On Interpolation to a Given Analytic Function By Analytic Functions of Minimum Norm

J. P. Evans, J. L. Walsh
1955 Transactions of the American Mathematical Society  
We shall consider here the following problem. Let the region Rx of the zplane contain the points 0u, 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
more » ... , 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.
doi:10.2307/1992843 fatcat:jj3kovfkabhbln73oxqe2nfbry