Fourier decay from a derivative jump ($\hat{f}(k)\supset \frac{[f^{(j)}{]}_a{e}^{-ika}}{(ik{)}^{j+1}}$)

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.

For $p>-1$ this formula is interpreted away from $k=0$ as a tempered distribution or by exponential regularization. It also supplies local high-frequency cusp terms after multiplying by a smooth cutoff. Even nonnegative integer $p$ give smooth polynomials and only contact terms at $k=0$, so their localized forms have no algebraic tail. For example, the cusp expansion of $(1+|x{|}^3{)}^{-1}$ gives $-12|k{|}^{-4}+29!|k{|}^{-10}+O(|k{|}^{-16})$.