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Real Analysis Qualifying Exam Problem
I think this should be an easy question, and I believe the answer should be in the positive, but I am not sure how to start. I would appreciate some help. Thank you.
Suppose that $f_j$ is a ...
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Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.
Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$
is the Fourier transform of a function in $L^1(R^d,m)$.
[Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
