Some properties of rational approximations of degree (k, 1) in the hardy spaceH 2(D)

We prove that the well-known interpolation conditions for rational approximations with free poles are not sufficient for finding a rational function of the least deviation. For rational approximations of degree (k, 1), we establish that these interpolation conditions are equivalent to the assertion that the interpolation pointc is a stationary point of the functionk(c) defined as the squared deviation off from the subspace of rational functions with numerator of degree k and with a given pole 1/¯c. For any positive integersk ands, we construct a functiong H₂(D) such thatR _k ,1(g)=R _k +s,1(g) > 0. whereR _k ,1(g) is the least deviation ofg from the class of rational function of degree (k, 1).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 251–259, August, 1998.The author is keenly grateful to N. S. Vyacheslavov, E. P. Dolzhenko, and V. G. Zinov for useful discussions.