Bivariate polynomials with a fixed leading term ${{x}^{m}}{{y}^{n}}$ , which deviate least from zero in the uniform or ${{L}^{2}}$-norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in ${{L}^{p}}$ , $1\,\le \,p\,\le \,\infty $ is studied on $D$ in the set of products of linear polynomials.

Let h be a harmonic function on ℝn of suitably restricted growth. It is known that if h vanishes, or is bounded, on the lattice ℤn−1 × {0}, then the same is true on ℝn−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.

The definiteness of the Peano kernel is proved for a functional associated with the mean-value property of Picone and Bramble and Payne for polyharmonic functions in the ball. An important corollary of this is that if a function f satisfying (−1)pΔpf>0 vanishes on p concentric spheres centered at 0, then f(0)>0. This generalizes a well-known property of subharmonic functions (which arise in the special case p = 1).