On interpolation to a given analytic function by analytic functions of minimum norm

We shall consider here the following problem.Let the region Rx of the zplane contain the points 0u, (0.1) /9,i, 022, 0U, 0Z2, 033, and let the function f(z) be analytic in these points.To study the convergence to f(z) of the sequence of functions gn(z); here gn(z) is analytic throughout Rx, coincides with f(z) in the points 0nl, 0n2, • • • , 0nn, and among all functions with these two properties has the least norm in Rx.This problem has been previously studied [6; 7] where norm is [lub |g"(z)|, z in A\], and is now to be studied ( §1) where norm is measured by a surface integral over Rlt or ( §2) a parametric integral over the boundary of Rx, or ( §3) a line integral over the boundary of A\.If the norm is measured by the integral of the square of the modulus, we obtain by this method an expansion of f(z) in a series of orthogonal functions, an expansion whose convergence properties we study ( §4) in some detail.The asymptotic behavior of these orthogonal functions themselves and of their zeros is investigated in §5. Interpolation by functions of minimum norm, surface integrals. IfAi is a given region, we define jQ'(Rx) (0 °°) lub [| F(z)\, z in A\].We define J&(Ri) as the subclass of Q(Ri) consisting of those functions of Jif(Rx) which coincide with the given f(z) in the points p\,i, 0n2, • • • , /8"".The functions of class J^n(Ri) form a normal family in Rx, and standard methods show that there exists at least one such function Fn(z) of minimum norm.The function Fn(z) is unique if 1 <g< oo, and also if q = oo and Rx is simply connected.If 5 is any point set, we denote its closure by 5.With the generic notation v.V) = [// l^)l8