 Half-line Fourier transform

A half-line Fourier transform separates the contributions supported on the positive and negative real axes. With the displayed sign convention, a bound $y(x)=O(e^{-{\gamma }_{+}x})$ as $x\to +\infty$ gives a holomorphic $Y_{+}$ in $\mathrm{Im}k>-{\gamma }_{+}$; a bound $y(x)=O(e^{-{\gamma }_{-}|x|})$ as $x\to -\infty$ gives a holomorphic $Y_{-}$ in $\mathrm{Im}k<{\gamma }_{-}$. The estimates hold uniformly on compact subsets of these half-planes and remain integrable after multiplication by any fixed power of $x$, which justifies complex differentiation under the integral. The overlap strip is the natural domain for a Wiener-Hopf equation. Holomorphic continuation of a transform beyond its convergence region is a separate question from convergence of the defining integral.