Fourier coefficient decay ($|a_n|+|b_n|=O(n^{-p})$)

The rate at which Fourier coefficients approach zero as their index increases. The differentiability of the periodic extension, including matching values and derivatives at its joins, controls this rate through integration by parts. An absolutely bounded tail $O(n^{-p})$ with $p>1$ gives a uniform truncation error $O(N^{1-p})$.