Exercise 9, § 3

The object of this exercise is to present another proof of Theorem 5 of V, p. 455. Let G be a locally compact commutative topological group, endowed with a Haar measure $\mu$. Let $\widehat{f}$ denote the Fourier transform of $f\in L^1$(G), and let $\mathscr{F}(G)\subset \mathscr{C}_0(\widehat{G})$ denote the image of the Fourier transform on $L^1(G)$.

a) The space $\mathscr{F}(G)$ is a dense subspace of $\mathscr{C}_0(G)$.

b) For every $f\in L^1$(G), the spectral radius of $f$ in the Banach algebra $L^1(G)$ is equal to $\|\widehat{f}\|_{\infty}$.

c) Let $\varphi \in$ Pos(G). It is a uniformly continuous and bounded function on G; for every $f\in L^1$(G), one has

d) Let $\lambda$ be the continuous linear form on $L^1(G)$ determined by $\varphi$, and put $b(f, g) =\lambda (f^**g)$ for every $(f, g)\in L^1(G)\times L^1(G)$. The mapping $b$ is a positive hermitian form on $L^1(G)$.

e) For every $f\in L^1$(G), one has $|\lambda (f)|^2\leqslant b(f, f)$.

f ) Let $f\in L^1(G)$ and $g=f^**f\in \mathscr{C}_b(G)$. Define $g_1=g$ and $g_{n+1}=g_n*g$ for every integer $n\geqslant 1$. For every integer $n\geqslant 1$, one has $|\lambda (f)|\leqslant \|g_n\|^{2^{-n}}_1$.

g) For every $f\in L^1$(G), one has $|\lambda (f)|\leqslant \|\widehat{f}\|_{\infty}$. There exists a continuous linear mapping $\widehat{\lambda}$ on $\mathscr{F}(G)$ such that $\lambda (f) =\widehat{\lambda}(\widehat{f})$ for every $f\in L^1(G)$. h) There exists a bounded measure $\nu$ on $\widehat{G}$ such that $\lambda (f) =\nu (\widehat{f})$ for every $f\in L^1(G)$; the measure $\nu$ is positive and one has

for every $x\in G$.