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The compact-support Fourier-Laplace transform is entire in each coordinate

Statement

Let $f \in C_{c}^{\infty} (R^{2}; C)$ and, for $z = (z_{1}, z_{2}) \in C^{2}$ , define $F (z) = \int_{R^{2}} f (x) e^{- 2 π i (x_{1} z_{1} + x_{2} z_{2})} d x$ using the compact-support Riemann integral on any rectangle containing $supp f$ . This value is independent of the containing rectangle. Each coordinate slice of $F$ is entire, with $\frac{\partial F}{\partial z_{j}} (z) = \int_{R^{2}} (- 2 π i x_{j}) f (x) e^{- 2 π i x \cdot z} d x$ for $j = 1, 2$ , and the restriction of $F$ to $R^{2}$ is the compact-support Fourier transform in the stated line-normal convention. The proof uses no Countable Choice.

Statement references

Compact-support Fourier transform convention

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1

Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography, §1.3.2

Facts & Assumptions

Given: A compactly supported smooth complex function $f$ on $R^{2}$ , a rectangle containing its support, and a complex frequency $z \in C^{2}$ .

[F1]

The Euclidean Radon transform on R²: the compact-support Fourier transform uses $e^{- 2 π i x \cdot ξ}$ and is defined by a compact-support Riemann integral.

[F2]

Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable: each continuous real function on a nondegenerate closed rectangle is Riemann integrable; apply this to real and imaginary parts.

[F3]

The complex exponential is entire and its complex derivative is itself: the complex exponential is entire with derivative itself, so $(e^{w} - 1) / w \to 1$ as $w \to 0$ .

[F4]

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ : the Riemann integral is linear and its absolute-value estimate bounds an integral by the rectangle volume times a uniform bound; for complex functions apply the real estimate to both components.

[F5]

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions: a function complex differentiable at every point of $C$ is entire.

Proof

1.1F1F2givenchoose

Choose a nondegenerate closed rectangle $B$ containing $supp f$ . For every $z \in C^{2}$ , the integrand $x \mapsto f (x) e^{- 2 π i x \cdot z}$ is continuous on $B$ , so [F2] makes its real and imaginary parts Riemann integrable. It vanishes outside $supp f$ ; hence enlarging $B$ changes the integral only by adding regions on which the integrand is zero. This proves existence and independence of the containing rectangle.

1.2F2F3F4given

Fix $j \in {1, 2}$ , fix the other complex coordinate, and let $h \in C ∖ {0}$ . By [F4], the coordinate difference quotient is the integral over $B$ of $f (x) e^{- 2 π i x \cdot z} (e^{- 2 π i x_{j} h} - 1) / h$ . The coordinate $x_{j}$ is bounded on $B$ ; [F3] therefore makes $(e^{- 2 π i x_{j} h} - 1) / h$ converge uniformly on $B$ to $- 2 π i x_{j}$ as $h \to 0$ . Multiplication by the fixed bounded continuous factor $f (x) e^{- 2 π i x \cdot z}$ preserves uniform convergence, and [F4] passes the real and imaginary integrals to their limits. Thus the coordinate derivative exists and is the displayed integral. Since the other coordinate and the point were arbitrary, every coordinate slice is complex differentiable everywhere.

2.1F1F5step 1.1step 1.2∎

By [F5], each coordinate slice is entire. At real $z = ξ$ , the defining integral is exactly the compact-support Fourier integral of [F1], so $F (ξ) = \hat{f} (ξ)$ . This proves all claims without using a Lebesgue/Riemann identification theorem or Countable Choice.

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Sources

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1 (standard reference, not scraped)
Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography (standard reference, not scraped)