In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We motivate the formulae for Fourier series. For simplicity we ignore here all questions concerning convergence, which are dealt with in Section 3 and the additional material.

**Fourier series**

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

But since

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.

**Taylor series**

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

we have

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

Then

Assume that is of the form (1). Then let

Then

* 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

and

which implies that

For a function as defined above we have

and hence