Local constancy principle

Fourier localization controls the variation of a demodulated envelope at reciprocal scales. Suppose the Fourier support lies in a rectangular box centered at ${\xi }_0$, with widths of order $N$ in spatial-frequency coordinates and $N^2$ in a time-frequency coordinate. Set $g(x,t)=e^{-i{\xi }_0\cdot (x,t)}f(x,t)$. Choose a Schwartz function whose transform is one on the centered box and rescale it to obtain a kernel $K_N$. Thenfor $d$ spatial coordinates. The convolution averages on spatial scale $N^{-1}$ and temporal scale $N^{-2}$. For bounded $g$, differentiating the kernel gives $\Vert {\partial }_{x_j}g{\Vert }_{\infty }\le CN\Vert g{\Vert }_{\infty }$ and $\Vert {\partial }_{t}g{\Vert }_{\infty }\le C{N}^2\Vert g{\Vert }_{\infty }$.

These are quantitative variation bounds, not literal constancy on boxes. Removing the center frequency is essential: multiplying a slowly varying envelope by $e^{iMx}$ can make its phase rotate rapidly while leaving its Fourier-support width unchanged. The principle controls a localized envelope or a smoothed magnitude, rather than forbidding carrier oscillations.