A 2π-periodic function can be identified as a function on circle, which is T = R/(2πZ). Some important properties of Fourier transform are
• The differentiation becomes a multiplication under Fourier transform. It is also equivalent to say that the differential operator is diagonalized in Fourier basis.
• The convolution becomes a multiplication under Fourier transform.
Differentiation
Lemma 5.2. If f ∈ C1[T], then
fb0k= ik ˆfk. Proof.
fb0k = 1 2π
Z 2π 0
f0(x)e−ikxdx
= 1
2π e−ikxf (x)
x=2π x=0 − 1
2π Z 2π
0
(−ik)e−ikxf (x) dx
= ik ˆfk.
Here, we have used the periodicity of f in the last step.
Convolution If f and g are in L2(T), we define the convolution of f and g by Here, we have used Fubini theorem.
Remarks
1. The above two lemmae are also valid for f, g are in L2. Their proofs are based on the L2 convergence of the Fourier series for nice functions and the fact that nice functions are dense in L2.
2. Many solutions of differential equations are expressed in convolution forms. For instance
−u00= f in T, its solution can be expressed as u = g ∗ f , where g is the Green’s function of
−d2/dx2on T.
3. In image processing, a blurred image is modelled by z(x) =
Z
k(x − y)f (y) dy where f (y) is the original image, z the blurred image, and
k(x) = 1
2πσ2e−|x|2/2σ2 the blur operator.
5.2. THE FOURIER BASIS IN L2(T) 91 Regularity and decay If f is smooth, then its Fourier coefficients decays very fast. Indeed, by taking integration by part n times, we have
fˆk = 1
Here, Dπ/kis a backward finite difference operator. Thus, ˆfkmeasures the oscillation of f at scale π/k. If f is smooth, then Dπ/kn f = O(|k|−n)g(x) with g being uniformly bounded in k. Thus, fˆk= O(|k|−n). Indeed we have better result:
Lemma 5.4. If f ∈ Cn(T), then ˆfk= o(|k|−n).
We shall only need to show that ˆfk → 0 as |k| → ∞ for continuous function f . The rest for high derivative cases can be obtained by taking integration by part. We have seen that
fˆk = 1
When f is continuous on T, it is uniformly continuous on T. Thus, for any > 0, we can find K > 0 such that for all |k| > K we have
1If fact, we shall see later from the Riemann-Lebesgue lemma that ˆfk= o(|k|−n).
From this, we obtain
| ˆfk| ≤ 1 2π
Z π
−π
f (x) − f (x − π/k)
2 e−ikx
dx < .
When f is not smooth, say in L1, we still have ˆfk → 0 as |k| → ∞. This is the following Riemann-Lebesgue lemma.
Lemma 5.5 (Riemann-Lebesgue). If f is in L1(a, b), then fˆk :=
Z b a
f (x)e−ikxdx → 0, as |k| → ∞.
Proof. 1. For f ∈ L1(a, b), we have
| ˆfk| ≤ kf k1for all k.
2. Any function f ∈ L1(a, b) can be approximated by a continuous function g ∈ C[a, b] in the L1sense. That is, for any > 0, there exists g ∈ C[a, b] such that kf − gk1< .
3. For g ∈ C[a, b], we have: for any > 0, there exists a K > 0 such that for |k| > K, we have
|ˆgk| < .
Combining these two, we get
| ˆfk| ≤ |ˆgk| + | ˆfk− ˆgk| ≤ |ˆgk| + kf − gk1 < 2.
Thus, | ˆfk| → 0 as |k| → ∞.
Remarks.
1. If f is a Dirac delta function, we can also define its Fourier transform fˆk = 1
2π Z π
−π
δ(x)e−ikxdx = 1 2π.
In this case, δ 6∈ L1 and ˆδk= 1/2π does not converge to 0 as |k| → ∞.
2. If f is a piecewise smooth function with finite many jumps, then it holds that ˆfk= O(1/k).
One may consider f has only one jump first. Then f is a superposition of a step function g and a smooth function h. We have seen that ˆhkdecays fast. For the step function g, we have ˆ
gk = O(1/k).
5.2. THE FOURIER BASIS IN L2(T) 93 5.2.3 Convergence Theory
Let denote the partial sum of the Fourier expansion by fN:
fN(x) :=
N
X
k=−N
fˆkeikx.
We shall show that under proper condition, fN will converge to f . The convergence is in the sense of uniform convergence for smooth functions, in L2 sense for L2functions, and in pointwise sense for BV functions.
Convergence theory for Smooth functions
Theorem 5.4. If f is a 2π-periodic, C∞-function, then for any n > 0, there exists a constant Cn such that is called the Dirichlet kernel. Using DN(x)dx = π, we have
fN(x) − f (x) = 1
The function g(x, t) := (f (x + t) − f (x))/ sin(t/2) =R1
0 f0(x + st) ds · t/ sin(t/2) is 2π periodic and in C∞. We can apply integration-by-part n times to arrive
fN(x) − f (x) = (N + 1
2)−n(−1)n/2 2π
Z π
−π
∂tng(t) sin((N +1 2)t) dt for even n. Similar formula for odd n. Thus, we get
sup
x
|fN(x) − f (x)| ≤ CN−n Z
∂tng(x, t) sin((tN + 1 2)t) dt
= O(N−n).
This completes the proof.
Remark. The constant Cn, which depends onR |g(n)| dt, is in general not big, as compared with the term N−n. Hence, the approximation (5.2) is highly efficient for smooth functions. For example, N = 20 is sufficient in many applications. The accuracy property (5.2) is called spectral accuracy.
5.2.4 L2 Convergence Theory
The L2convergence theory states that:
Theorem 5.5. If f ∈ L2(T), then the Fourier expansion fN(f ) → f in L2(T). In other word, {eikx|k ∈ Z} constitutes an orthonormal basis in L2(T).
Proof. In the proof below, I shall use the fact that C∞(T) is dense in L2(T). I shall not prove this theorem. We can prove the L2convergence by the following two equivalent arguments.
1. We have seen that C∞can be approximated by trigonometric polynomials. That is, C∞(T) ⊂ hU i, where U = {eikx|k ∈ Z}. From C∞(T) = L2(T), we get L2(T) = hU i.
2. Alternatively, we show that if f ∈ L2(T) and f ⊥ eikx for all k ∈ Z, then f = 0. For any f ∈ C∞(T), if (f, eikx) = 0 for all k ∈ Z, from its finite Fourier expansion fN, which is zero, converges to f , we get that f ≡ 0. Thus, U⊥∩ C∞(T) = {0}. For arbitrary f ∈ L2(T), suppose f ⊥ eikx for all k. We regularize f by f := ρ∗ f → f in L2(T). But the Fourier coefficients bfare
ρ\∗ f
k= (ρb)k( ˆf )k= 0.
Thus, f⊥ eikx for all k ∈ Z. This together with f∈ C∞(T) give f≡ 0. Since f → f in L2(T), we get f ≡ 0 also. We conclude that f ⊥ U implies f = 0.
5.2. THE FOURIER BASIS IN L2(T) 95 Remark By the general theorem of orthogonal basis in Hilbert space, we have seen that (a) U⊥= {0} ⇔ (b) hU i = L2(T) ⇔ (c) Parvesal equality: kf k2 =P
k| ˆfk|2. Yet, we shall state and prove them below.
The Fourier transform maps a 2π-periodic function f into its Fourier coefficients ( ˆfk)∞k=−∞. We may view the Fourier transform maps L2(T) space into `2 space. The function spaces L2 and
`2are defined below.
L2(T) := {f | f is 2π periodic and
This is the Bessel inequality. It says that the Fourier transform maps continuously from L2(T) to
`2(Z).
Theorem 5.6 (Isometry property). The Fourier transform is an isometry from L2(T) to `2(Z):
(f, g) =X
k
fˆkgˆk.
Proof. To show this, we first assume that f is a smooth function. We can apply the convergence theorem for f . This yields
(f, g) = 1
In the last equality, the summation in k converges fast and is independent of x (from smoothness of f ). This implies that we can interchange the integration in x and the summation in k.
To show this formula is also valid for all f, g ∈ L2, we approximate f by f := ρ∗ f , which are in C∞and converge to f in L2. The isometry property is valid for fand g: (f, g) = ( bf, ˆg).
As → 0,
|(f− f, g)| ≤ kf− f kkgk → 0, and
|( bf− ˆf , ˆg)| ≤ k bf− ˆf kkˆgk ≤ kf− f kkgk → 0.
The last inequality is from the Bessel inequality. Thus, we obtain (f, g) = ( ˆf , ˆg).
The isometry property says that the Fourier transformation preserves the inner product. When g = f in the above isometry property, we obtain the following Parseval identity.
Corollary 5.2 (Parseval identity). For f ∈ L2, we have kf k2=X
k
| ˆfk|2.
Theorem 5.7 (L2-convergence theorem). If f ∈ L2, then fN =
N
X
k=−N
fˆkeikx→ f in L2.
Proof. First, the sequence {fN} is a Cauchy sequence in L2. This follows from kfN − fMk = P
N ≤|k|<M| ˆfk|2 and the Bessel inequality. Suppose fN converges to g. Then it is easy to check that the Fourier coefficients of f − g are all zeros. From the Parvesal identity, we have f = g.
5.2.5 BV Convergence Theory
A function is called a BV function (or a function of finite total variation) on an interval (a, b), if for any partition π = {a = x0 < x1< · · · < xn= b},
kf kBV := sup
π
X
i
|f (xi) − f (xi−1)| < ∞.
An important property of BV function is that its singularity can only be jump discontinuities, i.e., at a discontinuity, say, x0, f has both left limit f (x0−) and right limit f (x0+).
Further, any BV function f can be decomposed into f = f0 + f1, where f0 is a piecewise constant function andf1is absolutely continuous (i.e. f1 is differentiable and f10 is integrable). The jump points of f0are countable. The BV-norm of f is exactly equal to
kf kBV =X
i
|[f (xi)]| + Z
|f10(x)| dx.
where xiare the jump points of f (also f0) and [f (xi)] := f (xi+) − f (xi−) is the jump of f at xi.
5.2. THE FOURIER BASIS IN L2(T) 97 Theorem 5.8 (Fourier inversion theorem for BV functions). If f is in BV (function of bounded variation), then
Gibbs phenomena In applications, we encounter piecewise smooth functions frequently. In this case, the approximation is not uniform. An overshoot and undershoot always appear across disconti-nuities. Such a phenomenon is called Gibbs phenomenon. Since a BV function can be decomposed into a piecewise constant function and a smooth function, we concentrate to the case when there is only one discontinuity. The typical example is the function
f (x) =
1 for 0 < x < π
−1 for − π < x < 0 The corresponding fN is
fN(x) = 1
First, we show that we may replace 2 sin(t/2)1 by 1t with possible error o(1/N ). This is because the function1t −2 sin(t/2)1 is in C1on [−π, π] and the Riemann-Lebesgue lemma. Thus, we have
Here, the function sinc(t) := sin(t)/t. It has the following properties: To see the latter inequality, we rewrite
Z ∞
where n = [z/π] + 1. Notice that the series is an alternating series. Thus, the series is bounded by its leading term, which is of O(1/z). Let us denote the integralRz
0 sinc(t) dt by Si(z).
To show that the sequence fN does not converge uniformly, we pick up x = z/(N + 1/2) with z > 0. After changing variable, we arrive
fN( z In general, for function f with arbitrary jump at 0, we have
fN( z distance of x and the nearest discontinuity is N−1+α, then the convergent rate at x is only O(N−α).
If the distance is O(1), then the convergent rate is O(N−1). This shows that the convergence is not uniform.
The maximum of Si(z) indeed occurs at z = π where 1
πSi(π) ≈ 0.58949 This yields
fN( π
N + 1/2) = f (0+) + 0.08949 (f (0+) − f (0−)).
Hence, there is about 9% overshoot. This is called Gibbs phenomenon.
5.2. THE FOURIER BASIS IN L2(T) 99 5.2.6 Fourier Expansion of Real Valued Functions
We have
Thus, when f is real valued,
fˆn= ˆf−n.
The functions {cos nx, sin nx} are orthogonal to each other. But 1
1. Derive the Fourier expansion formula for periodic functions with period L.
2. What is the limit of the above Fourier expansion formula as L → ∞.
3. Derive the Fourier expansion for the following functions: f (x) = |x| − 1/2 for |x| ≤ 1 and f is a periodic function with period 2.
4. What is the convergence rate of the above function in L2and pointwise convergence rate at x = 0?
5.3 Applications of Fourier expansion
5.3.1 Characterization of Sobolev spaces
Let Hm(T) be the completion of C∞(T) under the norm
kuk2Hm := kuk2+ ku0k2+ · · · + ku(m)k2. From bu0k= ik ˆuk, we get
ud(m)k= (ik)muˆk. From Parseval equality, we obtain
kuk2 =X
k∈Z
|ˆuk|2, · · · , ku(m)k2 =X
k∈Z
|k|2m|ˆuk|2.
Thus, we have
kuk2Hm =X
k∈Z
(1 + |k|2+ · · · + |k|2m)|ˆuk|2.
The regularity of u is characterized by u, ..., u(m) ∈ L2. On the other hand, it is also described by X
k∈Z
(1 + |k|2+ · · · + |k|2m)|ˆuk|2 < ∞,
which is an equivalent way to characterize the decay of ˆuk.
Remark. Notice that for a fixed m ≥ 0, the following quantities are equivalent (1 + |k|)2m∼ (1 + |k|2)m∼ (1 + |k|2m) for all k ∈ Z.
This means that there are positive constants Cisuch that
(1 + |k|)2m≤ C1(1 + |k|2)m≤ C2(1 + |k|2m) ≤ C3(1 + |k|)2mfor all k ∈ Z.
Thus, the Sobolev norm kuk2Hmwith m ≥ 0 is equivalent to X
k∈Z
(1 + |k|2m)|ˆuk|2.
We can also define Sobolev space with negative exponent m by kuk2Hm =X
k∈Z
(1 + |k|)2m|ˆuk|2.
When m is large enough, the Sobolev space Hm(T) can be embedded into C(T). This is the following theorem.
5.3. APPLICATIONS OF FOURIER EXPANSION 101 Theorem 5.9. For m > 1/2, we have Hm(T) ⊂ C(T).
Proof. 1. For any smooth function f , we have
|f (x)| =
5.3.2 Heat equation on a circle
We can solve the heat flow on circle exactly. This problem indeeds motivated Fourier invent the Fourier expansion. Let us consider
ut= uxx, x ∈ T
Since {einx|n ∈ Z} are independent, we get
˙
un= −n2un. Thus,
un(t) = un(0)e−n2t.
At t → 0, we expect un(0) = ˆfn. Thus, we define the function u(x, t) =X
n∈Z
fˆne−n2teinx.
In the following, we need to check:
1. u, utand uxxexist and ut= uxx for t > 0 and x ∈ T;
2. u(·, t) → f in L2(T) as t → 0+.
• Proof of (1). We show that ux exists here. It is clearly that P
n∈Zfˆne−n2teinx converges absolute and uniformly w.r.t. x for t > 0, as long as ˆfn grows at most algebraically in n.
SinceP
n∈Zine−n2tfˆneinxconverges absolute and uniformly w.r.t. x for t > 0. This implies u is differentiable in x and the differentiation can be interchange with the infinite summation:
∂xu = ∂x
X
n∈Z
fˆne−n2t=X
n∈Z
fˆne−n2tineinx.
Similar proof for the existence of uxx and utfor t > 0. Since the Fourier coefficients of ut
and uxxare identical on t > 0, we thus get ut= uxx.
• Proof of (2). Let us denote u(·, t) by T (t)f and itself Fourier transform ˆuk(t) by ˆT f , or T (t) bˆ f . T is a linear operator from L2(T) to itself, while bT (t) a linear operator in `2(Z). We have
T (t) bb fn= e−n2tfˆn.
Our goal is to prove T (t)f → f in L2(T). By the isometry property of the Fourier transform, this is equivalent to bT (t) bf → bf in `2(Z). We have
t→0+lim k bT (t) bf − bf k22 = lim
t→0+
X
n∈Z
|(e−n2t− 1)2| bfn|2
=X
n∈Z
t→0+lim |(e−n2t− 1)2| bfn|2 = 0.
The interchange ofP and lim here is due to the dominant convergence theorem and the convergence of
X
n∈Z
|(e−n2t− 1)2| bfn|2 ≤ 22X
n∈Z
| bfn|2< ∞.
is uniform w.r.t. t.
5.3.3 Solving Laplace equation on a disk We consider the Laplace equation
uxx+ uyy = 0
5.3. APPLICATIONS OF FOURIER EXPANSION 103 on the domain
Ω : x2+ y2 < 1, with the Dirichlet boundary condition:
u = f on ∂Ω.
In the polar coordinate, the equation has the form:
urr+1
rur+ 1
r2uθθ = 0.
The boundary condition is
u(1, θ) = f (θ), θ ∈ T.
The solution is expanded as
u(r, θ) =X
n∈Z
un(r)einθ. Plug this into the Laplace equation, we get
X
n∈Z
u00n+1
ru0n−n2 r2
einθ= 0.
This leads to
u00n+1
ru0n−n2
r2 = 0 for all n ∈ Z.
The two independent solutions are un = rnor un = r−n. However, the one with negative power will not satisfy the finiteness of u at r = 0. Thus, we obtain
u(r, θ) =X
n∈Z
anr|n|einθ. At r = 1, we get
f (θ) =X
n∈Z
aneinθ. Thus,
an= 1 2π
Z
T
f (θ)e−inθdθ.
For r < 1, the L2norm of the infinite seriesP
n∈Zanr|n|einθis bounded byP
n∈Z|an|2, uniformly in r < 1. Thus, from dominant convergence theorem, we have
r→1−lim u(r, ·) = f (·) in L2(T).
If we differentiate the infinite series in r term-by-term, we get X
n∈Z
an|n|r|n|−1einθ.
This infinite series converges absolutely and uniformly for r ≤ r0 for any fixed r0 < 1. This implies that u is differentiable in r and the differentiation can be performed term-by-term in the infinite series:
∂ru =X
n∈Z
an|n|r|n|−1einθ.
By the same argument, we get ∂rru and ∂θθu exist and u satisfies the Laplace equation in polar coordinate form.
Alternatively, we can write the above summation in convolution form:
u(r, θ) = Z
T
g(r, θ − φ)f (φ) dφ, where
g(r, θ) = 1 2π
X
n∈Z
r|n|einθ
= 1 2π
1
1 − reiθ + e−iθ 1 − e−iθ
= 1 2π
1 − r2 1 − 2r cos θ + r2
The function g is called the Poisson kernel. It is infinitely differentiable for r < 1. This implies g ∗ f ∈ C∞(Ω).
5.3.4 Hurwitz’s proof for isoperimetric inequality (see Hunter’s book)
The isoperimetric inequality involves to find the maximal area enclosed by a simple closed curve with given perimeter. If the perimeter is L, the area is A, then the isoperimeter inequality is
4πA ≤ L2.
The equality holds when the closed curve is a circle. There are many proofs of this inequality.
In 1902, Hurwitz provided a proof using Fourier expansion. Let us show his proof here as an application of Fourier expansion. Let the closed curve is given by (x, y) = (f (s), y(s)), where s is the arc length. We may assume the length of the curve is 2π, otherwise we rescale it by (x, y) by (2πx/L, 2πy/L). Since s is the arc length, we have
f (s)˙ 2+ ˙g(s)2= 1.
The area of the enclosed region is given by A = 1
2 Z
T
f (s) ˙g(s) − g(s) ˙f (s) ds.
Our goal is to maximize A subject to the perimeter constraint Z
T
f (s)˙ 2+ ˙g(s)2ds = 2π.
5.3. APPLICATIONS OF FOURIER EXPANSION 105 We expand f and g in Fourier series:
f (s) = For the area functional, we get
A
Subtracting these two series, we get 1 −A
Homework Derive the Euler-Lagrange for the constrained maximization problem:
max1
5.3.5 Von Neumann stability analysis for finite difference methods
In numerical PDEs, the stability analysis is a crucial step to the convergence theory of a numerical scheme. Below, I shall demonstrate the von Neumann stability analysis for heat equation in one dimension. It is a L2 stability analysis suitable for for (the interior part of) numerical PDEs with constant coefficients.
Let us consider the heat equation:
ut= uxx
in one dimension with initial data u(x, 0) = f (x). Let h = ∆x, k = ∆t be the spatial and temporal mesh sizes. Define xj = jh, j ∈ Z and tn = nk, n ≥ 0. Let us abbreviate u(xj, tn) by unj. We shall approximate unj by Ujn, where Ujnsatisfies some finite difference equations.
• Spatial discretization: The simplest one is to use the centered finite difference approximation for uxx:
uxx = uj+1− 2uj + uj−1
h2 + O(h2) := Dx,+Dx,−u + O(h2).
Here, the notation (Dx,+u)j := (u(xj+1)−u(xj))/h is the forward finite difference, (Dx,−u)j = (u(xj) − u(xj−1))/h the backward finite difference. You can check that
Dx,+Dx,−uj = a ((uj+1− uj) − (uj− uj−1)) /h2. The spatial discretization results in the following systems of ODEs
U˙j(t) = Uj+1(t) − 2Uj(t) + Uj−1(t) h2
or in vector form
U =˙ 1 h2AU where U = (U0, U1, ...)t, A = diag (1, −2, 1).
• Temporal discretization: We can apply numerical ODE solvers – Forward Euler method:
Un+1= Un+ k
h2AUn (5.4)
– Backward Euler method:
Un+1= Un+ k
h2AUn+1 (5.5)
– 2nd order Runge-Kutta (RK2):
Un+1− Un= k
h2AUn+1/2, Un+1/2= Un+ k
2h2AUn (5.6) – Crank-Nicolson:
Un+1− Un= k
2h2(AUn+1+ AUn). (5.7)
5.3. APPLICATIONS OF FOURIER EXPANSION 107 These linear finite difference equations can be solved formally as
Un+1= GUn where
• Forward Euler: G = 1 + hk2A,
• Backward Euler: G = (1 − hk2A)−1,
• RK2: G = 1 + hk2A +12 hk2
2
A2
• Crank-Nicolson: G = 1+
k 2h2A 1− k
2h2A
For the Forward Euler, We may abbreviate it as
Ujn+1= G(Uj−1n , Ujn, Uj+1n ), (5.8) where
G(Uj−1, Uj, Uj+1) = Uj + k
h2(Uj−1− 2Uj+ Uj+1)
Stability and Convergence for the Forward Euler method Our goal is to show under what condition can Ujn converges to u(xj, tn) as the mesh sizes h, k → 0. To see this, we first see the error produced by a true solution by the finite difference equation. Plug a true solution u(x, t) into (5.4). We get
un+1j − unj = k
h2 unj+1− 2unj + unj−1 + kτjn (5.9) where
τjn= Dt,+unj − (ut)nj − (D+D−unj − (uxx)nj) = O(k) + O(h2).
Let enj denote for unj − Ujn. Then subtract (5.4) from (5.9), we get en+1j − enj = k
h2 enj+1− 2enj + enj−1 + kτjn. (5.10) This can be expressed in operator form:
en+1 = Gen+ kτn. (5.11)
kenk ≤ kGen−1k + kkτn−1k
≤ kG2en−2k + k(kGτn−2k + kτn−1k)
≤ kGne0k + k(kGn−1τ0k + · · · + kGτn−2k + kτn−1k) Suppose G satisfies the stability condition
kGnU k ≤ CkU k
for some C independent of n. Then
kenk ≤ Cke0k + C max
m |τm|.
If the local truncation error has the estimate
maxm kτmk = O(h2) + O(k) and the initial error e0satisfies
ke0k = O(h2), then so does the global true error satisfies
kenk = O(h2) + O(k) for all n.
The above analysis leads to the following definitions.
Definition 5.3. A finite difference method is called consistent if its local truncation error τ satisfies kτh,kk → 0 as h, k → 0.
Definition 5.4. A finite difference scheme Un+1 = Gh,k(Un) is called stable under the norm k · k in a region(h, k) ∈ R if
kGnh,kU k ≤ CkU k for alln with nk fixed. Here, C is a constant independent of n.
Definition 5.5. A finite difference method is called convergence if the true error keh,kk → 0 as h, k → 0.
In the above analysis, we have seen that for forward Euler method for the heat equation, stability + consistency ⇒ convergence.
L2Stability – von Neumann Analysis Since we only deal with smooth solutions in this section, the L2-norm or the Sobolev norm is a proper norm to our stability analysis. For constant coefficient and scalar case, the von Neumann analysis (via Fourier method) provides a necessary and sufficient condition for stability. For system with constant coefficients, the von Neumann analysis gives a necessary condition for statbility. For systems with variable coefficients, the Kreiss’ matrix theorem provides characterizations of stability condition.
Below, we give L2stability analysis. We use two methods, one is the energy method, the other is the Fourier method, that is the von Neumann analysis. We describe the von Neumann analysis below.
Given {Uj}j∈Z, we define
kU k2=X
j
|Uj|2
5.3. APPLICATIONS OF FOURIER EXPANSION 109 and its Fourier transform
U (ξ) =ˆ 1 2π
XUje−ijξ.
The advantages of Fourier method for analyzing finite difference scheme are
• the shift operator is transformed to a multiplier:
T U (ξ) = ed iξU (ξ),ˆ where (T U )j := Uj+1;
• the Parseval equility
kU k2 = k ˆU k2
≡ Z π
−π
| ˆU (ξ)|2dξ.
If a finite difference scheme is expressed as Ujn+1= (GUn)j =
m
X
i=−l
ai(TiUn)j, then
U[n+1(ξ) = bG(ξ) cUn(ξ).
From the Parseval equality,
kUn+1k2 = k [Un+1k2
= Z π
−π
| bG(ξ)|2| cUn(ξ)|2dξ
≤ max
ξ | bG(ξ)|2 Z π
−π
| cUn(ξ)|2dξ
= | bG|2∞kU k2 Thus a sufficient condition for stability is
| bG|∞≤ 1. (5.12)
Conversely, suppose | bG(ξ0)| > 1, from bG being a smooth function in ξ, we can find and δ such that
| bG(ξ)| ≥ 1 + for all |ξ − ξ0| < δ.
Let us choose an initial data U0in `2such that cU0(ξ) = 1 for |ξ − ξ0| ≤ δ. Then k cUnk2 =
Z
| bG|2n(ξ)| cU0|2
≥ Z
|ξ−ξ0|≤δ
| bG|2n(ξ)| cU0|2
≥ (1 + )2nδ → ∞ as n → ∞
Thus, the scheme can not be stable. We conclude the above discussion by the following theorem.
Theorem 5.10. A finite difference scheme Ujn+1=
m
X
k=−l
akUj+kn with constant coefficients is stable if and only if
G(ξ) :=b
m
X
k=−l
ake−ikξ satisfies
−π≤ξ≤πmax | bG(ξ)| ≤ 1. (5.13)
Homeworks.
1. Compute the bG for the schemes: Forward Euler, Backward Euler, RK2 and Crank-Nicolson.
5.3.6 Weyl’s ergodic theorem
In discrete dynamical systems, the dynamics of xn is characterized by an iterative map xn+1 = F (xn). The simplest example is xn ∈ T and F (xn) = xn+ 2πγ. This is arisen from integrable system. It is the Poincare section map of the continuous integrable system:
(x, y) 7→ (x(t), y(t)) := (x + 2πωxt, y + 2πωyt), (x, y) ∈ T2, ωx, ωy ∈ R.
The cross section is taken to be y ≡ 0( mod 2π). Then the trajectory with y0= 0 will revisit y = 0 at time with 2πωyt = 2πn, that is, tn= n/ωy. The corresponding x(tn) = x + 2πnωx/ωy. If we call γ := ωx/ωy, then the map: x(tn) 7→ x(tn+1) is the above linear discrete map.
For a continuous function f : T → C, we are interested in two kinds of averages:
• phase space average: hf is:= 2π1 R
Tf (x) dx,
• time average: hf it:= limN →∞N +11 PN
n=0f (xn).
Theorem 5.11 (Weyl ergodic 1916). If γ is irrational, then
hf it(x0) = hf is (5.14)
for allf ∈ C(T) and for all x0 ∈ T.
Proof. 1. It is easy to check that (5.14) holds for all f = eimx: 1
N + 1f (xn) = 1 N + 1
N
X
n=0
eim(x0+2πnγ)
= eimx0 N + 1
N
X
n=0
e2πimγn
= eimx0 N + 1
1 − e2πimγ(N +1)
1 − e2πimγ
! .
5.3. APPLICATIONS OF FOURIER EXPANSION 111
Here, we have used e2πimγ 6= 1 for irrational γ. On the other hand, heimxis= 1
2π Z
T
eimxdx = 0.
Thus, heimxit= heimxis. With this, (5.14) also holds for all trigonometric polynomials.
2. The trigonometric polynomials are dense in C(T).
3. Given any f ∈ C(T), for any > 0, there exists a trigonometric polynomial p such that kf − pk∞< . We have Taking limit sup, we get
lim sup
Chapter 6
Compactness
6.1 Compactness in metric space
6.1.1 Motivation and brief history
1 The concept of compactness plays an important role in analysis. There are many notions for compactness. Here, I will mention the most fundamental two. The first one states that: a set is called compact if any its open cover has finite subcover. It is motivated from a fundamental question in analysis: on which domain a local property can also be a global property. For instance, on which domain a continuous function is indeed uniform continuous. This concept (open cover) was introduced by Dirichlet in his 1862 lectures, which were published in 1904. This concept was repeatedly introduced by Heine (1872), Borel (1895) and continuously developed by Cousin (1895), Lebesgue (1898), Alexandroff and Uryson (1929). The main theorem is the Heine-Borel theorem which states that a set in Rnis compact if and only if it is closed and bounded.
The concept of the second notion is called sequential compactness, which was started from the Bolzano(1817)-Weistrass(1857) theorem. It states that a bounded sequence in Rnalways has a convergence subsequence. It can be proven by bi-section method. The original Bolzano theorem was indeed a lemma to prove the extremal value theorem: a continuous function on a closed bounded interval always attains its extremal value. It was generalized to the function space (C[a, b], | · |∞) by Arzel`a(1882-1883)-Ascoli(1883-1884) which states that a bounded and equi-continuous family of functions has convergence subsequence. Its generalization to C(K) was done by Fr´echet (1906)
The concept of the second notion is called sequential compactness, which was started from the Bolzano(1817)-Weistrass(1857) theorem. It states that a bounded sequence in Rnalways has a convergence subsequence. It can be proven by bi-section method. The original Bolzano theorem was indeed a lemma to prove the extremal value theorem: a continuous function on a closed bounded interval always attains its extremal value. It was generalized to the function space (C[a, b], | · |∞) by Arzel`a(1882-1883)-Ascoli(1883-1884) which states that a bounded and equi-continuous family of functions has convergence subsequence. Its generalization to C(K) was done by Fr´echet (1906)