= Gaussian positivity for even extension moments
{c}
Let $\mu$ be a finite positive surface measure and $|f|\leq1$. Expand $|\widehat{f\mu}(x)|^{2k}$ and integrate against $e^{-\pi|x|^2/R^2}$. The <Fourier transform of a Gaussian> turns the oscillatory kernel into $R^d e^{-\pi R^2|\sum_{j\leq k}\omega_j-\sum_{j>k}\omega_j|^2}$, which is nonnegative. Taking absolute values of the bounded amplitudes gives an upper bound by the same moment with $f=1$. This weighted positivity is valid even when $f$ is merely measurable; it transfers a scalar surface-decay estimate to a local bounded-density moment estimate.
Back to article page
