Exercise 48, § 1

(a) Let $p\in [1,+\infty [$ be given. A subset X of $L^p(\mathbf{R})$ is relatively compact if and only if it is bounded and satisfies the conditions

lim$_{y\rightarrow 0}$ sup$_{f\in X}\|\boldsymbol{\gamma }(y)f-f\|_p= 0$

lim$_{R\rightarrow+\infty}$ sup$_{f\in X}\|(1-\varphi_R)f\|_p= 0$,

where $\varphi_R$ is the characteristic function of the interval $[-R,R]$ (cf. INT, VIII, p. 205, exercise 26).

(b) Let X be a bounded subset of $L^2(\mathbf{R})$. Prove that X is precompact if and only if

lim$_{R\rightarrow+\infty}$ sup$_{f\in X}\int_{\mathbf{R}}(1-\varphi_R)(|f|^2+|\widehat{f}|^2)dx= 0$.

(c) Let $(f_n)$ be a sequence in $L^2(\mathbf{R})$ orthonormal in $L^2(\mathbf{R})$ such that sup$_n|f_n|$ belongs to $L^2(\mathbf{R})$. Then sup$_n|\widehat{f}_n|$ does not belong to $L^2(\mathbf{R})$.