The Riemann Lebesgue lemma is the basic result which makes possible the study of pointwise
convergence of Fourier series. It is also a major result in other contexts and serves as a useful
example.
For the purpose of simple notation, let R
((a,b])
denote those functions f which are in
R
([a + δ,b])
for every δ > 0 and the improper integral
∫ b
lim f (x)dx exists.
δ→0 a+δ
Lemma 11.3.1Let f ∈ R
((a,b])
,f
(x)
≥ 0, where
(a,b)
is some finite interval and letε > 0. Then there exists an interval of finite length,
[a1,b1]
⊆
(a,b)
and a differentiablefunction h having continuous derivative such that both h and h^{′}equal 0 outside of
[a1,b1]
which has the property that 0 ≤ h
(x)
≤ f
(x)
and
∫ b
a |f − h|dx < ε (11.6)
(11.6)
Proof: First here is a claim.
Claim:There exists a continuous g which vanishes near a and b such that g ≤ f
and
∫ b
|f (x) − g(x)|dx < ε∕3.
a
where the integral is the improper Riemann integral defined by
∫ b ∫ b
(f (x)− g(x))dx ≡ lim (f (x)− g(x))dx
a δ→0+ a+δ
Proof of the claim: First let a_{0}> a such that
||∫ b ∫ b || ∫ a0
|| f (x)dx− f (x) dx||= |f (x )|dx < ε∕3.
|a a0 | a
Let
{x0,x1,⋅⋅⋅,xn}
be a partition of
[a0,b]
and let
∑n
ak (xk − xk−1)
k=1
be a lower sum such that
||∫ b ∑n ||
|| f (x)dx− ak (xk − xk−1)|| < ε∕6.
| a0 k=1 |
The sum in the above equals
∫ b∑n
akX [xk−1,xk)(x) dx
a0k=1
where
{
X (x) = 1 if x ∈ [xk−1,xk)
[xk−1,xk) 0 if x ∕∈ [xk−1,xk)
Now let ψ_{k} be a continuous function which approximates X_{[xk−1,xk)}
(x)
and vanishes near the
endpoints of [x_{k−1},x_{k}) as shown in the following picture in which δ is sufficiently
small.
provided h is small enough, due to the uniform continuity of g. (Why is g uniformly
continuous?) Also, since h satisfies 11.7, g_{h} and g_{h}^{′} vanish outside some closed interval,
[a1,b1]
⊆
(a0,b)
. Since g_{h} equals zero between a and a_{0}, this shows that for such
h,
| |
∫ b ||∫ b ∫ b ||
a |f (x)− gh(x)|dx = || a f (x)dx − a gh (x)dx||
|∫ ∫ | |∫ ∫ |
|| b b || || b b ||
≤ ||a f (x )dx− a0 f (x)dx|| +|| a0 f (x)dx − a0 gh(x)dx||
||∫ b ∫ b || ||∫ b ∫ b ||
≤ ε+ || f (x)dx − g(x)dx||+ || g(x)dx− gh (x)dx ||
3 | a0 a0 | | a0 a0 |
< ε∕3 + ε∕3+ ε∕3 = ε.
Letting h = g_{h} this proves the Lemma.
The lemma can be generalized to the case where f has values in ℂ. In this case,
∫ b ∫ b ∫ b
f (x)dx ≡ Re f (x)dx +i Im f (x)dx
a a a
and f ∈ R
((a,b])
means Ref,Imf ∈ R
((a,b])
.
Lemma 11.3.2Let
|f|
∈ R
((a,b])
, where
(a,b)
is some finite interval and let ε > 0.Then there exists an interval of finite length,
[a1,b1]
⊆
(a,b)
and a differentiable function hhaving continuous derivative such that both h and h^{′}equal 0 outside of
[a1,b1]
which has theproperty that
∫ b
|f − h|dx < ε (11.8)
a
(11.8)
Proof: For g a real valued bounded function, define