= Fourier decay from a derivative jump
{title2=$\widehat f(k)\supset\frac{[f^{(j)}]_a e^{-ika}}{(ik)^{j+1}}$}
For a piecewise smooth decaying function, a jump in its $j$th derivative at $a$ contributes the displayed algebraic term to its high-frequency <Fourier transform>. Distributionally, $D^{j+1}f$ contains $[f^{(j)}]_a\delta(x-a)$, and <integration by parts> or the derivative-transform identity supplies the factor $(ik)^{-j-1}$. Adequate integrability of the subsequent derivatives controls the remainder. Smooth localized polynomial terms give no algebraic cusp contribution.
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