Extension and approximation of functions subharmonic in a half plane. Impossibility of extension of plurisubharmonic functions

Abstract

We construct different extensions of functions subharmonic in a half plane to the whole plane. The results obtained are applied to the approximation of subharmonic functions of finite order in a half plane by the logarithm of the modulus of an entire function. It is shown that the problem of extension of a plurisubharmonic function may have no solution.