Harmonic approximation and Sarason's-type theorem

In the present paper, we study uniform approximation of bounded functions on an open subset of the Euclidean space d by means of harmonic functions arising as solutions of the classical or generalized Dirichlet problem. As a consequence of results obtained we establish a harmonic analogue of Sarason’s H + C – theorem; see e. g. [5]. We shall consider an arbitrary bounded open subset of , d ≥ 2. As usual, C(U) and C(∂U) denote the space of continuous functions on the closure U and the boundary ∂U of U , respectively. The Banach space of all h ∈ C(U) which are harmonic on U is denoted by H(U) and Hb(U) stands for the space of bounded harmonic functions on U . If f ∈ C(∂U), HUf denotes the Perron-Wiener-Brelot solution of the Dirichlet problem on U for the boundary condition f . Given a bounded function g on U , we define functions g∗, g ∗ on U by