Spectral synthesis in sobolev spaces, and uniqueness of solutions of the Dirichlet problem

By analogy with the classical spectral synthesis of Beurling (see e.g. [20]) we say tha t sets K with the approximation property in the theorem admit (m, q)-synthesis. Thus, in contrast to the situation in harmonic analysis, the conclusion here is tha t all closed sets in R ~ admit (m~ q)-synthesis, at least i / q > 2 1 / d . Among the consequences we mention the following uniqueness theorem for the Dirichlet problem. This is in fact an equivalent formulation of the result in the case q = 2. By way of illustration we only formulate the theorem in the simplest case. Generalizations to more general elliptic equations are immediate. See T. Kolsrud [21] for an extension to situations where u is defined only in G.