Positivity loss in bandlimited spectral reproduction on spheres

How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on ${\mathbb{S}}^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order ${\left(L\mathrm{/}(N+1)\right)}^2$ when $1\le L<N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée–Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.