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LCA Fourier inversion is not an everywhere statement for arbitrary L^1 functions

Statement

Let $G$ be a nondiscrete locally compact Hausdorff abelian group with Haar measure $m_{G}$ , and fix a Haar measure $m_{\hat{G}}$ on its dual. If $f \in L^{1} (G, m_{G})$ and $\hat{f} \in L^{1} (\hat{G}, m_{\hat{G}})$ , the inverse integral $F_{f} (x) : = \int_{\hat{G}} \hat{f} (γ) γ (x) d m_{\hat{G}} (γ)$ is defined at every $x \in G$ and depends only on the $L^{1}$ class of $f$ . For every $x_{0} \in G$ , that class has a measurable representative whose value at $x_{0}$ differs from $F_{f} (x_{0})$ and which agrees with any given representative away from $x_{0}$ . Thus the inverse integral cannot recover an arbitrary representative pointwise.

Facts & Assumptions

Given: A nondiscrete locally compact Hausdorff abelian group $G$ , Haar measure $m_{G}$ , a fixed Haar measure $m_{\hat{G}}$ on its dual, a class $f \in L^{1} (G, m_{G})$ with $\hat{f} \in L^{1} (\hat{G}, m_{\hat{G}})$ , a measurable representative $u$ of $f$ , and $x_{0} \in G$ .

[F1]

There is a compact neighborhood $K$ of the identity and an open neighborhood $U$ with $0 \in U \subseteq K$ (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Topological group: multiplication and inversion are continuous, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular). Haar measure is finite on compact sets, so $m_{G} (K) < \infty$ (Haar measure is positive on nonempty open sets and finite on compact sets).

[F2]

The $L^{1}$ space is the quotient by null functions, so measurable representatives agreeing almost everywhere determine the same class (The space $L^{p} (μ)$ as the quotient by null functions, Measure-null sets and almost-everywhere statements relative to a measure).

[F3]

The Fourier transform is defined on the $L^{1}$ class; equal representatives have equal transforms (The Fourier transform on an LCA group, The Pontryagin dual with the compact-open topology). Since every character has modulus one, the inverse integral is absolutely convergent at each point when $\hat{f} \in L^{1} (\hat{G}, m_{\hat{G}})$ .

Proof technique: direct.

Counterexample

1.1F1algebra

(Singletons are Haar-null.) Put $a : = m_{G} ({0})$ , which is finite because ${0}$ is compact. If $a > 0$ , translation invariance gives $m_{G} ({x}) = a$ for every $x \in G$ . The open neighborhood $U$ in [F1] is infinite: if it were finite, then $U ∖ {0}$ would be closed in the Hausdorff space $G$ , making ${0} = U \cap (G ∖ (U ∖ {0}))$ open and $G$ discrete. For every positive integer $N$ , choose $N$ distinct points of $U$ ; their disjoint singletons lie in $K$ , so finite additivity gives $N a \leq m_{G} (K)$ . Since this holds for all $N$ and $m_{G} (K) < \infty$ , $a = 0$ . Translation invariance then gives $m_{G} ({x}) = 0$ for every $x \in G$ .

2.1F2step 1.1

(A point change preserves the class.) Define $\tilde{u}$ to agree with $u$ off ${x_{0}}$ and choose its value at $x_{0}$ to be any complex number different from $u (x_{0})$ and from $F_{f} (x_{0})$ . Such a value exists because $C$ is infinite. By step 1.1, $u$ and $\tilde{u}$ agree almost everywhere, so [F2] gives $[u] = [\tilde{u}] = f$ .

3.1F3step 2.1

(The inverse integral cannot distinguish the representatives.) By [F3], $\hat{u} = \hat{\tilde{u}} = \hat{f}$ on $\hat{G}$ . Therefore both representatives give the same absolutely convergent inverse integral $F_{f} (x_{0})$ , while $\tilde{u} (x_{0}) \neq F_{f} (x_{0})$ by construction. This is an explicit failure of pointwise recovery for an arbitrary representative.

4.1step 1.1step 3.1∎

The modification leaves the $L^{1}$ class and its Fourier transform unchanged but changes the value at $x_{0}$ ; hence no inverse formula determined by the transform can hold everywhere for every representative.

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Sources

Manfred Einsiedler and Thomas Ward, Ergodic Theory with a View Towards Number Theory, Appendix C.2-C.3 (course-hosted full text) (standard reference, not scraped)