Lema Riemann-Lebesgue mai este numită şi Teorema lui Mercer sau Lema lui Riemann .
Teorema 1 . (Lema lui Riemann )
Dacă
f
:
[
0
,
2
π
]
→
R
{\displaystyle f: [0, 2 \pi] \rightarrow \mathbb R \!}
funcție integrabilă Riemann , atunci șirurile :
(
∫
0
2
π
f
(
x
)
sin
n
x
d
x
)
n
∈
N
∗
{\displaystyle \bigg ( \int_0^{2 \pi} f(x) \sin nx \; dx \bigg )_{n \in \mathbb N^*} \!}
(
∫
0
2
π
f
(
x
)
cos
n
x
d
x
)
n
∈
N
∗
{\displaystyle \bigg ( \int_0^{2 \pi} f(x) \cos nx \; dx \bigg )_{n \in \mathbb N^*} \!}
sunt convergente şi avem:
lim
n
→
∞
∫
0
2
π
f
(
x
)
sin
n
x
d
x
=
0
=
lim
n
→
∞
∫
0
2
π
f
(
x
)
cos
n
x
d
x
.
{\displaystyle \lim_{n \to \infty} \int_0^{2 \pi} f(x) \sin nx \; dx = 0 =\lim_{n \to \infty} \int_0^{2 \pi} f(x) \cos nx \; dx. \!}
Generalizare [ ] Vom face referire la câteva rezultate:
Teorema 2 .
Dacă
f
:
[
0
,
T
]
→
R
{\displaystyle f: [0, T] \rightarrow \mathbb R \!}
funcție continuă şi
g
:
[
0
,
∞
)
→
[
0
,
∞
)
{\displaystyle g: [0, \infty) \rightarrow [0, \infty) \!}
periodică de perioadă T , atunci:
lim
n
→
∞
∫
0
T
f
(
x
)
g
(
n
x
)
d
x
=
1
T
(
∫
0
T
f
(
x
)
d
x
)
(
∫
0
T
g
(
x
)
d
x
)
.
{\displaystyle \lim_{n \to \infty} \int_0^T f(x) g(nx) dx = \frac 1 T \bigg ( \int_0^T f(x) dx \bigg ) \bigg ( \int_0^T g(x) dx \bigg ). \!}
Teorema 3 .
Dacă
f
:
[
a
,
b
]
→
R
{\displaystyle f: [a, b] \rightarrow \mathbb R \!}
integrabilă Riemann şi
φ
:
[
c
,
d
]
→
[
a
,
b
]
{\displaystyle \varphi : [c, d] \rightarrow [a, b] \!}
derivabilă şi cu derivata integrabilă Riemann, atunci funcţia
(
f
∘
φ
)
φ
′
{\displaystyle (f \circ \varphi) \varphi' \!}
schimbării de variabilă ":
∫
a
b
f
(
x
)
d
x
=
∫
φ
−
1
(
a
)
φ
−
1
(
b
)
f
(
φ
(
t
)
)
φ
′
(
t
)
d
t
.
{\displaystyle \int_a^b f(x)dx= \int_{\varphi^{-1}(a)}^{\varphi^{-1}(b)} f(\varphi (t)) \varphi'(t) dt. \!}
Corolar .
Fie
α
∈
R
∗
{\displaystyle \alpha \in \mathbb R^* \!}
φ
:
[
c
,
d
]
→
[
a
,
b
]
,
φ
(
x
)
=
α
x
+
β
{\displaystyle \varphi : [c, d] \rightarrow [a, b], \; \varphi(x) = \alpha x + \beta \!}
f
:
[
a
,
b
]
→
{\displaystyle f: [a, b] \rightarrow \mathbb \!}
f
∘
φ
{\displaystyle f \circ \varphi \!}
∫
a
b
f
(
x
)
d
x
=
α
∫
φ
−
1
(
a
)
φ
−
1
(
b
)
f
(
φ
(
t
)
)
d
t
.
{\displaystyle \int_a^b f(x) dx = \alpha \int_{\varphi^{-1}(a)}^{\varphi^{-1}(b)} f(\varphi (t)) dt. \!}
Lema lui Riemann generalizată [ ] Teorema 4. (lema lui Riemann generalizată ).
Dacă
f
:
[
0
,
T
]
→
R
{\displaystyle f: [0, T] \rightarrow \mathbb R \!}
integrabilă Riemann şi
g
:
[
0
,
∞
)
→
R
{\displaystyle g: [0, \infty) \rightarrow \mathbb R \!}
T , astfel încât restricţia
g
|
[
0
,
T
]
{\displaystyle g|_{[0, T]} \!}
lim
n
→
∞
∫
0
T
f
(
x
)
g
(
n
x
)
d
x
=
1
T
(
∫
0
T
f
(
x
)
d
x
)
(
∫
0
T
g
(
x
)
d
x
)
.
{\displaystyle \lim_{n \to \infty} \int_0^T f(x) g(nx) dx = \frac 1 T \bigg ( \int_0^T f(x) dx \bigg ) \bigg ( \int_0^T g(x) dx \bigg ). \!}
(1) Demonstraţie .
Conform corolarului, funcţia dată de
g
→
g
(
n
x
)
,
x
∈
[
0
,
T
]
{\displaystyle g \rightarrow g(nx), \; x \in [0, T] \!}
(
∫
0
T
f
(
x
)
g
(
n
x
)
d
x
)
n
∈
N
∗
{\displaystyle \bigg ( \int_0^T f(x) g(nx)dx \bigg )_{n \in \mathbb N^*} \!}
Resurse [ ] ![{\displaystyle f:[0,2\pi ]\rightarrow \mathbb {R} \!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/1d43087e03f55d7daeb4c16762048b99447600b0)
![{\displaystyle f:[0,T]\rightarrow \mathbb {R} \!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/2c18baba4126cbe2128ac06cb620dd78ac9cf719)
![{\displaystyle f:[a,b]\rightarrow \mathbb {R} \!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/8c2e7eb0ac87050ed7850fe6a7fd1bb2e8256c20)
![{\displaystyle \varphi :[c,d]\rightarrow [a,b]\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/90f9c0e53acd720295ceebca96d7bb38003c7ff4)
![{\displaystyle \varphi :[c,d]\rightarrow [a,b],\;\varphi (x)=\alpha x+\beta \!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/46152cc842128954a936d5e7de20ea547a0c4015)
![{\displaystyle f:[a,b]\rightarrow \mathbb {\!} }](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/399371550319e56c626ff9dfe5ae3b40e945619a)
![{\displaystyle g|_{[0,T]}\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/85aba1f4bd381d1390104468024b07a1ce688973)
![{\displaystyle g\rightarrow g(nx),\;x\in [0,T]\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/3a588384ca6b5f0af9a279235ccfaaf8103c9825)
