Convergence of Fourier Series
Basic Results From Orthogonality
Outline
The following three results are valid in any finitedimensional inner product space, and the infinitedimensional spaces `2 (Z), and R.
I. Key Concepts
• The Pythagorean Theorem: If X and Y are
orthogonal, then
II. Summary
III. Basic Results From Orthogonality
kX + Y k2 = kXk2 + kY k2
IV. Special Infinite-Dimensional Vector Spaces
• The Cauchy-Schwartz Inequality:
V. Approximation Theorems
|hX, Y i| = kXkkY k
VI. Mean-Square Convergence
• The Triangle Inequality
VII. Proof of Mean-Square Convergence
VIII. Other Results
kX + Y k ≤ kXk + kY k
Key Concepts
Special Infinite Dimensional Vector
Spaces
• Orthogonality
• The space of all (double-sided) sequences
• Completeness
(. . . , a−n , . . . , a−1 , a0 , a1 , . . . , an , . . . )
• Hilbert Space
such that
Summary
∞
X
|an |2 < ∞
n=−∞
The mean-square of a Riemann integrable function
defined on [a, b] is the integral
Z b
1
|f (t)|2 dt
b−a a
– Addition & Scalar Multiplication:
{an } + c{bn } = {an + cbn }
– Inner Product & Norm:
This short chapter is devoted to answering the
question of whether the Fourier series of an integrable function converges in the mean-square sense:
Z π
1
|f (t) − SN (f )(t)|2 dt → 0
2π −π
han , bn i =
∞
X
an b n
n=−∞
kan k =
∞
X
n=−∞
– Denoted by `2 (Z)
as N → ∞.
1
!1/2
2
|an |
Mean-Square Convergence
– A Hilbert space
• The space of all complex-valued Riemann integrable functions on the circle
Let f be a Riemann integrable
function on
P∞
the circle with f (θ) ∼ n=−∞ an e−inθ . Then
we have:
– Addition & Scalar Multiplication:
(f + cg)(x) = f (x) + cg(x)
(a) Mean-square convergence of the
Fourier series:
Z π
1
|f (t) − SN (f )(t)|2 dt → 0
2π −π
– Inner Product & Norm:
Z 2π
1
f (θ)g(θ) dθ
hf, gi =
2π 0
Z 2π
1/2
1
2
kf k =
kf (θk
2π 0
as N → ∞.
(b) Parseval’s identity:
– Denoted by R
1
2π
– A pre-Hilbert space
Z
π
2
|f (θ)| dθ =
−π
∞
X
|an |2
n=−∞
Parseval’s identity is a consequence of the meansquare convergence of the Fourier series.
Approximation Theorems
• Best Approximation:
Proof of Mean-Square Convergence
For any sequence of complex numbers
{cn }|n|≤N , we have
Fix > 0. Since f is integrable on the circle, there
exists a continuous function g such that
kf − SN (f )k ≤ f −
X
cn einθ
sup |g(θ| ≤ sup |f (θ| = B
|n|≤N
θ∈[0,2π]
and
– Proved from the observation that for all n ∈ Z
[f − SN (f )] ⊥ en .
(Here en (x) = e
inx
θ∈[0,2π]
π
2 π
|f (θ) − g(θ)| dθ <
9B
−π
Z
This continuous function will be the key to the
proof. We want to prove that there exists N ∈ N
such that n ≥ N implies kf − SN k < . The answer is the degree of the trigonometric polynomial
P such that kg − P k < /3. Since g is continuous
)
2
on the circle, such a polynomial exists:
Z 2π
1
2
kf − gk =
|f (θ) − g(θ)|2 dθ
2π 0
Z
B 2π
|f (θ) − g(θ)| dθ
≤
π 0
2
<
9
So, if M is the degree of P , then N > M implies
eiN θ + P
kf − SN k ≤ f −
3
≤ kf − P k +
3
≤ kf − gk + kg − P k +
3
≤ + + =
3 3 3
Notice how I played with the epsilons! The best
approximation lemma tells us that the first equality holds if P and SN have the same degree, which
isn’t the case. So I had to add that extra (/3)eiN θ
to make the degrees match and be able to use the
lemma. But I couldn’t have done this without g
approximating f and P approximating g. It feels
amazing being able to approximate a function. I
feel like a god. This concludes the proof.
Other Results
• Riemann-Lebesgue Lemma
• Generalized Parseval’s Identity
There’s actually two more sections in the chapter.
It’s about pointwise convergence, and an example
of a continuous function with a diverging Fourier
series. I haven’t read them yet tho.
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