Convexity preserving convolution operators (Q916949)

Let K: \({\mathbb{R}}\to {\mathbb{R}}\) and \(\gamma\) : \({\mathbb{R}}\to {\mathbb{R}}^ 2\) be piecewise smooth, \(2\pi\)-periodic functions, with K(x)\(\geq 0\) for all x in \({\mathbb{R}}\). Define K*\(\gamma\) to be the obvious convolution product: \[ K*\gamma (x)=(\pi)\int^{2\pi}_{0}K(x-t)\gamma (t)dt. \] It is shown that a necessary and sufficient condition for the transform \(\gamma\mapsto K*\gamma\) to map convex curves to convex curves is that the curve \((K',K)\) be positively convex. The necessary part of this result is attributed to Loewner.