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L1 Fourier inversion with an integrable transform

Statement

Assume countable choice. If $f \in L^{1} (R^{n}; C)$ and $\hat{f} \in L^{1}$ , then $g (x) = \int \hat{f} (ξ) e^{2 π i x \cdot ξ} d ξ$ is bounded and continuous, equals f almost everywhere, and equals the specified value at every Lebesgue point of f.

Facts & Assumptions

Given: $n \geq 1$ , $f, \hat{f} \in L^{1}$ and The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Gaussian Fourier means recover each Lebesgue value (Gaussian Fourier summability at Lebesgue points).

[F2]

An $L^{1}$ transform is bounded and uniformly continuous (The L1 transform is bounded and uniformly continuous).

[F3]

Dominated convergence permits passage under an integral with a single integrable majorant (Dominated convergence).

[F4]

Almost every point of a locally integrable function is a Lebesgue point under countable choice (Almost every point is a Lebesgue point of a locally integrable function).

Proof

1.1

Since $\hat{f} \in L^{1}$ , F2 applied to it shows that $g (x) = F (\hat{f}) (- x)$ is bounded and uniformly continuous. For fixed $x$ and the explicit sequence $t_{m} = 1 / (m + 1)$ , $m \in N$ , the damped integrands converge pointwise to $\hat{f} (ξ) e^{2 π i x \cdot ξ}$ and have majorant $∣ \hat{f} ∣$ . F3 therefore gives $S_{t_{m}} f (x) \to g (x)$ .

F2F3given

2.1

At any Lebesgue point with value a, F1 gives the same sequence limit a, so $g (x) = a$ . Since $f \in L^{1}$ is locally integrable, F4 gives such points with a=f(x) outside a null set (for complex inputs apply the real conclusion to both components and bound the complex oscillation by their sum). Hence $g = f$ almost everywhere. In particular g represents the original $L^{1}$ class. Countable choice is precisely inherited from F1 and F4; no statement about all values of an arbitrary representative follows.

F1F4step 1.1∎

Depends on

Used by

Uniqueness of the L1 Fourier transform Corollary
Null-set modifications defeat everywhere representative recovery Counterexample
Poisson kernel transform and Abel summability on the line Example
Fourier inversion on Schwartz space Theorem
Fourier transform of a product with one integrable transform Theorem

Dependency tree · two levels

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Sources

Gerald Teschl, Topics in Real and Functional Analysis (2017) (standard reference, not scraped)