Fourier Analysis

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5 Fourier Analysis

All bases are equal, but some are more equal then others.

As we saw already any separable Hilbert space posses an orthonormal basis (infinitely many of them indeed). Are they equally good? This depends from our purposes. For solution of differential equation which arose in mathematical physics (wave, heat, Laplace equations, etc.) there is a proffered choice. The fundamental formula: d/dx e^ax=ae^ax reduces the derivative to a multiplication by a. We could benefit from this observation if the orthonormal basis will be constructed out of exponents. This helps to solve differential equations as was demonstrated in Subsection 0.2.

7.40pm Fourier series: Episode II

Today’s TV listing

5.1 Fourier series

Now we wish to address questions stated in Remark 9. Let us consider the space L₂[−π,π]. As we saw in Example 3 there is an orthonormal sequence e_n(t)=(2π)^−1/2e^int in L₂[−π,π]. We will show that it is an orthonormal basis, i.e.

f(t)∈ L₂[−π,π] ⇔ f(t)=

∞

∑

k=−∞

⟨ f,e_k ⟩e_k(t),

with convergence in L₂ norm. To do this we show that CLin{e_k:k∈ℤ}=L₂[−π,π].

Let CP[−π,π] denote the continuous functions f on [−π,π] such that f(π)=f(−π). We also define f outside of the interval [−π,π] by periodicity.

Lemma 1 The space CP[−π,π] is dense in L₂[−π,π].

Figure 9: A modification of continuous function to periodic

Proof. Let f∈L₂[−π,π]. Given є>0 there exists g∈ C[−π,π] such that ||f−g||<є/2. From continuity of g on a compact set follows that there is M such that | g(t) |<M for all t∈[−π,π].

We can now replace g by periodic g′, which coincides with g on [−π,π−δ] for an arbitrary δ>0 and has the same bounds: | g′(t) |<M, see Figure 9. Then

⎪⎪
⎪⎪

g−g′

⎪⎪
⎪⎪

₂²=

∫

π−δ

⎪
⎪

g(t)−g′(t)

⎪
⎪

² d t ≤ (2M)²δ.

So if δ<є²/(4M)² then ||g−g′||<є/2 and ||f−g′||<є. □

Now if we could show that CLin{e_k: k ∈ ℤ} includes CP[−π,π] then it also includes L₂[−π,π].

Notation 2 Let f∈CP[−π,π],write

f_n=

∑

k=−n

⟨ f,e_k ⟩ e_k , for n=0,1,2,… (24)

the partial sum of the Fourier series for f.

We want to show that ||f−f_n||₂→ 0. To this end we define nth Fejér sum by the formula

F_n=

f₀+f₁+⋯+f_n

n+1

, (25)

and show that

⎪⎪
⎪⎪

F_n−f

⎪⎪
⎪⎪

_∞ → 0.

Then we conclude

⎪⎪
⎪⎪

F_n−f

⎪⎪
⎪⎪

₂=

⎛
⎜
⎜
⎝

∫

−π

⎪
⎪

F_n(t)−f

⎪
⎪

⎞
⎟
⎟
⎠

1/2

≤ (2π)^1/2

⎪⎪
⎪⎪

F_n−f

⎪⎪
⎪⎪

_∞→ 0.

Since F_n∈Lin((e_n)) then f∈CLin((e_n)) and hence f=∑_−∞^∞⟨ f,e_k ⟩e_k.

Remark 3 It is not always true that ||f_n−f||_∞→ 0 even for f∈CP[−π,π].

Exercise 4 Find an example illustrating the above Remark.

The summation method used in (25) us useful not only in the context of Fourier series but for many other cases as well. In such a wider framework the method is known as .

It took 19 years of his life to prove this theorem

5.2 Fejér’s theorem

Proposition 5 (Fejér, age 19) Let f∈CP[−π,π]. Then

F_n(x)

2π

∫

−π

f(t) K_n(x−t) d t, where

(26)

K_n(t)

n+1

∑

k=0

∑

m=−k

e^imt,

(27)

is the Fejér kernel.

Proof. From notation (24):

f_k(x)

∑

m=−k

⟨ f,e_m ⟩ e_m(x)

∑

m=−k

∫

−π

f(t)

e^−imt

√

2π

 d t 

e^imx

√

2π

∫

−π

f(t)

∑

m=−k

e^im(x−t) d t.

Then from (25):

F_n(x)

n+1

∑

k=0

f_k(x)

n+1

2π

∑

k=0

∫

−π

f(t)

∑

m=−k

e^im(x−t) d t

2π

∫

−π

f(t)

n+1

∑

k=0

∑

m=−k

e^im(x−t) d t,

which finishes the proof. □

Lemma 6 The Fejér kernel is 2π-periodic, K_n(0)=n+1 and can be expressed as:

K_n(t)=

n+1

sin²

(n+1)t

sin²

, for t∉2πℤ. (28)

1

z⁻¹ 1 z

z⁻² z⁻¹ 1 z z²

⋮ ⋮ ⋮ ⋮ ⋮ ⋱

Table 1: Counting powers in rows and columns

Proof. Let z=e^it, then:

K_n(t)

n+1

∑

k=0

(z^−k+⋯+1+z+⋯+z^k)

n+1

∑

j=−n

(n+1−

⎪
⎪

)

z^j,

by switch from counting in rows to counting in columns in Table 1. Let w=e^it/2, i.e. z=w², then

K_n(t)

n+1

(w⁻²ⁿ+2w⁻²ⁿ⁺²+⋯+(n+1)+nw²+⋯+w²ⁿ)

n+1

(w⁻ⁿ+w⁻ⁿ⁺²+⋯+wⁿ⁻²+wⁿ)²

(29)

n+1

⎛
⎜
⎜
⎝

w⁻ⁿ⁻¹−wⁿ⁺¹

w⁻¹−w

⎞
⎟
⎟
⎠

Could you sum a geometric progression?

n+1

⎛
⎜
⎜
⎜
⎜
⎜
⎝

2isin

(n+1)t

2isin

⎞
⎟
⎟
⎟
⎟
⎟
⎠

if w≠ ± 1. For the value of K_n(0) we substitute w=1 into (29). □

Figure 10: A family of Fejér kernels with the parameter m running from 0 to 9 is on the left picture. For a comparison unregularised Fourier kernels are on the right picture.

The first eleven Fejér kernels are shown on Figure 10, we could observe that:

Lemma 7 Fejér’s kernel has the following properties:

K_n(t)≥0 for all t∈ ℝ and n∈ℕ.
∫_−π^πK_n(t) d t=2π.
For any δ∈ (0,π)

−δ

∫

−π

+
π

∫

δ

K_n(t) d t → 0 as n→ ∞.

Proof. The first property immediately follows from the explicit formula (28). In contrast the second property is easier to deduce from expression with double sum (27):

∫

−π

K_n(t) d t

∫

−π

n+1

∑

k=0

∑

m=−k

e^imt d t

n+1

∑

k=0

∑

m=−k

∫

−π

e^imt d t

n+1

∑

k=0

2π

2π,

since the formula (19).

Finally if | t |>δ then sin²(t/2)≥ sin²(δ/2)>0 by monotonicity of sinus on [0,π/2], so:

0≤ K_n(t) ≤

(n+1) sin²(δ/2)

implying:

0≤

∫

δ≤

⎪
⎪

≤ π

K_n(t)  d t ≤

1(π−δ)

(n+1) sin²(δ/2)

→ 0 as n→ 0.

Therefore the third property follows from the squeeze rule. □

Theorem 8 (Fejér Theorem) Let f∈CP[−π,π]. Then its Fejér sums F_n (25) converges in supremum norm to f on [−π,π] and hence in L₂ norm as well.

Proof. Idea of the proof: if in the formula (26)

F_n(x)=

2π

∫

−π

f(t) K_n(x−t) d t,

t is long way from x, K_n is small (see Lemma 7 and Figure 10), for t near x, K_n is big with total “weight” 2π, so the weighted average of f(t) is near f(x).

Here are details. Using property 2 and periodicity of f and K_n we could express trivially

f(x)= f(x)

2π

x+π

∫

x−π

K_n(x−t)  dt =

2π

x+π

∫

x−π

f(x) K_n(x−t) d t.

Similarly we rewrite (26) as

F_n(x)=

2π

x+π

∫

x−π

f(t) K_n(x−t) d t,

then

⎪
⎪

f(x)−F_n(x)

⎪
⎪

2π

⎪
⎪
⎪
⎪

x+π

∫

x−π

(f(x)−f(t)) K_n(x−t) d t

⎪
⎪
⎪
⎪

≤

2π

x+π

∫

x−π

⎪
⎪

f(x)−f(t)

⎪
⎪

K_n(x−t) d t.

Given є>0 split into three intervals: I₁=[x−π,x−δ], I₂=[x−δ,x+δ], I₃=[x+δ,x+π], where δ is chosen such that | f(t)−f(x) |<є/2 for t∈ I₂, which is possible by continuity of f. So

2π

∫

I₂

⎪
⎪

f(x)−f(t)

⎪
⎪

K_n(x−t) d t≤

2π

∫

I₂

K_n(x−t) d t <

And

2π

∫

I₁⋃ I₃

⎪
⎪

f(x)−f(t)

⎪
⎪

K_n(x−t)  dt

≤

⎪⎪
⎪⎪

_∞

2π

∫

I₁⋃ I₃

K_n(x−t)  dt

⎪⎪
⎪⎪

_∞

∫

δ<

⎪
⎪

<π

K_n(u)  du

if n is sufficiently large due to property 3 of K_n. Hence | f(x)−F_n(x) |<є for a large n independent of x. □

Remark 9 The above properties 1–3 and their usage in the last proof can be generalised to the concept of approximation of the identity. See § 15.4 for a further example.

We almost finished the demonstration that e_n(t)=(2π)^−1/2e^int is an orthonormal basis of L₂[−π,π]:

Corollary 10 (Fourier series) Let f∈ L₂[−π,π], with Fourier series

∞

∑

n=−∞

⟨ f,e_n ⟩e_n=

∞

∑

n=−∞

c_ne^int where c_n=

⟨ f,e_n ⟩

√

2π

√

2π

∫

−π

f(t)e^−int d t.

Then the series ∑_−∞^∞⟨ f,e_n ⟩e_n=∑_−∞^∞c_ne^int converges in L₂[−π,π] to f, i.e

lim

k→ ∞

⎪⎪
⎪⎪
⎪⎪
⎪⎪

f−

∑

n=−k

c_ne^int

⎪⎪
⎪⎪
⎪⎪
⎪⎪

₂=0.

Proof. This follows from the previous Theorem, Lemma 1 about density of CP in L₂, and Theorem 15 on orthonormal basis. □

Remark 11 There is a reason why we had used the Fejér kernel and the Cezàro summation F_n (25) instead of plain partial sums f_n (24) of the Fourier series. It can be shown that point-wise convergence f_n → f does not hold for every continuous function f, cf. Cor. 41.

5.3 Parseval’s formula

The following result first appeared in the framework of L₂[−π,π] and only later was understood to be a general property of inner product spaces.

Theorem 12 (Parseval’s formula) If f, g∈L₂[−π,π] have Fourier series f=∑_n=−∞^∞c_ne^int and g=∑_n=−∞^∞d_ne^int, then

⟨ f,g ⟩=

∫

−π

f(t)

g(t)

 d t=2π

∞

∑

−∞

c_n

d_n

. (30)

More generally if f and g are two vectors of a Hilbert space H with an orthonormal basis (e_n)_−∞^∞ then

⟨ f,g ⟩=

∞

∑

k=−∞

c_n

d_n

, where c_n=⟨ f,e_n ⟩, d_n=⟨ g,e_n ⟩,

are the Fourier coefficients of f and g.

Proof. In fact we could just prove the second, more general, statement—the first one is its particular realisation. Let f_n=∑_k=−nⁿ c_ke_k and g_n=∑_k=−nⁿ d_ke_k will be partial sums of the corresponding Fourier series. Then from orthonormality of (e_n) and linearity of the inner product:

⟨ f_n,g_n ⟩=⟨

∑

k=−n

c_ke_k,

∑

k=−n

d_ke_k ⟩=

∑

k=−n

c_k

d_k

This formula together with the facts that f_k→ f and g_k→ g (following from Corollary 10) and Lemma about continuity of the inner product implies the assertion. □

Corollary 13 A integrable function f belongs to L₂[−π,π] if and only if its Fourier series is convergent and then ||f||²=2π∑_−∞^∞| c_k |².

Proof. The necessity, i.e. implication f∈L₂ ⇒ ⟨ f,f ⟩=||f||²=2π∑| c_k |² , follows from the previous Theorem. The sufficiency follows by Riesz–Fisher Theorem. □

Remark 14 The actual rôle of the Parseval’s formula is shadowed by the orthonormality and is rarely recognised until we meet the wavelets or coherent states. Indeed the equality (30) should be read as follows:

Theorem 15 (Modified Parseval) The map W: H → l₂ given by the formula [Wf](n)=⟨ f,e_n ⟩ is an isometry for any orthonormal basis (e_n).

We could find many other systems of vectors (e_x), x∈ X (very different from orthonormal bases) such that the map W: H → L₂(X) given by the simple universal formula

[Wf](x)=⟨ f,e_x ⟩ (31)

will be an isometry of Hilbert spaces. The map (31) is oftenly called wavelet transform and most famous is the Cauchy integral formula in complex analysis. The majority of wavelets transforms are linked with group representations, see our postgraduate course Wavelets in Applied and Pure Maths.

Heat and noise but not a fire?

Answer:

5.4 Some Application of Fourier Series

We are going to provide now few examples which demonstrate the importance of the Fourier series in many questions. The first two (Example 16 and Theorem 17) belong to pure mathematics and last two are of more applicable nature.

Example 16 Let f(t)=t on [−π,π]. Then

⟨ f,e_n ⟩=

∫

−π

te^−int d t=

⎧
⎪
⎪
⎨
⎪
⎪
⎩

(−1)ⁿ

2π i

n≠ 0

n=0

(check!),

so f(t)∼ ∑_−∞^∞(−1)ⁿ (i/n) e^int. By a direct integration:

⎪⎪
⎪⎪

₂²=

∫

−π

t² d t=

2π³

On the other hand by the previous Corollary:

⎪⎪
⎪⎪

₂²=2π

∑

n≠ 0

⎪
⎪
⎪
⎪

(−1)ⁿi

⎪
⎪
⎪
⎪

²=4π

∞

∑

n=1

n²

Thus we get a beautiful formula

∞

∑

n²

π²

Here is another important result.

Theorem 17 (Weierstrass Approximation Theorem) For any function f∈C[a,b] and any є>0 there exists a polynomial p such that ||f−p||_∞<є.

Proof. Change variable: t=2π(x−a+b/2)/(b−a) this maps x∈[a,b] onto t∈[−π,π]. Let P denote the subspace of polynomials in C[−π,π]. Then e^int∈$P_^$ for any n∈ℤ since Taylor series converges uniformly in [−π,π]. Consequently P contains the closed linear span in (supremum norm) of e^int, any n∈ℤ, which is CP[−π,π] by the Fejér theorem. Thus $P_^$⊇ CP[−π,π] and we extend that to non-periodic function as follows (why we could not make use of Lemma 1 here, by the way?).

For any f∈C[−π,π] let λ=(f(π)−f(−π))/(2π) then f₁(t)=f(t)−λ t∈ CP[−π,π] and could be approximated by a polynomial p₁(t) from the above discussion. Then f(t) is approximated by the polynomial p(t)=p₁(t)+λ t. □

It is easy to see, that the rôle of exponents e^int in the above prove is rather modest: they can be replaced by any functions which has a Taylor expansion. The real glory of the Fourier analysis is demonstrated in the two following examples.

Figure 11: The dynamics of a heat equation:

x—coordinate on the rod,

t—time,

T—temperature.

Example 18 The modern history of the Fourier analysis starts from the works of Fourier on the heat equation. As was mentioned in the introduction to this part, the exceptional role of Fourier coefficients for differential equations is explained by the simple formula ∂_x e^inx= ine^inx. We shortly review a solution of the heat equation to illustrate this.

Let we have a rod of the length 2π. The temperature at its point x∈[−π,π] and a moment t∈[0,∞) is described by a function u(t,x) on [0,∞)×[−π,π]. The mathematical equation describing a dynamics of the temperature distribution is:

∂ u(t,x)

∂ t

∂² u(t,x)

∂ x²

or, equivalently,

⎛
⎝

∂_t−∂_x²

⎞
⎠

u(t,x)=0. (32)

For any fixed moment t₀ the function u(t₀,x) depends only from x∈[−π,π] and according to Corollary 10 could be represented by its Fourier series:

u(t₀,x)=

∞

∑

n=−∞

⟨ u,e_n ⟩e_n=

∞

∑

n=−∞

c_n(t₀)e^inx,

where

c_n(t₀)=

⟨ u,e_n ⟩

√

2π

√

2π

∫

−π

u(t₀,x)e^−inx d x,

with Fourier coefficients c_n(t₀) depending from t₀. We substitute that decomposition into the heat equation (32) to receive:

⎛
⎝

∂_t−∂_x²

⎞
⎠

u(t,x)

⎛
⎝

∂_t−∂_x²

⎞
⎠

∞

∑

n=−∞

c_n(t)e^inx

∞

∑

n=−∞

⎛
⎝

∂_t−∂_x²

⎞
⎠

c_n(t)e^inx

∞

∑

n=−∞

(c′_n(t)+n²c_n(t))e^inx=0 .

(33)

Since function e^inx form a basis the last equation (33) holds if and only if

c′_n(t)+n²c_n(t)=0 for all n and t. (34)

Equations from the system (34) have general solutions of the form:

c_n(t)=c_n(0)e^−n²t for all t∈[0,∞), (35)

producing a general solution of the heat equation (32) in the form:

u(t,x)=

∞

∑

n=−∞

c_n(0)e^−n²te^inx =

∞

∑

n=−∞

c_n(0)e^−n²t+inx, (36)

where constant c_n(0) could be defined from boundary condition. For example, if it is known that the initial distribution of temperature was u(0,x)=g(x) for a function g(x)∈L₂[−π,π] then c_n(0) is the n-th Fourier coefficient of g(x).

The general solution (36) helps produce both the analytical study of the heat equation (32) and numerical simulation. For example, from (36) obviously follows that

the temperature is rapidly relaxing toward the thermal equilibrium with the temperature given by c₀(0), however never reach it within a finite time;
the “higher frequencies” (bigger thermal gradients) have a bigger speed of relaxation; etc.

The example of numerical simulation for the initial value problem with g(x)=2cos(2*u) + 1.5sin(u). It is clearly illustrate our above conclusions.

Figure 12: Two oscillation with unharmonious frequencies and the appearing dissonance. Click to listen the blue and green pure harmonics and red dissonance.

Figure 13: Graphics of G5 performed on different musical instruments (click on picture to hear the sound). Samples are taken from Sound Library.

Figure 14: Fourier series for G5 performed on different musical instruments (same order and colour as on the previous Figure)

(a) (b)
(c)
Figure 15: Limits of the Fourier analysis: different frequencies separated in time

Example 19 Among the oldest periodic functions in human culture are acoustic waves of musical tones. The mathematical theory of musics (including rudiments of the Fourier analysis!) is as old as mathematics itself and was highly respected already in Pythagoras’ school more 2500 years ago.

The earliest observations are that

The musical sounds are made of pure harmonics (see the blue and green graphs on the Figure 12), in our language cos and sin functions form a basis;
Not every two pure harmonics are compatible, to be their frequencies should make a simple ratio. Otherwise the dissonance (red graph on Figure 12) appears.

The musical tone, say G5, performed on different instruments clearly has something in common and different, see Figure 13 for comparisons. The decomposition into the pure harmonics, i.e. finding Fourier coefficient for the signal, could provide the complete characterisation, see Figure 14.

The Fourier analysis tells that:

All sound have the same base (i.e. the lowest) frequencies which corresponds to the G5 tone, i.e. 788 Gz.
The higher frequencies, which are necessarily are multiples of 788 Gz to avoid dissonance, appears with different weights for different instruments.

The Fourier analysis is very useful in the signal processing and is indeed the fundamental tool. However it is not universal and has very serious limitations. Consider the simple case of the signals plotted on the Figure 15(a) and (b). They are both made out of same two pure harmonics:

On the first signal the two harmonics (drawn in blue and green) follow one after another in time on Figure 15(a);
They just blended in equal proportions over the whole interval on Figure 15(b).

This appear to be two very different signals. However the Fourier performed over the whole interval does not seems to be very different, see Figure 15(c). Both transforms (drawn in blue-green and pink) have two major pikes corresponding to the pure frequencies. It is not very easy to extract differences between signals from their Fourier transform (yet this should be possible according to our study).

Even a better picture could be obtained if we use windowed Fourier transform, namely use a sliding “window” of the constant width instead of the entire interval for the Fourier transform. Yet even better analysis could be obtained by means of wavelets already mentioned in Remark 14 in connection with Plancherel’s formula. Roughly, wavelets correspond to a sliding window of a variable size—narrow for high frequencies and wide for low.


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Last modified: November 6, 2024.