Minimal H 2 Interpolation in the Carathéodory Class

For (c,,..., c„) in C, let C(cx,..., c„) denote the class of functions/(z) = 1 + cxz + ■ ■ • + cnz" + 2*>_„+ia*z* which are analytic and satisfy Re/(z) > 0 in the unit disc. The unique function of least H2 norm in C(cx,. .., c„) is explicitly determined. In this paper we consider a minimal interpolation problem at the origin, for the class H2 n C, where H2 denotes the well-known Hardy space for the unit disc, and C is the Carathéodory class of functions f(z) = 1 + cxz + c2z2 + • • • which are analytic and satisfy Re/(z) > 0 in |z| < 1. In particular, given n complex numbers cx, . . . ,cn, we wish to find the function / in H2 n C, of the form f(z) = 1 + cxz + • • • + cnz" + f akzk (1) k = n+l with least H2 norm. For each n, expansion (1) defines a mapping v„:f->(cx,...,cn) of C onto some compact set C„ c C", which is called the nth coefficient body for C. The following result is due to C. Carathéodory and O. Toeplitz (see [2]): Theorem A. Cn is a convex, compact body in C. To each point in the interior ofC„ there correspond infinitely many functions in C; but each boundary point of Cn corresponds to only one f in C. The boundary points correspond to functions of the form "1 1 + akz f(z) = Z i _ „ fe *=1 1 akZ where 1 < m < n; \ak\ = 1, ¡ik > 0 (k = 1, . . ., m), and 2TM nk = 1. From Theorem A it follows that a function corresponding to a boundary point of C„ has infinite H2 norm. On the other hand, any interior point corresponds to infinitely many functions in H2 n C. Indeed, if (c,, . . ., cn) is interior to C„, choose 0 < r < 1 such that (bx, . . . , bn) E C„, where bk = Received by the editors November 21, 1977 and, in revised form, January 17, 1978. AMS (MOS) subject classifications (1970). Primary 30A38, 30A76; Secondary 42A04, 49B35.