→ doubleN :3. (f*η_ε) → f 10/01 21:08
→ doubleN :4.變數變換 10/01 21:08
→ doubleN :5.請參考泛函書籍 10/01 21:09
1.
Suppose that |f|^p is Riemann (improper) integrable over the domain Ω
p
(but not continuous) , does f belong to L (Ω)？
To be more precise , for such an f , is there a sequence
f belonging to C(Ω) so that
k
p
lim ∫ |f (x) - f(x)| dx = 0 ？
k→∞ Ω k
2. Is there any function which is not Riemann (improper) integrable but is
a ||˙|| limit of continuous function ？
p
n p
3. Suppose Ω is contained in R , and f belongs to L (Ω) . Is there a
∞ p
sequence of smooth functions {f_k} so that f → f in L (Ω)
k=1 k
as k→∞ ？
n
4. For Ω contained in R open , for ε>0 sufficiently small , define the
open subset Ω of Ω by
ε
Ω = { x belongs to Ω | dist(x , boundry of Ω) > ε} .
ε
∞ n
Define η belonging to C (R ) by
{ C(e^(1/(|x|^2 - 1))) if |x| < 1
η(x) = {
{ 0 if |x| ≧ 1
with constant C > 0 chosen such that ∫ η(x) dx = 1 .
R^n
For ε> 0 , the standard sequence of mollifiers is defined by
η = (ε^(-n))(η(x/ε)) .
ε
Prove that ∫ η (x) dx = 1
R^n ε
_______
and supp(η ) is contained in B(0,ε)
ε
5. Prove that all norms on R^n are equivalent .
以上這5題實變
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Q.Q
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