As pointed out in the post on Section 2 at the beginning, in order to represent a function we need an orthonormal bases with respect to the inner product
From Section 2 we know that the functions
are orthogonal with respect to the inner product
the set of functions
is not orthonormal. Hence, by rescaling them, i.e. by considering the functions
and we obtain an orthonormal basis. Let
Then, using the same approach as in the beginning of Section 2 we obtain
Notice that by setting
we obtain exactly the formulae for the Fourier series. For example for we have
The other formulae can be obtained analoguously.
In the following we shall ignore questions concerning convergence.
Assume that is infinitely times differentiable and let . Then we can calculate a Taylor series expansion
One can motivate this formula by observing that for a function
and so on. In general we obtain
where denotes the th derivative of and and . Therefore we obtain the formula
Hence the derivation of the formula for Taylor series is different from that of Fourier series. However, one could use a similar approach as for Fourier series to obtain an expansion in polynomials.
Consider now the inner product
Notice that the polynomials are not orthogonal to each other since
There is, however, a set of orthogonal polynomials defined on (notice that for intervals different from you need different polynomials). These are called Legendre polynomials and are denoted by . The first few Legendre polynomials are
and in general
You can find the graph of the first few Legendre polynomials here.
The inner product of the Legendre polynomials is given by
Let us define
Assume that is of the form (1). Then let
Example Let . Then
Then we have
Taylor series and inner product
As we have seen above, Taylor series are not based on an expansion of functions which are orthonormal with respect to the inner product since they are based on the functions However, one can create an inner product by defining these functions as orthonormal. For functions
where are real numbers, let
We define the associated norm norm . The functions are now orthonormal, since
which implies that
For a function as defined above we have