Derivatives of the L^p cosine transform

Abstract

The $L^p$-cosine transform of an even, continuous function $f\in C_e(\Sn)$ is defined by: $$H(x)=\int_{\Sn}|\ip{x}ξ|^pf(ξ) dξ,\quad x\in {\R}^n.$$ It is shown that if $p$ is not an even integer then all partial derivatives of even order of $H(x)$ up to order $p+1$ (including $p+1$ if $p$ is an odd integer) exist and are continuous everywhere in ${\R}^n\backslash\{0\}$. As a result of the corresponding differentiation formula, we show that if $f$ is a positive bounded function and $p>1$ then $H^{1/p}$ is a support function of a convex body whose boundary has everywhere positive Gauss-Kronekcer curvature.