Haar normalisations on the circle and the integers

(The dual identifications.) A character $\chi :\mathbb{Z}\to \mathbb{T}$ is determined by $z:=\chi (1)$, and each $z\in \mathbb{T}$ gives ${\gamma }_z(k)=z^k$. This is a group isomorphism $\mathbb{T}\to \widehat{\mathbb{Z}}$. Its inverse is evaluation at $1$, while each evaluation $z\mapsto z^k$ is continuous; [F2] therefore makes this a homeomorphism. For a character $\chi :\mathbb{T}\to \mathbb{T}$, lift $t\mapsto \chi (e^{2\pi it})$ to $\mathbb{R}$ and apply [F1]; periodicity forces the resulting frequency to be an integer. Thus every character is $z\mapsto z^n$ for a unique $n\in \mathbb{Z}$. Since the compact-open topology on this dual is uniform on $\mathbb{T}$, and ${\sup }_{z\in \mathbb{T}}\mathrm{\mid }{z}^n-z^m\mathrm{\mid }=2$ for $n\mathrm{\ne }m$, $\widehat{\mathbb{T}}$ is discrete.

(Inversion on $\mathbb{T}$.) Let $f\in L^1(\mathbb{T})$ with $a_n:=\widehat{f}(n)$ satisfying ${\sum }_n\mathrm{\mid }{a}_n\mathrm{\mid }<\mathrm{\infty }$. The series $g(z):={\sum }_{n\in \mathbb{Z}}{a}_n{z}^n$ converges uniformly; [F3] shows its Fourier coefficients are $a_n$. Its Fejer means are ${\sigma }_{N}f(z)={\sum }_{\mathrm{\mid }n\mathrm{\mid }\le N}(1-\frac{\mathrm{\mid }n\mathrm{\mid }}{N+1})a_n{z}^n\mathrm{.}$ Absolute summability implies these weighted sums converge uniformly to $g$: first bound the tail by ${\sum }_{\mathrm{\mid }n\mathrm{\mid }>M}\mathrm{\mid }{a}_n\mathrm{\mid }$, then let $N\to \mathrm{\infty }$ on the finite central sum. By [F4], ${\sigma }_{N}f\to f$ in $L^1$; uniform convergence also gives ${\sigma }_{N}f\to g$ in $L^1$. Uniqueness of limits yields $f=g$ almost everywhere. Thus counting measure on $\mathbb{Z}$ gives the displayed dual inversion formula.

The measures in [F3] are Haar measures on the two groups. Under the identifications of step 1.1, the Fourier transforms are $\widehat{f}(z)={\sum }_{k\in \mathbb{Z}}f(k)z^{-k}$ for $f\in {\mathrm{\ell }}^1(\mathbb{Z})$ and $\widehat{f}(n)={\int }_{\mathbb{T}}f(z)z^{-n}d{m}_{\mathbb{T}}(z)$ for $f\in L^1(\mathbb{T})$.

(Inversion on $\mathbb{Z}$.) For $f\in {\mathrm{\ell }}^1(\mathbb{Z})$ the series for $\widehat{f}$ converges absolutely and uniformly, hence is integrable. Termwise integration and [F3] give, for each $j\in \mathbb{Z}$, ${\int }_{\mathbb{T}}\widehat{f}(z)z^{j}d{m}_{\mathbb{T}}(z)={\sum }_{k\in \mathbb{Z}}f(k){\int }_{\mathbb{T}}{z}^{j-k}d{m}_{\mathbb{T}}(z)=f(j)\mathrm{.}$ Thus normalized arc measure gives the displayed inversion formula for counting measure on $\mathbb{Z}$.

Steps 1.1 and 2.1 identify the dual groups and Haar measures, step 3.1 proves inversion for $\mathbb{Z}$, and step 1.2 proves Fourier-series inversion on $\mathbb{T}$ for summable Fourier coefficients. These explicit pairs give the usual normalization conventions.