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 sequence of functions in $L^1(\mathbb{R})$ satisfying

(i) $\|f_j\|_\infty \leq 5$ and $f_j(x) = 0$ for $|x| \geq 10, j \in \mathbb{N}.$

(ii) the $f_j$ converge pointwise to $f \in L^1(\mathbb{R})$.

Does it follow that the $\hat{f_j}$ converge pointwise to $\hat{f}$?

  • 7
    $\begingroup$ Try applying the dominated convergence theorem with the uniform bound $|f_j(x)| \leq g(x)$ for $g(x) = 5 I_{(-10,10)}(x)$. Here $I_{(-10,10)}$ is the indicator function of the set $(-10,10)$. Also, I have a serious case of deja vu right now. $\endgroup$ – A Blumenthal Mar 7 '13 at 6:41
  • 2
    $\begingroup$ You went from asking "What is the derivative of $f(x) = 5 \cos (1.2 x)$?" to asking this in 1 hour!?!?!? Well done! $\endgroup$ – Quinn Culver Mar 7 '13 at 22:18
  • $\begingroup$ 'You' can be plural. $\endgroup$ – zyx Jul 19 '13 at 19:51
  1. It's enough to show that $\lVert f_n-f\rVert_{L^1}\to 0$.

  2. In order to do that, we can use Egoroff's theorem: fix $\varepsilon>0$, and $S_\varepsilon\subset [0,1]$ such that $[-10,10]\setminus S$ has a measure $<\varepsilon$ and there is uniform convergence of $f_n$ to $f$ on $S$. Since all the $f_n$ and $f$ are supported by $[-10,10]$, we get
    $$\int_{\Bbb R}|f_n-f|d\lambda=\int_{[-10,10]}|f_n-f|d\lambda\leqslant 20\sup_{x\in S}|f_n(x)-f(x)|+10\varepsilon.$$


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