Fourier transform
Complex domain

The integral for the Fourier transform f ^ (ξ) = ∫ − ∞ ∞ e − i 2 π ξ t f (t) d t {\displaystyle {\hat {f}}(\xi)=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt} can be studied for complex values of its argument ξ. Depending on the properties of f, this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of ξ = σ + iτ, or something in between.

The Paley–Wiener theorem says that f is smooth (i.e., n -times differentiable for all positive integers n) and compactly supported if and only if f̂ (σ + iτ) is a holomorphic function for which there exists a constant a > 0 such that for any integer n ≥ 0, | ξ n f ^ (ξ) | ≤ C e a | τ | {\displaystyle \left\vert \xi ^{n}{\hat {f}}(\xi)\right\vert \leq Ce^{a\vert \tau \vert }} for some constant C. (In this case, f is supported on.) This can be expressed by saying that f̂ is an entire function which is rapidly decreasing in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).

(If f is not smooth, but only L, the statement still holds provided n = 0.) The space of such functions of a complex variable is called the Paley—Wiener space. This theorem has been generalised to semisimple Lie groups.

If f is supported on the half-line t ≥ 0, then f is said to be "causal" because the impulse response function of a physically realisable filter must have this property, as no effect can precede its cause. Paley and Wiener showed that then f̂ extends to a holomorphic function on the complex lower half-plane τ < 0 which tends to zero as τ goes to infinity. The converse is false and it is not known how to characterise the Fourier transform of a causal function.
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