Due March 9
Problem 1
Let f be integrable, and for each α>0 let Eα={x:|f(x)|>α}. Then
∫|f|=∫∞0m(Eα)dα.
Problem 2 (Riemann-Lebesgue Lemma)
For f∈L1(\Rd), let
ˆf(ξ)=∫\Rdf(x)e−2πix⋅ξdx
be its Fourier transform. Then ˆf(ξ)→0 as |ξ|→0.
Problem 3
Let f,g∈L1(\Rd).
- (x,y)↦f(x−y)g(y)∈L1(\Rd×\Rd).
- The convolution f∗g(x)=∫\Rdf(x−y)g(y)dy is well defined for a.e. x.
- ||f∗g||L1≤||f||L1||g||L1.
- ^f∗g(ξ)=ˆf(ξ)ˆg(ξ).
Problem 4
There does not exist I∈L1(\Rd) such that, for all f∈L1(\Rd),
f∗I=f.
Problem 5
Let fn→f in measure.
- If fn≥0, then ∫f≤lim inf
- If there exists g\in L^1 such that |f_n|\le g for all n, then f_n\to f in L^1 and \displaystyle \int f_n \to \int f.
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