From b134e501265aff29545835e19fe8a814f99f8f41 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 27 Sep 2026 18:55:13 -0400 Subject: [PATCH] Added some harmonic analysis. --- .vscode/project.code-snippets | 7 +- src/harmonic/commutative/dual.tex | 106 +++++++++++++++++++++++++++++ src/harmonic/commutative/index.tex | 4 ++ src/harmonic/lcg/algebra.tex | 59 ++++++++++++++++ 4 files changed, 175 insertions(+), 1 deletion(-) create mode 100644 src/harmonic/commutative/dual.tex create mode 100644 src/harmonic/commutative/index.tex diff --git a/.vscode/project.code-snippets b/.vscode/project.code-snippets index ad0c0e0..1b3b624 100644 --- a/.vscode/project.code-snippets +++ b/.vscode/project.code-snippets @@ -181,5 +181,10 @@ "scope": "latex", "prefix": "scaf", "body": ["\\hyperref[scaffolded]{definition:measure-scaffold}$0"] - } + }, + "Locally Compact Abelian Group": { + "scope": "latex", + "prefix": "lcag", + "body": ["locally compact Abelian group$0"] + }, } diff --git a/src/harmonic/commutative/dual.tex b/src/harmonic/commutative/dual.tex new file mode 100644 index 0000000..7ef5094 --- /dev/null +++ b/src/harmonic/commutative/dual.tex @@ -0,0 +1,106 @@ +\section{Pontryagin Duality} +\label{section:lacg-dual} + + +\begin{definition}[Unitary Character] +\label{definition:unitary-character} + Let $G$ be a locally compact Abelian group and $\phi: G \to \mathbb{T}$, then $\phi$ is a \textbf{unitary character} of $G$ if $\phi$ is a continuous homomorphism. + + The set $\wh G$ of all unitary characters of $G$, equipped with pointwise operations and the topology of compact convergence, is the \textbf{dual group} of $G$. + + For any $\phi \in \wh G$ and $x \in G$, denote $\phi(x) = \dpn{x, \phi}{G}$. +\end{definition} + +\begin{theorem} +\label{theorem:unitary-character-isomorphism} + Let $G$ be a locally compact Abelian group and $\phi \in \wh G$. Under the identification that $\mathbb{T} = \partial B_\complex(0, 1)$ and $B(\complex) = \complex$, the representation of $L^1(G; \complex)$ determined by $\phi$ takes the form + \[ + \pi_\phi: L^1(G; \complex) \to \complex \quad \dpn{f, \pi_\phi}{L^1(G; \complex)} = \int_G f(x)\dpn{x, \phi}{G} dx + \] + + and the mapping $\wh G \to \Omega(L^1(G; \complex))$ defined by $\phi \mapsto \pi_\phi$ is a homeomorphism. Moreover, for any $f \in L^1(G; \complex)$ with $\pi_\phi(f) \ne 0$, $\dpn{x, \phi}{G} = \pi_\phi(L_xf)/\pi_\phi(f)$ for all $x \in G$. +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 4.3]{FollandHarmonic}}}. ] + (Continuous): Let $\seqf{f_j} \subset L^1(G; \complex)$ and $\eps > 0$. By inner regularity of the Haar measure, there exists $K \subset G$ compact such that $\int_{G\setminus K}|f_j| < \eps$ for all $1 \le j \le n$. In which case, for any $\phi, \psi \in \wh G$ with $\sup_{x \in K}|\phi(x) - \psi(x)| < \eps$, + \begin{align} + \abs{\dpn{f_j, \phi - \psi}{L^1(G; \complex)}} &\le \int_{K}|f_j \cdot (\phi - \psi)| + 2\int_{G \setminus K}|f_j| \\ + &\le \eps \norm{f_j}_{L^1(G; \complex)} + 2\eps + \end{align} + + (Injective): By (3) of \autoref{definition:representation-l1-group-algebra}, $\phi$ is uniquely determined by $\pi_\phi$. + + (Surjective): Let $\pi \in \Omega(L^1(G; \complex))$. Since $G$ is Abelian, $L^1(G; \complex)$ is also Abelian by \autoref{proposition:convolution-measures-property}. As such, for any $x \in G$ and $f, g \in L^1(G; \complex)$, $f * (L_xg) = L_x(f * g) = (L_xf) * g$ and $\pi(L_xf)\pi(g) = \pi(f)\pi(L_xg)$. In particular, if $\pi(f), \pi(g) \ne 0$, then $\pi(L_xf)/\pi(f) = \pi(L_xg)/\pi(g)$. Let\footnote{It may be tempting to construct $\phi$ directly as an element of $L^\infty$. However, this requires either reducing to the $\sigma$-compact case, or expanding the measurable sets of the Haar measure.} + \[ + \phi: G \to \complex \quad x \mapsto \frac{\pi(L_xf)}{\pi(f)} + \] + + then $\phi$ is defined independently of the choice of $f$. By \hyperref[continuity of translation]{proposition:haar-translation}, $\phi$ is a continuous homomorphism from $G$ to $\complex \setminus \bracsn{0}$. Moreover, by \autoref{proposition:multiplicative-less-unit}, $\norm{\pi}_{L^1(G; \complex)^*} \le 1$, and $\phi$ is bounded. Given that $\phi$ is a homomorphism, $\phi(G) \subset \partial B_\complex(0, 1)$, and $\phi \in \wh G$. + + Since $\pi \ne 0$, there exists $\psi \in L^1(G; \complex)$ with $\pi(\psi) = 1$. Using \hyperref[Riemann sums]{lemma:convolution-distribute-functional}, + \begin{align*} + \pi(f) &= \pi(f)\pi(\psi) = \pi(f * \psi) = \int_G f(y)\pi(L_y\psi) dy \\ + &= \int_G f(y)\frac{\pi(L_y\psi)}{\pi(\psi)}dy = \int_G f(y) \dpn{y, \phi}{G} dy + \end{align*} + + (Embedding): By \autoref{proposition:multiplicative-less-unit}, every multiplicative linear functional on $L^1(G; \complex)$ is contractive. Hence $\Omega(L^1(G; \complex))$ is equicontinuous. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the topology of compact convergence on $\Omega(L^1(G; \complex))$ coincides with the weak* topology. + + Let $\phi, \psi \in \wh G$. Since $\phi \ne 0$, there exists $f \in L^1(G; \complex)$ such that $\pi_\phi(f) \ne 0$. If $|\pi_\phi(f) - \pi_\psi(f)|$ is sufficiently small, then $\pi_\psi(f) \ne 0$ as well. Let $K \subset G$ be compact, then $\bracsn{L_xf|x \in K}$ is compact in $L^1(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}. In which case, + \[ + \sup_{x \in K}|(\phi - \psi)(x)| = \sup_{x \in K}\abs{\frac{\pi_\phi(L_xf)}{\pi_\phi(f)} - \frac{\pi_\psi(L_xf)}{\pi_\psi(f)}} + \] + + Thus for any $\eps > 0$, $\bracsn{\pi_\psi|\psi \in \wh G, \sup_{x \in K}|\phi(x) - \psi(x)| \le \eps}$ is a neighbourhood of $\pi_\phi$ in the weak* topology, and the mapping $\psi \mapsto \pi_\psi$ is an embedding. +\end{proof} + +\begin{proposition} +\label{proposition:unitary-character-orthonormal} + Let $G$ be a compact Abelian group with $|G| = 1$, then $\wh G$ is an orthonormal set in $L^2(G; \complex)$. +\end{proposition} +\begin{proof}[Proof, {{\cite[Proposition 4.4]{FollandHarmonic}}}. ] + Let $\phi, \psi \in \wh G$ be distinct, then there exists $x_0 \in G$ with $\dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}^{-1} \ne 1$. In which case, + \begin{align*} + \int_G \phi \ol{\psi} &= \int_G \dpn{x, \phi}{G}\dpn{x, \psi}{G}^{-1} dx \\ + &= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}\int_G \dpn{x - x_0, \phi}{G}\dpn{x - x_0, \psi}{G} dx \\ + &= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G} \int \phi \ol{\psi} + \end{align*} + + so $\phi \perp \psi$. +\end{proof} + +\begin{proposition} +\label{proposition:pontryagin-compact-discrete} + Let $G$ be a locally compact Abelian group. If $G$ is discrete, then $\wh G$ is compact. If $G$ is compact, then $\wh G$ is discrete. +\end{proposition} +\begin{proof}[Proof, {{\cite[Proposition 4.5]{FollandHarmonic}}}. ] + If $G$ is discrete, then $L^1(G; \complex)$ is a unital Banach algebra. By \autoref{definition:multiplicative-linear-functional-space}, $\wh G \iso \Omega(L^1(G; \complex))$ is compact. + + If $G$ is compact, then $\one \in L^1(G; \complex)$, and $\bracsn{f \in L^\infty(G; \complex)|\int f| > 1/2}$ is weak*-open. By \autoref{proposition:unitary-character-orthonormal}, for any $\phi \in \wh G \setminus \bracsn{\one}$, $\int \phi = 0$. Therefore $\bracsn{1}$ is open in $\wh G$, and $\wh G$ is discrete. +\end{proof} + +\begin{proposition} +\label{proposition:pontryagin-product} + Let $\seqi{G}$ be locally compact Abelian groups where $G_i$ is compact for all but finitely many $i \in I$, then + \[ + \braks{\prod_{i \in I}G_i}^\wedge \iso \bigoplus_{i \in I}\wh G_i + \] +\end{proposition} + +\begin{theorem} +\label{theorem:pontryagin-dual-usual-suspects} + \begin{enumerate} + \item $\wh \real \iso \real$ with dual pairing $\dpn{x, \xi}{\real} = e^{2\pi i \xi x}$. + \item $\wh \mathbb{T} \iso \integer$ with dual pairing $\dpn{x, n}{\mathbb{T}} = x^n$. + \item $\wh \integer = \wh \mathbb{T}$ with dual pairing $\dpn{n, x}{\mathbb{T}} = x^n$. + \end{enumerate} +\end{theorem} +\begin{proof}[Proof, {{\cite[Proposition 4.6]{FollandHarmonic}}}. ] + (1): Let $\phi \in \wh \real$, then $\dpn{0, \phi}{\real} = 1$. Thus there exists $\alpha > 0$ with $A := \int_0^\alpha \dpn{x, \phi}{\real}dx \ne 0$. In which case, + \[ + A \dpn{x, \phi}{G} = \int_0^\alpha \dpn{x + y, \phi}{\real} dy = \int_x^{x + \alpha} \dpn{y, \phi}{\real}dy + \] + + so $\phi \in C^1(\real)$ and there exists $c \in \complex$ with $\phi' = c\phi$. Therefore $\dpn{x, \phi}{\real} = e^{cx}$ for all $x \in \real$. +\end{proof} + + + diff --git a/src/harmonic/commutative/index.tex b/src/harmonic/commutative/index.tex new file mode 100644 index 0000000..71a2874 --- /dev/null +++ b/src/harmonic/commutative/index.tex @@ -0,0 +1,4 @@ +\chapter{Locally Compact Abelian Groups} +\label{chap:lcag} + +\input{./dual.tex} diff --git a/src/harmonic/lcg/algebra.tex b/src/harmonic/lcg/algebra.tex index 7e72bfd..7a1ee59 100644 --- a/src/harmonic/lcg/algebra.tex +++ b/src/harmonic/lcg/algebra.tex @@ -306,6 +306,64 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req and $f * \phi_U \to f$ in $L^p(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}. \end{proof} + + +\begin{lemma}[Riemann Sums] +\label{lemma:convolution-distribute-functional} + Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$, then for any $f, g \in L^1(G; \complex)$, + \[ + \dpn{f * g, \phi}{L^1(G; \complex)} = \int_G f(y)\dpn{L_yg, \phi}{L^1(G; \complex)} dy + \] +\end{lemma} +\begin{proof} + Let $K \subset G$ be compact and $\eps > 0$. By continuity of translation, there exists a partition $K = \bigsqcup_{j = 1}^n U_j$ of $K$ and $\seqf{y_j}$ such that: + \begin{enumerate} + \item For each $1 \le j \le n$, $y_j \in U_j$. + \item For each $1 \le j \le n$ and $y \in U_j$, + \[ + \normn{L_yg - L_{y_j}g}_{L^1(G; \complex)} \vee \normn{\dpn{L_yg, \phi}{L^1(G; \complex)} - \dpn{L_{y_j}g, \phi}{L^1(G; \complex)}} < \eps + \] + \end{enumerate} + + Since $g \in L^1(G; \complex)$, $\bracsn{g \ne 0}$ is $\sigma$-finite. By \hyperref[Tonelli's Theorem]{theorem:fubini-tonelli}, + \begin{align*} + \norm{\one_K * g - \sum_{j = 1}^n|U_j| L_{y_j}g}_{L^1(G)} &= \int_G\abs{\one_K * g(x) - \sum_{j = 1}^n|U_j| L_{y_j}g(x)}dx \\ + &\le \iint_{G \times G}\abs{\one_K(y)L_yg(x) - \sum_{j = 1}^n\one_{U_j}(y) L_{y_j}g(x)} dy dx \\ + &\le \sum_{j = 1}^n \int_G\int_{U_j}|L_yg(x) - L_{y_j}g(x)|dydx \\ + &\le \sum_{j = 1}^n \eps |U_j| = \eps |K| + \end{align*} + + Similarly, + \[ + \abs{\int_G \one_K(y)\dpn{L_yg, \phi}{L^1(G; \complex)}dy - \sum_{j = 1}^n |U_j|\dpn{L_{y_j}g, \phi}{L^1(G; \complex)}} \le \eps |K| + \] + + By linearity, + \[ + \angles{\sum_{j = 1}^n |U_j|L_{y_j}g, \phi}_{L^1(G; \complex)} = \sum_{j = 1}^n |U_j| \dpn{L_{y_j}g, \phi}{L^1(G; \complex)} + \] + + so + \[ + \abs{\dpn{\one_K * g, \phi}{L^1(G; \complex)} - \int \one_K(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy} \le 2\eps |K| (1 \vee \norm{\phi}_{L^1(G; \complex)^*}) + \] + + As the above holds for all $\eps > 0$, $\dpn{\one_K * g, \phi}{L^1(G; \complex)} = \int_G\one_K(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy$. + + By inner regularity of the Haar measure, + \[ + \dpn{\one_B * g, \phi}{L^1(G; \complex)} = \int_G\one_B(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy + \] + + for all $B \in \cb_\complex$. By \hyperref[density of simple functions]{proposition:lp-simple-dense}, + \[ + \dpn{f * g, \phi}{L^1(G; \complex)} = \int_G f(y)\dpn{L_y g, \phi}{L^1(G; \complex)} dy + \] + + for all $f \in L^1(G; \complex)$. + +\end{proof} + \begin{theorem} \label{theorem:lcg-group-algebra-ideal} Let $G$ be a locally compact group and $J \subset L^1(G; \complex)$ be a closed subspace, then $J$ is a left ideal if and only if $L_xJ \subset J$ for all $x \in G$. @@ -331,3 +389,4 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req \end{proof} +