Abstract
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 functionΩk(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 thatRk,1(g)=Rk+s,1(g) > 0. whereRk,1(g) is the least deviation ofg from the class of rational function of degree ≤ (k, 1).
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