, how we can determine the coefficients ck . Again we obtain this by finding the extremum
(minimum)
{ }
01 )
( )
( 2
1
1
2 ⎥ =
⎦
⎢ ⎤
⎣
⎡
− −
∂
∂
∫
− x
x dx f
x ck gn
Chebyshev polynomials - interpolation
... to obtain ...
n k
x x dx
T x f
ck k , 0,1,2,...,
1 ) ( ) 2 1 (
1
2 =
=
∫
− π −... surprisingly these coefficients can be calculated with FFT techniques, noting that
n k
d k f
ck 2 (cos )cos , 0,1,2,...,
0
=
= π
∫
π ϕ ϕ ϕ... and the fact that f(cosϕ) is a 2π-periodic function ...
n k
d k f
ck = 1
∫
(cos )cos , = 0,1,2,...,−
ϕ ϕ π ϕ
π
π
... which means that the coefficients ck are the Fourier coefficients of the periodic function F(ϕ)=f(cos ϕ)!
Chebyshev - discrete functions
N i
xi π
= cos
... leading to the polynomial ...
in this particular case the coefficients are given by
2 / ,...
2 , 1 , 0 ,
) cos(
) 2 (cos
1
* f k k N
c N
N
j
j
k = ∑ j =
=
ϕ ϕ
∑=
+
= m
k
k k
m x c T c T x
g
1
*
* 0
* ( )
2 ) 1
( 0
... what happens if we know our function f(x) only at the points
... with the property
N 0,1,2,..., j
j/N) cos(
x at )
( )
( j
* x = f x = π =
gm
Chebyshev - collocation points - |x|
f(x)=|x| => f(x) - blue ; gn (x) - red; xi - ‘+’
-1 -0 .8 -0 .6 -0 .4 -0 .2 0 0 .2 0 .4 0 .6 0 .8
0 0 .2 0 .4 0 .6 0 .8 1
N = 8
0.2 0.4 0.6 0.8 1
N = 1 6
8 points
16 points
Chebyshev - collocation points - |x|
f(x)=|x| => f(x) - blue ; gn (x) - red; xi - ‘+’
32 points
128 points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8 1
N = 32
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.
0 0.2 0.4 0.6 0.8 1
N = 128
Chebyshev - collocation points - x
2f(x)=x2 => f(x) - blue ; gn (x) - red; xi - ‘+’
8 points
64 points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8 1 1.2
N = 8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 1.2
N = 64
The interpolating function gn (x) was shifted by a small amount to be visible at all!
Chebyshev vs. Fourier - numerical
f(x)=x2 => f(x) - blue ; gN (x) - red; xi - ‘+’
This graph speaks for itself ! Gibb’s phenomenon with Chebyshev?
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
N = 16
0 2
-5 0 5 10 15 20 25 30 35
N =
Chebyshev Fourier
Chebyshev vs. Fourier - Gibb’s
f(x)=sign(x-π) => f(x) - blue ; gN (x) - red; xi - ‘+’
Chebyshev Fourier
-1 -0.5 0 0.5 1
-1.5 -1 -0.5 0 0.5 1 1.5
N = 16
0 2
-1.5 -1 -0.5 0 0.5 1 1.5
N = 16
Chebyshev vs. Fourier - Gibb’s
f(x)=sign(x-π) => f(x) - blue ; gN (x) - red; xi - ‘+’
Chebyshev Fourier
-1 -0.5 0 0.5 1
-1.5 -1 -0.5 0 0.5 1 1.5
N = 64
0 2 4
-1.5 -1 -0.5 0 0.5 1 1.5
N = 6
Fourier vs. Chebyshev
Fourier Chebyshev
N i
xi = 2π i
xi πN
= cos
periodic functions limited area [-1,1]
) sin(
),
cos( nx nx
ϕ ϕ cos
), cos(
) (
=
= x
n x
Tn
{ }
) 2 cos(
1
) sin(
) cos(
2 ) 1 (
* 1
1
*
*
*
*
0
kx a
kx b
kx a
a x
g
m m
k m
k k
+
+ +
=
∑−
=
∑=
+
= m
k
k k
m x c T c T x
g
1
*
* 0
* ( )
2 ) 1
( 0
collocation points
domain
basis functions
interpolating function
Fourier vs. Chebyshev (cont’d)
Fourier Chebyshev
coefficients
some properties
∑
∑
=
=
=
=
N
j
j j
N
j
j j
kx x
N f b
kx x
N f a
k k
1
*
1
*
) sin(
) 2 (
) cos(
) 2 (
∑=
= N
j
j
j k
N f ck
1
* 2 (cos ϕ )cos( ϕ )
• Gibb’s phenomenon for discontinuous functions
• Efficient calculation via FFT
• infinite domain through periodicity
• limited area calculations
• grid densification at boundaries
• coefficients via FFT
• excellent convergence at boundaries
• Gibb’s phenomenon
The Fourier Transform Pair
∫
∫
∞
∞
−
−
∞
∞
−
=
=
ω ω
ω π
ω
ω
d e
F t
f
dt e
t f F
t i
t i
) (
) (
) 2 (
) 1
(
Forward transformInverse transform
Note the conventions concerning the sign of the exponents and the factor.
Some properties of the Fourier Transform
Defining as the FT: f (t) ⇒ F(ω)
¾ Linearity
¾ Symmetry
¾ Time shifting
¾ Time differentiation
) ( )
( )
( )
( 2 1 2
1 t bf t aF ω bF ω
af + ⇒ +
) ( 2 )
(−t ⇒ πF −ω f
) ( )
(t t eω F ω
f + Δ ⇒ i Δt
) ( ) ) (
( iω F ω
t t
f n
n
n ⇒ −
∂
∂
Differentiation theorem
¾ Time differentiation ( ) ( iω) F(ω) t
t
f n
n
n ⇒ −
∂
∂
Convolution
∫
∫
∞∞
−
∞
∞
−
−
=
−
=
∗ ( ) ( ') ( ') ' ( ') ( ') ' )
(t g t f t g t t dt f t t g t dt
f
The convolution operation is at the heart of linear systems.
Definition:
Properties: f (t)∗ g(t) = g(t)∗ f (t) )
( )
( )
(t t f t
f ∗∂ =
∫
=
∗H t f t dt t
f ( ) ( ) ( )
H(t) is the Heaviside function:
The convolution theorem
A convolution in the time domain corresponds to a multiplication in the frequency domain.
… and vice versa …
a convolution in the frequency domain corresponds to a multiplication in the time domain
) ( )
( )
( )
( t g t F ω G ω
f ∗ ⇒
) ( )
( )
( )
( t g t F ω G ω
f ⇒ ∗
The first relation is of tremendous practical implication!
Summary
¾ The Fourier Transform can be derived from the problem of approximating an arbitrary function.
¾ A regular set of points allows exact interpolation (or derivation) of arbitrary functions
¾ There are other basis functions (e.g., Chebyshev polynomials, Legendre polynomials) with similar properties
¾ These properties are the basis for the success of the spectral element method
