Recent Progress in Ployhedral Harmonics

Polyhedral harmonics is a subject which deals with the problem of characterizing the continuous functions satisfying the mean value property with respect to a polytope. The main feature of it is the finite dimensionality of the space of polyhedral harmonic functions. The theory involves not only analysis but also algebra and combinatorics, and has a. rather different flavor from classical harmonic analysis. This paper aims at providing a survey on the subject, focusing on the author’s recent results.