Lecture 27 MIT 3.016 (Fall 2005) © W. Craig Carter 2003-2005

Basis and Orthogonal Functions

This is a generalization of the Fourier series. The orthogonality of the trignometric functions
(i.e., Sin[2 π n x] Sin[2 π m x]dx =

	if n=m
0	if n≠m

)
was the key to the ``trick''  that permitted calculation of the Fourier coefficients.
This is a generalization of dot or scalar vector product, but for functions.  The norm that is defined above for the trignometric functions is fairly simple, it is the l2-norm:
f(x) • g(x) ≡  ∫ f(x) g(x) dx        (l2-norm)
However, there can be many other types of norms, for instance a Gaussian weighted l2-norm
f(x) • g(x) ≡  ∫ f(x) g(x)

dx (Gaussian weighted l2-norm)
and many many others.

Both the norm and the domain of integration must be defined.

To represent a function in terms of a sum of basis functions with coefficients, a orthogonality relation must be obtained for the basis functions.
We will do a few examples of orthogonality relations, the coefficients can be found as a straightforward extension of the Fourier method.

Establish by example the orthogonality relations for the Lengendre Polynomials

In[115]:=

Out[115]=

{0, 0, 0, 0, 2/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/11, 0, 0, 0, 0, 0}

Calculate the norm as a function of the order of the Legendre polynomial

In[116]:=

Out[116]=

{2, 2/3, 2/5, 2/7, 2/9, 2/11, 2/13, 2/15, 2/17, 2/19, 2/21, 2/23, 2/25, 2/27, 2/29, 2/31}

Make a guess at what the norm looks like

In[117]:=

Out[117]=

{2, 2/3, 2/5, 2/7, 2/9, 2/11, 2/13, 2/15, 2/17, 2/19, 2/21, 2/23, 2/25, 2/27, 2/29, 2/31}

The orthogonality relation is plausibly: LegendreP[i,x] LegendreP[j,x]dx =

	if \| i\|=\|j\|
0	if \|i \|≠\|j\|

In fact, this is correct

Representation of a function in terms of the Legendre basis

This demonstrates how to take a specific set (basis) of orthogonal functions and represent an arbitrary function in terms of an infinite sum of terms involving the basis functions, each with its own coefficient. This is a direct analog to what we did with Fourier series.

A function to calculate the coefficients

In[118]:=

$LegendreCoefficient[f_, i_] := (2 i + 1)/2Integrate[f[x] LegendreP[i, x], {x, -1, 1}]$