- #1
Yaelcita
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Homework Statement
Prove that [tex]f(x)=\sin(\pi x)/(\pi x)[/tex] is in [tex]L^2(R)[/tex] but not in [tex]L^1(R)[/tex]
This is in a chapter of the book dealing with Inverse Fourier Transform
f is in [tex]L^1[/tex] if [tex]\int|f|<\infty[/tex]
f is in [tex]L^2[/tex] if [tex]\sqrt{\int|f|^2}<\infty[/tex]
Homework Equations
I just have no idea how to do it
The Attempt at a Solution
Because the question was in a chapter on Fourier Transform, and many of the theorems there only work for L1 functions, I thought maybe I could prove the conclusion doesn't hold for [tex]\sin(\pi x) /(\pi x)[/tex], so it's not in L1. But so far I haven't been very successful, because every integral I come across is extremely complicated and I have no idea how to solve it. Also, I have nothing to prove it is in L2.
So now I've turned to trying to prove it directly, from the definition of L1 and L2, but I'm still getting nowhere. I just don't know how to solve the integrals.