Notes on Fourier analysis

1.1. Definition

When solving heat equations with linear and homogeneous boundary conditions, we obtained product solutions in terms of infinite series of sines and cosines. Joseph Fourier developed his ideas on the convergence of these trigonometric series while studying heat flow.

We now study Fourier series of a periodic function. Suppose is a function defined on the real line such that for all , then we say that is -periodic. We shall assume that is Riemann integrable on every bounded interval; this will be the case if is bounded and is continuous except at finitely many points in each bounded interval. We allow is be complex-valued (we will see why later). We wish to obtain the following series expansion:

The series expansion of can also be written in terms of complex exponential functions . We use the following properties:

Equation (1) can be rewritten as

where

We can consider (5) as the sum of two infinity series, one going from to and one going from to .

Alternatively, we can express the coefficients in the following way:

How can the coefficient be expressed in terms of ? Let’s first assume that the series is term by term integrable, we multiply equation (5) by , is an integer, and integrate both sides from to . We obtain

Note that

Hence, the only non-zero term after the integration is the term , and thus we have

Relabeling the term as we get

This means that we can also find , and easily

for

To recapitulate: if has a series expansion of the form in (1), and if the series converges decently such that term by term integration is permissible, then the coefficients and are given by (14), (15), (12) respectively. But now if is any Riemann-integrable periodic functions, the integrals in (14), (15), and (12) make perfectly good sense, and we use them to define the coefficients and . We are able to give the formal definition as follows:

Definition 1.1 Suppose is periodic with period and integrable over . The numbers and defined in (14) and (15), or the number defined in (12), are called the Fourier coefficients of , and the corresponding series

are called the Fourier series of .

We also present the following Lemmas:

Lemma 1.2 With reference to the formulas (14) and (15),

if f is even,

if f is odd,

For Fourier series of a -periodic function, the constant term of series is

which is equivalent to the mean value of in the interval . Thus,

Lemma 1.3 The constant term in the Fourier series of a -periodic function is the mean value of on an interval of length .

We now turn to the discussion of periodic functions with an arbitrary period . Similarly, we obtain the following definition of the Fourier series.

Definition 1.4 For a function with period the Fourier coefficients of are

and the corresponding series

are called the Fourier series of function .

Definition 1.5 Let be a function defined on the interval . The periodic extension of is that function, denoted by , which satisfies for

for each integer k.

For an even function over , the Fourier series for has the form

Formula (25) is called the cosine series of over .

For an odd function over , the Fourier series for has the form

Formula (26) is called the sine series of over .

Definition 1.6 Let the function be defined on . The even extension and the odd extension of are the following functions

1.2. A convergence theorem

Now that we have formally introduced the Fourier series. However, one important question remains unresolved: How do we know whether Fourier series converges for all function ? Proving the convergence of the Fourier series is not a simple matter. Firstly, let’s derive an estimate of the Fourier coefficients that will be needed to establish convergence.

Theorem 2.1 (Bessel’s Inequality) If is -periodic and Riemann integrable on , and the Fourier coefficients are defined by (16), then

Proof We use the following property of complex numbers,

and let

Divide both sides of (31) by and integrate from to , we obtain the following using formula (12),

It is obvious that the left-hand-side of (32) is nonnegative, thus

Letting , we have

Bessel’s inequality can also be stated in terms of and defined by (16). By equation (7), for we have

so that

Therefore,

The significance of (37) is that the series , and are all convergent. As a consequence, we obtain the following result.

Corollary 2.2 The Fourier coefficients , , and all tend to as (and as in the case of ).

Proof , and are the th term of convergent series, so they all tend to as ; hence so do , , and .

In fact, corollary 2.2 is a special case of the Riemann-Lebesgue lemma. Next, we define the class of functions with which we shall be working.

Definition 2.3 Suppose , we say that a function on the closed interval is piecewise continuous provided that (i) f is continous on except perhaps at finitely many points ;(ii) at each of the points the left-hand side and right-hand side limits of ,

and

exists. (If (or ) are one of the exception points , then we require only the left-hand (or the right-hand) limit to exist). Thus, we say that is piecewise continuous on the if f is continuous there except for finitely many finite jump continuities.

Definition 2.4 A function on the closed interval is piecewise smooth if and its first derivative are both piecewise continuous.

Pictorially, is piecewise smooth if its graph is a smooth curve except for finitely many jumps (where is discontinuous) and corners (where is discontinuous). We do not allow infinite discontinuities or sharp cusps.

A very useful tool in Fourier analysis is the Dirichlet’s kernel for the th partial sum

of the Fourier series for . Substitute as defined in (16) into (31), we get

Note that we changed the variable of integration from to for later convenience. We obtain the following

by replacing with . The sum is not affected since it ranges from to . Now, letting , we have

We can write (34) as

The function is called the th Dirichlet kernel. We introduce the following properties of the Dirichlet kernal:

1. 
2. 

Proof For (1), we use the property to obtain

For (2), we recognize that is the sum of a finite geometric series:

Since , we have

Multiplying top and bottom by ,

using the property , we get

Lemma 2.5 For any ,

Proof We integrate both sides of property (45) from to ,

We are now ready to present the main convergence theorem. It says that the Fourier series of a function that is piecewise smooth on every bounded interval converges pointwise to , provided that we redefine at its points of discontinuities to be the average of its left and right hand limits.

Theorem 2.6 If is periodic and piecewise smooth on , and is defined by (40), then

for every . In particular, for every at which is continuous.

Proof  By formula (50), we have

and hence by equation (44)

We wish to show that for each fixed , equation (56) tends to 0 as . By (47) we can write it as

where

We see that is a well-behaved function except when near . However, since and , by L’Hôpital’s rule,

Likewise, approaches as approaches from the right. Thus, is piecewise continuous on . By Bessel’s inequality, the Fourier coefficient

tends to 0 as . However, we know that formula (57) is just . Thus, equation (55) tends to as . This is what we needed to show.

Theorem 2.6 says that the Fourier series of a -periodic piecewise smooth function converges to everywhere, provided that is (re)defined at each of its points of discontinuity to be the average of its left- and right-hand limits there. With this understanding, we have the following uniqueness theorem.

Corollary 2.7 If and are 2-periodic and piecewise smooth, and and have the same Fourier coefficients, then .

Proof and are both the sums of the same Fourier series.