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. Let w=*g(z) map the interior of the analytic Jordan curve C conformally into the interior of the circle \w\ = 1 . We shall say that the function/(s), analytic interior to C, is of class Ep there if fcr\f( )\\dz\ is bounded for r< 1, where Cr is the curve |g(z)| =r. A function is of class Hp if it is analytic for \z\ <1 , and of class Ep there. We are taking p > 0. Of the functions ƒ(z), analytic interior to C, of class Ep there, with f(a)=*A (z = a a point interior to C, A an arbitrary constant), let F0(z) be the one 1 which minimizes the integral /c|/(^)| , |d0|. Let Pn(z) be the minimizing polynomial of degree n with Pn(a)=-4, for this integral. We shall prove that the sequence Pn(z), n = 0, 1, 2, • • • , converges maximally to F0(z) on the closed set T, consisting of C and its interior, and then derive some extensions. We use the term maximal convergence in the sense of J. L. Walsh [3, p. 80]. We denote by p the maximum value of R such that the minimizing function can be extended so as to be analytic a*nd single-valued interior to TRf as used by Walsh [3, p. 80].

Approximation in the sense of least 𝑝th powers with a single auxiliary condition of interpolation