On functions harmonic in a circle, with special reference to Poisson representation
Yûsaku Komatu · Proceedings of the Japan Academy Series A Mathematical Sciences · 1952
1. We consider a family of functions harmonic in the unit circle of the z--re*O-plane.It is well known that the Dirichlet problem, i.e. the first boundary value problem on harmonic func- tions, for the unit circle is solved by the Poisson integral formula u0(z)= I----f(f)df, in the sense that, f() being prescribed as any boundary value function integrable for 0 < <2=, the function uo(z) defined by the formula is harmonic in Izl <1 and tends to f() almost everywhere in 0 < <2= as z tends to e along a Stolz path.The Poisson formula, especially in ease of bounded boundary values, is characterized by its special behavior that, if f() is re- stricted by f<f()_f for 0 <2, then the function Uo(Z) as- sociated to f() by the formula submits to the same restriction