MATH 321: Real Variables II
Lecture #28
University of British Columbia
Lecture #28:
Instructor:
Scribe:
March 17, 2008
Dr. Joel Feldman
Peter Wong
Theorem. (Parseval’s Theorem – Convergence of Fourier Series in the mean or L2 ) If f, g : R → C are
• 2π-periodic.
• Riemann integrable on [−π, π].
with
cn =
1
2π
Z
π
f (x)e−inx dx,
dn =
1
2π
−π
Z
π
g(x)e−inx dx
−π
and
SN (f, x) =
n
X
cm eimx ,
m=−n
then
qR
π
2
(a) limN →∞ kf (x) − SN (f, x)k2 = 0 where khk2 =
−π |h(x)| dx,
Rπ
P∞
1
1
2
2
kf k22 = 2π
(b) 2π
n=−∞ |cn | ,
−π |f (x)| dx =
Rπ
P
1
(c) 2π
(f, g) = ∞
n=−∞ cn dn where (f, g) = −π f (x)g(x) dx.
Step 1: There is a continuous functions h : R → C that is 2π-periodic such that kf − hk2 < 2ε .
(This was done in Problem Set 7 #3)
PM
eimx such that kh − P k2 < 2ε . In class, we
Step 2: There is a trigonometric polynomial P (x) =
n=−M pm
√
showed that there is a P (x) such that |h(x) − P (x)| < ε/ 2π for all x ∈ R
Z π
=⇒ kh − P k22 =
|h(x) − P (x)|2 dx < ε2 .
{z
}
|
−π
Proof.
ε2 /2π
Step 3: Proof of (a).
By the “best approximation theorem,” for any N ≥ M ,
kf − SN k2 ≤ kf − P k2 ≤ kf − hk2 + kh − P k2 <
ε
2
+
ε
2
+ ε.
Step 4: Proof of (b).
Last class we showed the Bessel’s inequality
2
1
2π kf k2
≥
∞
X
n=−∞
|cn |2
with equality if and only if SN convergences to f in the mean.
Step 5: Proof of (c).
(f, g) =
1
4
(Polarization Linearity
Problem Set 9 #2)
X
|cn + dn |2 − |cn − dn |2 + i|cn + idn |2 − i|cn − idn |2
kf + gk2 − kf − gk2 + ikf + igk2 − ikf − igk2
=
2π
4
=
2π
4
X
(cn + dn )(cn + dn ) − (cn − dn )(cn − dn ) + i(cn + dn )(cn − idn ) − i(cn − dn )(cn + idn )
=
2π
4
X
X
cn d n
4cn dn = 2π
n
n
n
n
2
MATH 321: Lecture #28
Lemma. (A formula for SN for any f ∈ R on [−π, π])
SN (f, x) =
N
X
cn einx =
n=−N
N
X
1
2π
Z
π
f (t)e−int einx dt =
−π
n=−N
1
2π
Z
π
N
X
f (t)ein(x−t) dt =
−π n=−N
1
2π
Z
π
−π
f (t)DN (x − t) dt
where if y 6= 0,
DN (y) =
N
X
einy =
n=−N
=
e−iN y − ei(N +1)y
1 − eiy
(since a + ar + · · · + arm =
a(1−r m+1 )
)
1−r
−2i sin(N + 21 )y
(e−iN y − ei(N +1)y )e−iy/2
sin(N + 1/2)y
=
=
y
iy
−iy/2
−2i sin( 2 )
sin(y/2)
(1 − e )e
MATH 321: Real Variables II
Lecture #29
University of British Columbia
Lecture #29:
Instructor:
Scribe:
March 19, 2008
Dr. Joel Feldman
Peter Wong
Last class, we saw
Lemma. For f : R → C being 2π-periodic and Riemann integrable on [−π, π], where
Z π
1
cn = 2π
f (x)e−inx dx,
−π
SN =
N
X
cn einx ,
n=−N
Dn =
n
X
e−int
m=−n
we have
Sn (x) =
1
2π
Z
π
−π
Z
=
1
− 2π
=
Z
1
2π
f (t)Dn (x − t) dt =
Z
x−π
f (x − τ )Dn (τ ) (−dτ ),
x+π
Z
x−π
x+π
π
−π
1
2π

2n + 1, if t = 2kπ, k ∈ Z
= sin[(n + 21 )t]

, if t 6= 2kπ for any k ∈ Z
sin(t/2)
1
2π
f (x − τ )Dn (τ ) dτ =
(change of variable: τ = x − t)
x+π
x−π
f (x − τ )Dn (τ ) dτ,
f (x − t)Dn (t) dt
(f (x − τ )Dn (τ ) is 2π-periodic)
(change of variable: t = τ )
Properties of Dn (t)
1. 2π-periodic
2. As a function to t, it is even
3. Continuous
Rπ
4. −π Dn (t) dt = 2π since
Z
π
Dn (t) dt =
−π
Also,
Z
n
X
m=−n
Z
(
π
X
0,
m 6= 0
eimt dt =
2π,
m
=0
−π
−π
π
0
−π
Dn (t) dt =
Z
π
Dn (t) dt = π
0
Theorem. Let f : R → C be 2π-periodic and Riemann integrable on [−π, π]. Also, let x ∈ R, δ > 0 and M < ∞.
(a) If |f (x + t) − f (x)| ≤ M |t| for all |t| < δ, then Sn (x) → f (x) as n → ∞.
(b) If f (x+ ) and f (x− ) exist, and that
|f (x + t) − f (x+ ) ≤ M |t|
|f (x + t) − f (x− ) ≤ M |t|
then limn→∞ Sn = 21 [f (x+ ) + f (x− )]
for all 0 < x < δ
for all − δ < x < 0
f (x− )
f (x− )
2
MATH 321: Lecture #29
Proof. Part (a) is a special case of (b) with f (x+ ) = f (x− ) = f (x)
Sn (x) − 21 f (x+ ) − 12 f (x− )
Z π
Z 0
Z π
+
1
1
1
= 2π
f (x − t)Dn (t) dt − 2π
f (x )Dn (t) dt − 2π
f (x− )Dn (t) dt
−π
−π
0
Z π
1
g(t) sin[(n + 21 )t] dt (after setting integrands to 0 at t = 0)
= 2π
−π
where
g(t) =
So
Sn (x) − 21 f (x+ ) − 12 f (x− ) =
1
2π
( f (x−t)−f (x+ )
,
sin(t/2)
f (x−t)−f (x− )
,
sin(t/2)
Z
π
−π
if − π ≤ t < 0,
if 0 ≤ t ≤ π,
1
1 i( 2 )t
g(t) 2i
e
eint dt −
1
2π
Z
π
−π
1
1 −i( 2 )t
g(t) 2i
e
e−int dt
since
Lemma. g(t) is Riemann integrable.
Proof. Next lecture.
At the end of the class, we have decided to study manifolds as the final (supplementary) topic in the course.
Course notes on manifolds and tempered distributions will be posted on the web shortly. Also, we have no lectures
on this Friday and next Monday. Enjoy the long weekend!