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.
3.4.1 Convergence theory for Smooth function
Theorem 3.3. If f is a 2π-periodic, C∞-function, then for any n > 0, there exists a constant Cn
such that
3.4. CONVERGENCE THEORY 71 and in C∞. We can apply integration-by-part n times to arrive
fN(x) − f (x) = (N + 1 for even n. Similar formula for odd 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 (3.3) is highly efficient for smooth functions. For example, N = 20 is sufficient in many applications. The accuracy property (3.3) is called spectral accuracy.
3.4.2 L2 Convergence Theory
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 `2space. The function spaces L2and `2are defined below.
L2(T) := {f | f is 2π periodic and Z π
−π
|f (x)|2dx < ∞}
with the inner product
(f, g) := 1
An important fact is that all L2-function can be approximated by smooth functions. Such a smooth function can be obtained by convoling f with a smooth function, called mollifier. Let ρ ∈ C∞(T), which is positive in a neighborhood of 0 and is zero elsewhere, andR
Tρ(x) dx = 1.
The space `2(Z) is defined as
`2(Z) := {(ak)∞k=−∞ |
∞
X
k=−∞
|ak|2< ∞}.
with inner product (a, b) :=P
kakbk.
It is easy to check that eikxare orthogonal in L2. From this, we have for any N , 0 ≤ (f − fN, f − fN) = kf k2− X
|k|≤N
| ˆfk|2.
Or equivalently,
X
|k|≤N
| ˆfk|2 ≤ kf k2. (3.4)
This is called the Bessel inequality. It says that the Fourier transform maps continuously from L2(T) to `2(Z).
Theorem 3.4 (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 2π
Z π
−π
f (x)g(x) dx
= 1
2π Z π
−π
X
k
fˆkeikxg(x) dx
= X
k
fˆkgˆk.
To show this formula is valid for all f, g ∈ L2, we notice that any function in L2 can be approxi-mated by smooth functions.
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.
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.
3.4. CONVERGENCE THEORY 73 Corollary 3.3 (Parseval identity). For f ∈ L2, we have
kf k2 =X
k
| ˆfk|2.
Theorem 3.5 (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 − fMk2 = P
N ≤|k|<M| ˆfk|2and the Bessel inequality. Suppose fN converges to g. We see that (f − f\N)k= 1
2π Z
T
(f − fN)(x)e−ikxdx = 0 if |k| < N.
Thus, for each fixed k, taking N → ∞, we get
(f − g)\ k= 0.
This holds for any k ∈ Z. Thus, the Fourier coefficients of f − g are all zeros. From the Parvesal identity, we have f = g.
3.4.3 BV Convergence Theory
A function is called a BV function on an interval (a, b), that is, function of finite total variation, 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. f1is differentiable and f10 is integrable). The jump points of f0 are 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. Theorem 3.6 (Fourier inversion theorem for BV functions). If f is in BV (function of bounded variation), then
fN(x) :=
N
X
k=−N
fˆkeikx→ 1
2(f (x+) + f (x−)).
Proof. Recall that
3.4.4 Pointwise estimate of rate of convergence
In applications, we encounter piecewise smooth functions frequently. In this case, the approxima-tion is not uniform. An overshoot and undershoot always appear across discontinuities. Such a phenomenon is called Gibbs phenomenon. Since a BV function can be decomposed into a piece-wise 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:
Z ∞ 0
sinc(t) dt = π/2.
3.4. CONVERGENCE THEORY 75 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 calledGibbs phenomenon.
Homeworks
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 L2 and pointwise convergence rate at x = 0?
3.4.5 Fourier Expansion of Real Valued Functions We have
Thus, when f is real-valued,
fˆn= ˆf−n.
The functions {cos nx, sin mx} are orthogonal to each other. Further, 1
The Parseval equality reads 1