On the Mean Convergence of Biharmonic Functions.

Let t denote the unit circle in the complex plane. given a function f (element of)lp (t), one uses t usual (harmonic) poisson kernel p ((zeta),z) for the unit disk to define the poisson integral of f, namely h=p[f]. here we consider the biharmonic poisson kernel f((zeta),z) for the unit disk to define the notion of f-integral of a given function f(element) lp (t); this associated biharmonic function will be denoted by u=f=[f]. we then consider the dilations ur(z)=u(rz) for z(element of) t and 0 =< r <1. the main result of this paper indicates that the dilations ur are convergent to f in the mean, or in the norm of lp(t).