Recent progress in polyhedral harmonics
Let \(\Omega\subset\mathbb{R}^n\) be a domain and \(\mu\) a probability measure with compact support \(K\subset\mathbb{R}^n\). A continuous function \(f:\Omega \to\mathbb{R}\) is said to have the \(\mu\)-mean value property if \(f\) satisfies, \[ f(x) :=\int_K f(x+ry)d \mu(y),\quad \forall x\in\Omega \text{ and }0<r<r_x, \tag{1} \] where \(r_x\) is a sufficiently small constant such that \((x+rK)\subset \Omega\) for each \(r\). Flatto and Friedman-Littman raised the problem of characterising the class of such functions \(f\), which will be denoted in the sequel by \(H_\mu(\Omega)\). The paper under review begins with some historical remarks, in which results given by Friedman-Littman are stated in Section 2. In the remaining Sections 3 to 7 the author provides a survey of works concerning the case where \(K\) is a polytope in \(\mathbb{R}^n\), focusing on the recent contributions to the subject given by himself. A closed convex polytope is a finite intersection of closed half-planes in \(\mathbb{R}^n\) which is bounded and has non-empty interior. By definition an \(n\)-dimensional polytope \(P\) is a finite union of closed convex polytopes. Note that such a set may be neither convex nor connected. For \(d=0,1,\dots,n\) let \(P(d)\) be the \(d\)-skeleton of \(P\), i.e., the union of all its faces of dimension \(<d\) and let, \[ m_d:= {1\over\bigl|P(d)\bigr|} \mu_d, \tag{2} \] where \(\mu_d\) is the Euclidean measure on \(P(d)\) with total mass denoted by \(|P(d)|\). Let \(H_{P(d)} (\Omega)\) be the class of all continuous functions in \(\Omega\) which satisfies the \(m_d\)-mean value property \((=P(d)\)-mean value property). The author discusses in Section 3 the following fundamental results due to himself: (i) The restriction map \(H_{P(d)} (\mathbb{R}^n)\to H_{P(d)} (\Omega)\) is an isomorphism and hence \(H_{P(d)}(\Omega)\) is independent of the domain of definition \(\Omega\). (ii) \(H_{P(d)} (\Omega)\) is a finite-dimensional linear space of polynomials. (iii) A basis of \(H_{P(d)} (\Omega)\) can be taken from homogeneous polynomials. (iv) Let \(\delta= ({d\over dx_1},\dots, {d\over dx_n})\) then \(H_{P(d)}(\Omega)\) admits a structure of \(R[\delta]\)-module, where \(R[\delta]\) is the ring of (p.d.e.) with constant coefficients. (v) If the complete symmetry group \(G\subset O(n)\) of \(P\) is irreductible, then \(H_{P(d)}(\Omega)\) is a finite-dimensional space of harmonics monomials. In Section 4 a characterization of the \(P(d)\)-mean value property in terms of (p.d.e.) is stated. Section 5 is devoted to the case of polytopes with symmetry. The results indicate a deep connection between the dimension of \(H_{P(d)}\) and the classification of regular polytopes given by Coxeter. The author investigates in section 6 the case of polytopes without assumptions on symmetry, starting with the case of mean value property over triangles. He claims that the case of a polygon or an \(n\)-dimensional simplex are to be considered next. The last Section 7 is concerned with four open problems on the structure of \(H_{P(d)}\) as an \(R(\delta)\)-module.
scientific article; zbMATH DE number 1506491