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).