On the mean value property for polyharmonic functions

We derive differential relations between spherical and solid means of continuous functions. Next we use the relations to give inductive proofs of the mean value property for polyharmonic functions and its converse in arbitrary dimension.     

A generalized mean value property for polyharmonic functions

A well known property of a harmonic function in a ball is that its value at the centre equals the mean of its values on the boundary. Less well known is the more general property that its value at any point x equals the mean over all chords through x of its values at the ends of the chord, linearly interpolated at x. In this paper we show that a similar property holds for polyharmonic functions of any order when linear interpolation is replaced by two-point Hermite interpolation of odd degree.     

On the mean value property for polyharmonic functions in the ball

We obtain the mean value property for the normal derivatives of a polyharmonic function with respect to the unit sphere. We find the values of a polyharmonic function and its Laplacians at the center of the unit ball expressed via the integrals of the normal derivatives of this function over the unit sphere.     

On the quasiarithmetic mean in a mean value property and the associated functional equation

The functional equation

Polytopes and the mean value property

LetP be any (not necessarily convex nor connected) solid polytope in then-dimensional Euclidean space ℝ_n, and letP(k) be thek-skeleton of P. LetH _P(k) be the set of all continuous functions satisfying the mean value property with respect toP (k). For anyk = 0,1,...,n, we show thatH _P(k) is a finite-dimensional linear space of polynomials. This settles an open problem posed by Friedman and Littman [37] in 1962. Moreover, we show that ifP admits ample symmetry, thenH _P(k) is a finite-dimensional linear space of harmonic polynomials. Some interesting examples are also given     

On a generalized asymptotic mean value property

In this paper we study the spaces of continuous functionsf on ² satisfying

Invariant mean value property and harmonic functions

We give conditions on the functions and u on [image omitted] such that if u is given by the convolution of and u, then u is harmonic on [image omitted].     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

Regular simplices, symmetric polynomials and the mean value property

LetP be ann-dimensional regular simplex in ℝⁿ centered at the origin, and let P(k) be thek-skeleton ofP fork = 0, 1,…,n. Then the set of all continuous functions in ℝⁿ satisfying the mean value property with respect to P(k) forms a finite-dimensional linear space of harmonic polynomials. In this paper the function space is explicitly determined by group theoretic and combinatorial arguments for symmetric polynomials.     

Semilattices with the strong endomorphism kernel property

An endomorphism on an algebra ${\mathcal{A}}$ is said to be strong if it is compatible with every congruence on ${\mathcal{A}}$ , and ${\mathcal{A}}$ is said to have the strong endomorphism kernel property provided every congruence on ${\mathcal{A}}$ , other than the universal congruence, is the kernel of a strong endomorphism on ${\mathcal{A}}$ . In this note, we characterize those semilattices that have this property.     

Martin compactification for discrete potential theory and the mean value property

A converse of the well-known theorem on themean value property of harmonic functions is given. It is shown that a positive measurable function is harmonic if it possesses arestricted mean value property. Earlier proofs obtained using the probabilistic techniques were given by Veech, Heath and Baxter. Our approach is based on a Martin type compactification built up with the help of some quite elementarya priori inequalities foraveraging kernels.     

Bisection property of the kernel

Maschler, Peleg and Shapley make use of the bisection property of the kernel to provide an interpretation of the kernel for n-person game with grand coalition. We develop the similar results for any n-person game with coalition structure.