Werner Haussmann

On the mean value property of harmonic functions and best harmonic ¹-approximation

Etude de la propriete de la moyenne des fonctions harmoniques. Existence, unicite et caracterisation de la meilleure approximation L 1 harmonique des fonctions strictement sous harmoniques

APPROXIMATION BY HARMONIC FUNCTIONS

Abstract For a given continuous and bounded (resp. integrable) function f defined on a domain in IRn, we consider the problem of best approximation of f by harmonic functions, mainly in the sense of the Chebyshev norm (resp. the L1-norm). In particular, we report about the characterization of best approximants. The most important tools are approximability (i.e. density) theorems as well as mean–value and inverse mean-value properties of harmonic functions.

Uniqueness Inequality and Best Harmonic L1-Approximation

If v1 and V2 are two best L1-approximants to f ∈ L1(X) from a vector subspace V ⊂ L1(X), then (f-v1) (f-V2) ≥ 0 a.e. on X, where (X, A, μ) is a given measure space. This simple inequality helps to derive the uniqueness of a best harmonic L1-approximant to a given subharmonic function under weak assumptions. In addition, an existence theorem for best harmionic L1-approximants is given.

Incomplete lattice sets that control the behaviour of entire harmonic functions

Let h be a harmonic function on ℝ n of suitably restricted growth. It is known that if h vanishes, or is bounded, on the lattice ℤ n −1 × {0}, then the same is true on ℝ n −1 × {0}. This paper presents sharp results which show that, if n [ges ] 3, then the same conclusions can be drawn even if information about h is missing on a substantial proportion of the lattice points. As corollaries we obtain uniqueness and Liouville-type theorems for harmonic, and also polyharmonic, functions which improve results by several authors.