3 Fourier Series 3.1 Introduction Although it was not apparent in the early historical development of the method of separation of variables, what we are about to do is the analog for function spaces of the following basic observation, which explains why the standard basis i = e1, j = e2, and k = e3, in 3-space is so useful. The basis vectors are an orthonormal set of vectors, which means
e1 e2 =0, e1 e3 =0, e2 e3 =0, · · · e1 e1 =1, e2 e2 =1, e3 e3 =1. · · · Any vector v in 3-space, has a unique representation as
v = b1e1 + b2e2+b3e3.
Furthermore, the coefficients b1,b2, and b3 are easily computed from v :
b1 = v e1,b2 = v e2,b3 = v e3. · · · Just as was the case for the laterally insulated heat-conducting rod and for the small transverse vibrations of a string, whenever the method of separation of variables is used to solve an IBVP, you reach a point where certain data given by a function f (x) must be expanded in a series of one of the following forms in which L>0:
∞ nπx f (x)= b sin , n L n=1 FourierX Sine Series Expansion
a ∞ nπx f (x)= 0 + a cos , 2 n L n=1 Fourier CosineX Series Expansion
a ∞ nπx nπx f (x)= 0 + a cos + b sin , 2 n L n L n=1 XFourier³ Series Expansion ´ or, more generally, ∞ f (x)= fnϕn (x) n=1 EigenfunctionX Expansion where ϕn (x) are the eigenfunctions of an EVP that arises in the separation of variables process.
18 Just for the record the Fourier sine and cosine series and the general Fourier series expansions arise from the EVPs
X00 λX =0, X (0) =− 0,X(L)=0, ½ X00 λX =0, − X0 (0) = 0,X0 (L)=0, ½ X00 λX =0, − . X (0) = X (2L) ,X0 (0) = X0 (2L) ½ In the context of heat conduction in a laterally insulated homogeneous rod, the first EVP comes up when the rod’s ends are held at temperature zero; the second when the ends of the rod are insulated; and the third when the rod is bent into a heat ring (doughnut) whose central circle has circumference 2L. To set the stage for how to proceed with the study of Fourier series and more general eigenfunction expansions, let’s review the corresponding situation for Taylor series and Taylor series expansions. Let f (x) be a function with infinitely many derivatives on an interval containing x =0. Suppose we suspect that f (x) hasapowerseriesexpansion,say,
∞ n f (x)= fnx on some interval I containing 0. n=0 X Then term-by-term differentiation of the power series shows that
f (n) (0) f = . n n! Now, suppose we do not know that f (x) hasapowerseriesexpansion,butwe hope that it does. It is natural to form the infinite series
∞ f (n) (0) xn n! n=0 X Taylor series of f about 0 and to try and prove that this series| converges{z } at least for x near 0 and has sum f (x) . If all this works out we have obtained a power series expansion for f (x) . So there are two basic questions:
TS 1. For which x (if any) does the Taylor series of f (x) converge? TS 2. For which x (if any) is f (x) the sum of its Taylor series?
When f (x) is the sum of its Taylor series over some interval we call
∞ f (n) (0) f (x)= xn the Taylor series expansion of f about 0. n! n=0 X
19 3.2 Fourier Series Evidently, Fourier sine and cosine series can be regarded as special cases of a general Fourier series so we will discuss Fourier series first and specialize the results to get information about Fourier sine and cosine series. We start by inquiring whether a rather general function f (x) has a series expansion of the form
a ∞ nπx nπx f (x)= 0 + a cos + b sin , 2 n L n L n=1 X ³ ´ over some interval, say the full real line. If so, the coefficients an and bn must depend on the function f (x) in some way. A formal calculation, based on orthogonality relations satisfied by cos (nπx/L) and sin (nπx/L) , leads to
1 L nπx an = f (x)cos dx, L L L Z− 1 L nπx bn = f (x)sin dx. L L L Z−
We call an and bn the Fourier coefficients of f (x) . The trigonometric series formed using the Fourier coefficients,
a ∞ nπx nπx 0 + a cos + b sin , 2 n L n L n=1 X ³ ´ is called the Fourier series of f (x) . The notation
a ∞ nπx nπx f (x) 0 + a cos + b sin 2 n L n L ∼ n=1 X ³ ´ means that the series on the right is the Fourier series of f (x) . Thesametwo questions posed for Taylor series are relevant for Fourier series:
FS 1. For which x (if any) does the Fourier series of f (x) converge? FS 2. For which x (if any) is f (x) the sum of its Fourier series?
When f (x) is the sum of its Fourier series on some interval, we call
a ∞ nπx nπx f (x)= 0 + a cos + b sin 2 n L n L n=1 X ³ ´ the Fourier series expansion of f (x) on that interval. Evidently an understanding of Fourier series involves some familiarity with properties of periodic functions.
20 3.3 Elementary Properties of Periodic Functions All functions in this section are real-valued and defined on the real line. Let p =0. Afunctionf (x) is periodic with period p (p periodic for short or just periodic6 if the period is understood) if f (x + p)=f (x) for all x in its domain. Fact 1. If f (x) is p periodic, then f (x) is np periodic for every integer n =0; that is, every nonzero integer multiple± of a period is a period. 6 Fact 2. If f (x) and g (x) are p periodic, then αf (x)+βg (x) is p periodic for all scalars α and β; that is, the p periodic functions are a function space. Fact 3. Every nonconstant, continuous p periodic function has a smallest pos- itive period, called its fundamental period (or primitive period). Every period of the function is an integer multiple of the fundamental period. Fact 4. If f (x) is p periodic and integrable over [0,p] , then it is integrable over any interval [a, b] of length p and b p f (x) dx = f (x) dx. Za Z0 Fact 5. Any property of a p periodic function can be recast as an equivalent property of a 2π periodic function and conversely because p f (x) is p periodic g (x)=f x is 2π periodic. ⇐⇒ 2π ³ ´ Consequently, theoretical results about Fourier series are often established in the 2π periodic case. The corresponding result for any other period follows free of charge from the foregoing equivalence (change of variable).
3.4 Even and Odd Functions All functions in this section are real-valued and defined on a subset of the real line. A function f is even if its graph is symmetric with respect to the y-axis; equivalently, f is even iff f ( x)=f (x) for all x in the domain of f. − A function f is odd if its graph is symmetric with respect to the origin; equiv- alently, f is odd iff f ( x)= f (x) for all x in the domain of f. − − This terminology is used because even and odd functions share some (not all) algebraic properties of even and odd numbers: For functions Even (Even)(Even)=Even, = Even, Even Odd (Odd)(Odd)=Even, = Even, Odd (Even) (Even)=Even, (Odd) (Odd)=Odd, ± ±
21 What can you say about (Even) (Odd)? ± Fact 1. Most functions are neither even nor odd. Fact 2. Every function defined on a symmetric interval about the origin is the sum of an even and an odd function. Fact 3. Properties of even and odd functions are especially useful in evaluating certain integrals over intervals symmetric about the origin:
L L f even = f (x) dx =2 f (x) dx, ⇒ L 0 Z− Z L f odd = f (x) dx =0. ⇒ L Z− Fact 4. The Fourier series of an even 2L periodic function is a cosine series. Fact 5. The Fourier series of an odd 2L periodic function is a sine series.
3.5 Fourier Sine and Cosine Series in Action In a number of situations in which separation of variables is used to solve an IBVP, you reach a situation where you need to expand a given function f (x) defined only for 0 x L into either a Fourier sine series or a Fourier cosine series. This is easy≤ to do≤ if you remember Facts 4 and 5 in Sec. 3.4.
If you need a cosine series, just extend the given function f (x) to an even function on L x L by defining f (x)=f ( x) for L x 0 and then make the extended− ≤ function≤ 2L periodic by requiring− that− f≤(x +2≤ L)=f (x) for all x. The extended function will be an even periodic function and, hence, its Fourier series will be a Fourier cosine series; you just use that series on 0 x L. ≤ ≤ (Picky point: The even extension of f to L x L is a new function as is the 2L periodic extension of the even extension− ≤ of ≤f; nevertheless, being sane, we denote all three of these functions by f. Alternatively, you decide that all along you really meant that f was the even 2L periodic extension of your given data and there is no picky point.)
If you need a sine series, just extend the given function f (x) so it is an odd function on the punctured interval ( L, L) / 0 , which is the interval ( L, L) with the origin removed, by defining−f (x)={ f}( x) for L 22 Now there is a nasty, not-so-picky point, to deal with. That is why the weak and strong inequalities and the removing of the origin were needed in the foregoing paragraph. The extension process needed to produce an odd periodic extension of f (x) must be done this way and can turn what was a nice smooth function on 0 x L into a discontinuous function on ( , ) whose Fourier series is needed.≤ When≤ this happens it carries with it unpleasant−∞ ∞ convergence behavior at the endpoints of the interval 0 x L. These unpleasantries go away if the function f (x) satisfies what we called≤ ≤ natural compatibility conditions. Carrying out the foregoing odd or even periodic extension program leads to: Fourier Sine Series on 0 x L. ≤ ≤ L ∞ nπx 2 nπx f (x) b sin ,b= f (x)sin dx. n L n L L ∼ n=1 0 X Z Fourier Cosine Series on 0 x L. ≤ ≤ L a ∞ nπx 2 nπx f (x) 0 + a cos ,a= f (x)cos dx. 2 n L n L L ∼ n=1 0 X Z We will apply the results of this section to complete the solution of the two IBVPs, one for the heat equation and one for the wave equation, that we been working on for some time. 3.6 Convergence of Fourier Series Before discussing convergence questions associated with Fourier series expan- sions, we will calculate explicitly two Fourier series that are useful in applica- tions and play a role in the proofs of the basic convergence theorems, proofs we will skip. Example 1. Find the Fourier series expansion of the 2π periodic function given by f (x)=x for π α ∞ nπx nπx 0 + α cos + β sin 2 n L n L n=1 X ³ ´ converges for all x its sum is a 2L periodic function on the real line. In typical applications, we need a Fourier series expansion of a function f (x) that is definedonanintervaloflengthL or 2L. It turns out that the most convenient way to arrive at the validity of such expansions is to first consider 23 thecasewhenf (x) is defined on the entire real line and is 2L periodic. We make that assumption now. What follows is a short list of very useful results about Fourier series ex- pansions. The statements include the words piecewise smooth and uniformly convergent. We will discuss these terms in class. Theorem 1 If the Fourier series of a 2L periodic, continuous function f (x) converges uniformly over a period, then this Fourier series has sum f (x): a ∞ nπx nπx f (x)= 0 + a cos + b sin for all x. 2 n L n L n=1 X ³ ´ Corollary 1 Let f (x) be 2L periodic and continuous. If the series of mag- nitudes of the Fourier coefficients an and bn are convergent, then the Fourier series of f (x) converges absolutely| | and uniformly| | to f (x) . P P Theorem 2 (Dirichlet) Let f be 2L periodic, continuous, and piecewise smooth, then its Fourier series converges absolutely and uniformly to f on ( , ) . −∞ ∞ Here is a useful refinement of Dirichlet’s theorem. Theorem 3 (Dirichlet) If f is 2L periodic and piecewise smooth, then the Fourier series of f converges to f (x )+f (x+) − for each x in ( , ) . 2 −∞ ∞ Consequently, if f is continuous at the point x, the Fourier series converges to f (x) . Moreover, the convergence to f (x) is absolute and uniform on any closed interval that contains no points of discontinuity of f. Vocabulary: Afunctionf (x) has a jump discontinuity at x if both one-sided limits f (x ) = lim f (ξ) and f (x+) = lim f (ξ) − ξ x ξ x ξ Afunctionf (x) is piecewise smooth on an interval [a, b] if both f (x) and f 0 (x) are piecewise continuous on [a, b] . Afunctionf (x) is piecewise smooth if it is piecewise smooth on every finite subinterval of the real line. 24 Pointwise and Uniform Convergence: A sequence of functions gn (x) with common domain D converges pointwise on D to g (x) if For each FIXED x in D, lim gn (x)=g (x) . n →∞ A sequence of functions gn (x) with common domain D converge uniformly on D to g (x) if for every ε>0 (a measure of error) The ε-tube about the graph of g contains the graph of gn for all n sufficiently large. The ε-tube about the graph of g is (x, y): y g (x) <εfor all x in the domain of g . { | − | } Convergence properties of infinite series are defined in terms of corresponding properties of its partial sums. Thus, if n∞=1 hn (x) has partial sums sn (x) , where sn (x)=h1 (x)+h2 (x)+ + hn (x) , we say ··· P ∞ hn (x) converges pointwise on D if its sequence of partials sums n=1 X sn (x) converges pointwise on D; and ∞ hn (x) converges uniformly on D if its sequence of partials sums n=1 X sn (x) converges uniformly on D. 3.7 Concluding Remarks Complex Form of a Fourier Series. Let f (x) be a reasonable function de- fined on L to L. Use of the Euler identities to replace sines and cosine by complex− exponential terms in the usual Fourier series of f (x) leads to a complex representation for the series: L ∞ 1 f (x) c einπx/L where c = f (x) e inπx/L dx. n n 2L − ∼ n= L X−∞ Z− 25 Notice the sum extends from to and that the constant term, here −∞ ∞ c0, no longer needs to be treated in a special way. How Many Fourier Series of f (x) on 0 x L are there? In the prob- lems we solved by separation of variables,≤ we≤ ultimately needed to expand a function f (x) defined on 0 x L into either a sine series or a cosine series. To get such a series we≤ were≤ forced to extend f (x) to be either odd or even on L to L. There are other ways to extend f (x) to a function on L to L.−Virtually any other type of extension will lead to a Fourier series− for f (x) that has both sine and cosine terms. Assuming the original f (x) is continuous and has a continuous derivative on 0 x L, and the extension to L to L is at all reasonable, the extended periodic≤ ≤ version of f (x) will be− piecewise smooth and all of the Fourier series you can form in this way will converge to f (x) for 0 26