Derivatives of the L^p cosine transform

Details Derivatives of the L^p cosine transform Derivatives of the L^p cosine transform

2001-11-27

arxiv.org/abs/math/0111272v1

Mathematics

Metric Geometry

LaTeX 14 pages. To appear in `Advances in Mathematics`. Current email address: yossi_lonke@creoscitex.com

Scientific paper

The $L^p$-cosine transform of an even, continuous function $f\in C_e(\Sn)$ is defined by: $$H(x)=\int_{\Sn}|\ip{x}{\xi}|^pf(\xi) d\xi,\quad x\in {\R}^n.$$ 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.

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