version of 23 October 1997
III.  Fourier series.  Introduction.† 

As mentioned in the previous section, perhaps the most important set of orthonormal functions is the set of sines and cosines (2.8).  These are what is known as a complete orthonormal set for the square-integrable functions on the interval [0,L].  They are also complete and orthonormal on any interval [a,b] with b-a = L.  

Definition III.1.  Square-integrable functions on [a,b] are functions f(x) for which 
	
The set of square-integrable functions is usually denoted L2, and we shall see that it is an inner-product space.  Later, we will also speak of square-integrable functions on regions in two or three dimensions, in which case we have multiple integrals over those regions.

Roughly speaking, a function on a finite interval is square integrable unless it is infinite somewhere.  It can be very discontinuous, and in fact can even be slightly infinite - like the function ln(x), or even |x|-1/3.  Most familiar functions are square-integrable.

Definition III.2.  An orthonormal set {en(x)} is complete (on some fixed set of values of x) if for any square integrable function f(x) and any e>0, there is a finite linear combination 

In other words, any reasonable function can be approximated as well as you wish (in the mean-square sense) by finite sums of the set.  Indeed, we will say that it is the limit of an infinite series: 
	
The sense in which this infinite sum converges is, of course, in mean square.  An equivalent way to describe completeness is: An orthonormal set {en(x)} is complete if the statement that
	
implies f(x) = 0 a.e.  This is often a practical way to show that a set is incomplete, because you just have to exhibit a nonzero function which is orthogonal to the entire set.

Unfortunately, it is not as easy to prove completeness as it is to disprove it.  It is a theorem that each of the sets of trigonometric functions (2.6)-(2.9) is complete and orthonormal, but the techniques for proving such a theorem are beyond the scope of this course.  To understand the issue better, consider a finite linear combination of an orthonormal basis, What is the norm of f?  Because orthogonality makes the cross terms vanish, a little calculation shows us that
	
This is an extension of Pythagoras's theorem, since it says that the square of the hypotenuse is the sum of the squares of the lengths of the sides, if the sides are at right angles; only in function space, lots of things can all be at right angles.  Suppose we had left out some of the components.  Then we would have 
            
This inequality is still true if N = •, and is known as Bessel's inequality.  The problem of completeness is that it is not easy to tell if we have included all the basis elements necessary to make both sides equal.  We can leave many of them out and still have an infinite number left; in the set of Fourier functions (2.8), for example, we could leave out all the sine functions and still have all the cosine functions left.  

The set (2.8) is particularly useful for periodic  functions, that is, functions such that f(x+L) = x for some fixed length L, called the period, and all x.  Since each of the functions in the set is periodic with period L, any linear combination of them is also periodic with the same period.  The completeness just alluded to means that every periodic function can be resolved into the trigonometric functions of the same period.  If the independent variable is time (which you might prefer to denote t rather than x), a periodic function of the form sin(2npt/L) or cos(2npt/L) may be detected by your ear and perceived as a pure musical tone with frequency n/L.  Any periodic sound wave can be resolved into pure musical tones.  If you are given a sound wave f(t), which is periodic with period L you can extract its components of frequency k/L with formulae (2.12)-(2.14), setting m or n = k.  There are two such components, one with the sine function and the other with the cosine.  This degree of freedom corresponds to the phase of the sound wave, because of the trig identity:
					(3.1)

The intensity (power) carried by the component of a sound wave at pure frequency k/L is proportional to  

In the next chapter this resolution into pure frequencies is carried out for the square wave and the results are plotted, among other things.  You may wish to glance at those plots now to get an intuitive feel for how a Fourier series can approximate a function.

The powerful theorem behind the Fourier series III.3.  We shall now carefully formulate a theorem which justifies the use of Fourier series for square-integrable functions and tells us several useful facts.  Historically, parts of this theorem were contributed by Fourier, Parseval, Plancherel, Riesz, Fischer, and Carleson, and in most sources it is presented as several theorems.  For proofs and further details, refer to Rudin’s Real Analysis or Rogosinski’s Fourier Series.
If f(x) is square-integrable on an interval [a,b], then
a)  All of the coefficients am and bn are definite numbers uniquely given by (2.12)-(2.14).
b)  All of the coefficients am and bn depend linearly on f(x)
c)  The series 

converges to f(x) in the mean-square sense.  In other words,

We shall express this by writing
		(3.2)
(Remember, however, that this series converges in a mean sense, and not necessarily at any given point.)  It is also true that it converges a.e.
d)  			(3.3)
and the right side is guaranteed to converge.
e)  If g is a second square-integrable function, with Fourier coefficients 

				(3.4)
This is known as the Parseval formula.
Conversely, given two square-summable sequences am and bn, i.e., real or complex numbers such that 
	 
is finite, they determine a square integrable function f(x) uniquely a.e. such that (2.12)-(2.14) and statements b)-d) hold.

Let’s stand back and think about what this big theorem tells us.  Square-integrable functions are very general, so this is telling us that any reasonable function can be approximated arbitrarily well, in the r.m.s. sense, by a “trigonometric series.”  We have a formula to generate the coefficients, and in fact a full correspondence between the square-integrable functions and the square-summable sequences.  The square-integrable functions L2 form an inner product vector space.  The set of double sequences {am,bn} is also a vector space with an inner product given by (3.4).  You can think of each object in this space as a vector with an infinite number of components, some of which are denoted am and others bn.  

Here are some examples showing why mean-square approximation is not always good enough.  At a later stage we shall discuss when Fourier series converge at individual points.  We shall see that if f(x) and f'(x) are continuous near x, the Fourier series converges at x.

Examples III.4.

1.  The indicator function  cS(x) of a set S takes on the value 1 when x is in S and 0 otherwise.  The sequence of indicator functions fn = cS(x) for S = the sets Sn = [0, 1/n] converges to the 0 function in the mean-square sense, but for each n there is always an interval near 0 such that  |fn - 0| = 1.
2.  This one is more disturbing than Example 1.  The function fn is still always an indicator function cS(x), but the sets Sn are as follows
	n=1		0 £ x £ 1
	n=2		0 £ x £ 1/2
	n=3		1/2 £ x £ 1
	n=4		0 £ x £ 1/4
	n=5		1/4 £ x £ 1/2
	n=6		1/2 £ x £ 3/4
	n=7		3/4 £ x £ 1
	n=8		0 £ x £ 1/8
etc.  This sequence also converges to the zero function in the mean-square sense.  For any fixed x, fn(x) will always have either the value 0 or 1, but it will never settle down definitively to 0 as n Æ •.  There are an infinite number of n’s for which fn(x) = 1.

If you look at the various Fourier series that are plotted in the next chapter, the crazy stuff of Example 2 doesn’t happen.  In fact, the convergence is very good except at the ends of the intervals or at places where the function is discontinuous.  If you look at the periodic extension of a function, you see that the end of an interval is a place where there is likely to be a discontinuity, and it is when this happens that the series did not converge at the end of the interval.  (A good example in chapter IV is f(x) = x on the basic interval [0,L].)  What we observe is described by a general theorem, which we now formulate.

A function is said to be piecewise continuous  (some say sectionally continuous) if it is continuous except at a discrete set of jump points, where it at least has an identifiable value on the left and a different one on the right.  Here is a formal way to state this:

Definition III.5.  A function f(x) is piecewise continuous  on a finite interval a£x£b if it is continuous except at a finite number of points a =: x0 < x1 <… < xn := b, and all the one-sided limits
	
and
	
exist (except that we only assume the limit from above at a and the limit from below at b).

Here the up-arrow indicates that the limit is taken for values of x tending to xk from below, and the down-arrow indicates that the limit is taken for values of x tending to xk from above.

What we see from the examples is that where a function has a discontinuity, the Fourier series, when truncated to a large but finite number of terms, takes on a value between the right and left limits.  The theorem says that the Fourier series finds the average of the two possibilities.

Theorem III.6.  Suppose that f(x) and f'(x) are piecewise continuous on a finite interval [a,b].  Then the Fourier series (3.2) converges at every value of x between a and b as follows:
	
At the end points, we have:
	
The limit at the end points is reasonable, because when the function is extended periodically, they are effectively the same point.  And a particular consequence is that:  At places where such a function is continuous, the Fourier series does indeed converge to the function.

In addition, if the function is continuous on the interval [a,b], and f(a) = f(b), then we can state a bit more, namely that the Fourier series converges uniformly  to the function.  This means that the error can be estimated independently of x:

Theorem III.7.  Suppose that f'(x) is piecewise continuous, f(x) itself is continuous on a finite interval [a,b], and f(a) = f(b).  Then

The condition that f(a) = f(b) is again reasonable if you think of f as a periodic function extending beyond the interval [a,b] - the extended function would be discontinuous at the end points if f(a) did not match f(b).

Exercises.

III.1.  Find the entire 
	Fourier (sine and cosine) series 
	Fourier sine series (see appendix)
	Fourier cosine series (see appendix)
	Fourier exponential series
on the interval 0 £ x £ L for the function  f(x) = x for 0 £ x £ L/2, otherwise 0.  Discuss what happens if the series is evaluated outside the interval 0 £ x £ L.

III.2.  Numerically find the first eight terms in the  
	Fourier series
	Fourier sine series
	Fourier cosine series
	Fourier exponential series
on the interval 0 < x < 2p for the function  f(x) = cosh(cos(x))
III.3.  Find the Fourier series of the following functions:
	
	(i)	cosh(2x), -p £ x £ p
	(ii)	x|x|,  -p £ x < p.  Notice that this is an odd function.
	(iii)	1 - |x|, -1 £ x £ 1.
	(iv)	|sin(x)|, 0 £ x £ p.
	(v)	2 - 2 cos(p x), -1 £ x £ 1.
	(vi)	f(x) = x for 0 £ x < L/2, 0 for L/2 £ x < L

(Implicitly, the functions are extended periodically from these basic intervals.)

III.4.  When we calculate the Fourier series for a square pulse and for x mod L in chapter IV, we find that am = 0 for all m ≥1.  Explain why, using ideas about symmetry.

III.5.  a)  Is it possible for a sequence of functions to converge in the r.m.s. sense for 0£x£1, to converge at every point of the interval 0£x£1/2, but not converge at any point of the interval 1/2<x£1?  Give an example or explain why not.
b)  Is it possible for a sequence of square-integrable functions to converge at every point of the interval 0£x£1, but not converge in the r.m.s. sense?  Give an example or explain why not.

III.6.  Use the Parseval formula (3.4) to calculate
	<3 + cos(x) + sin(x) + 2cos(2x) + 2sin(2x) + 3cos(3x),  1- sin(2x) - sin(4x)> 
(standard inner product for 0£x£2p), without doing any integrals.

Appendix on Fourier sine and cosine series.

There are three distinct versions of the Fourier series in common use, the "full" Fourier series, the Fourier sine series, and the Fourier cosine series. They are based on slightly different choices of an orthogonal basis set, whether (2.6) for the Fourier sine series, (2.7) for the Fourier cosine series, or (2.8) for the full Fourier series. Notice that while the full set (2.8) contains both sines and cosines, it does not contain as many frequencies as either of the other sets, since the trig functions depend on  2 pnx/L rather than all integer multiples of p x/L. We shall also discuss an "exponential" Fourier series, which uses complex functions of the form  exp(2pix/L),  but this is actually the same series as the full Fourier series:  It has simply been written in a convenient way using exponential functions. 

The three different sets do an equally good job of approximating functions in the r.m.s. sense, but their relationship to symmetries is different. For this reason, different sets of Fourier functions will arise in our study of partial differential equations, depending on the kind of "boundary conditions" which are physically appropriate for a given problem. 

The full Fourier series consists of functions which are periodic, and are particularly good for representing or approximating periodic functions, that is, functions for which 
	f(x + L) = f(x). 
Examples include sounds built on a basic frequency with overtones and any physical function depending on an angular variable such as longitude. The Fourier sine series, in contrast, is well adapted to functions which are zero at x=0 and x=L, since all the functions sin(npx/L) have this property. A related but less obvious property that begs for the use of sine functions is an odd symmetry: f(-x) = - f(x).  If the function you wish to approximate is an odd function, then the sines are a good candidate for a basis set. Notice that the sines are not only odd under reflection about 0, but also about the other end of the interval, x=L. Fourier cosine series are better suited for even functions. A Fourier sine series represents a function as
	 					(3.a1)  
for 0 < x < L; there are no cosine terms at all.  A Fourier cosine series, on the other hand, represents a function as
					(3.b1)
(No sine terms this time.)  In most applications a = 0 < x < L, but this is not strictly necessary. If the lower limit for x is a > 0, then either of these series can still be used if we change variables from x to x-a, so the basis functions are sin(np(x-a)/L) or, respectively, cos(np(x-a)/L).

The formulae for the coefficients an and bn are the same as in (2.14), except that the alternative Fourier sine series use all integer multiples of px/L, where L=b-a, not just the even multiples:  For the sine series,
					(3.a2)
while for the cosine series, 
	
while for m>0,
	 						(3.b2)

Just as for the full Fourier series, the Fourier sine functions are a complete set, and they satisfy some useful identities, which I have collected in a theorem (which is almost identical to the one for the full series): 

The powerful theorem behind the Fourier sine series III.3a. 
If f(x) is square-integrable on an interval [a,b], then:
a) All of the coefficients bn of (3.a2) are well defined.

b) All of the coefficients bn depend linearly on f(x).

c) The series 
	
converges to f(x) in the r.m.s. sense. In other words,
	
We shall express this by writing
	
(Remember, however, that this series converges in a mean sense, and not necessarily at any given point.)  It is also true that it converges a.e.

d)	
and the right side is guaranteed to converge.

e) If g is a second square-integrable function, with Fourier sine coefficients 

This is known as the Parseval formula for the Fourier sine series.

Conversely, given a square-summable sequence bn, i.e., real or complex numbers such that 
	
is finite, they determine a square integrable function f(x) uniquely a.e. such that all the statements of this theorem hold.

In addition, if f(x) is piecewise continuous, then Theorem III.6 likewise holds, except at the end points of the interval. If 0 < x < L, the Fourier sine series will converge to the average of the right and left limiting values of f at x. When x=0 or L, however, the Fourier sine series will always converge to 0, since all the sine functions are 0 at those two points. 

The Fourier cosine functions are also a complete set, and they satisfy similar useful relationships:

The powerful theorem behind the Fourier cosine series III.3b. 

If f(x) is square-integrable on an interval [a,b], then

a) All of the coefficients am of (3.b2) are well defined.

b) The series 
	
converges to f(x) in the r.m.s. sense. In other words,
	
We shall express this by writing
	
As before, it is also true that it converges a.e.

c)	
and the right side is guaranteed to converge.

d) All of the coefficients am depend linearly on f(x)

e) If g is a second square-integrable function, with Fourier cosine coefficients 

This is known as the Parseval formula for the Fourier cosine series.

Conversely, given a square-summable sequence am, i.e., real or complex numbers such that 

is finite, they determine a square integrable function f(x) uniquely a.e. such that all the statements of this theorem hold.

In addition, if f(x) is piecewise continuous, then Theorem III.6 likewise holds, except at the end points of the interval. If 0 < x < L, the Fourier cosine series will converge to the average of the right and left limiting values,   When x=0 or L, however, the Fourier cosine series will always converge to the one-sided limit of f at that point, . 

When we study partial differential equations beginning in chapter VI, Fourier sine or cosine functions will arise depending on the boundary conditions.  If we know that the functions we seek are equal to 0 at the end points 0 and L, then we can ensure this fact by writing them as sums of sines of the form sin(npx/L).  Similarly, if we know that the derivatives are equal to 0 at the end points, we can accomplish this with a sum of cosines of the form  cos(mpx/L).