Some properties of rational approximations of degree \((k,1)\) in the Hardy space \(H_2({\mathcal D})\) (Q1282534)

Let \(H_2({\mathbf D})\) be the Hardy space on the open unit disc \({\mathbf D}\). Let \(R_{k,1}\) denote the set of all rational functions in \(H_2({\mathbf D})\) of degree \((k,1)\), and let the subspace in \(H_2({\mathbf D})\) of rational functions of the form \[ r(z)=\sum_{j=0}^{k-1}a_jz^j + {az^k\over 1-\bar c z} \] of degree \((k,1)\) with a given \(c\in {\mathbf D}\setminus \{0\}\) be denoted by \(R_k(c)\). For a given \(f\in H_2({\mathbf D})\) the interpolation conditions \(f(0)=r(0)\), \(f'(0)=r'(0)\),\dots, \(f^{(k-1)}(0)=r^{(k-1)}(0)\), \(f(c)=r(c)\) are necessary and sufficient for \(r\in R_k(c)\) to coincide with the best approximation function in \(R_k(c)\). If, moreover, this rational function \(r\) satisfies \(\| f-r\| =\inf_{u\in R_{k,1}} \| f - u\|\), then the additional interpolation condition \(f'(c)=r'(c)\) be fulfilled. The author observes that all mentioned above interpolation conditions are insufficient for finding the best approximation in the class \(R_{k,1}\), and establishes that in \(R_{k,1}\) these conditions are equivalent to the assertion that the interpolation point \(c\) is a stationary point of the function \(\Omega_k(c)\) defined as the squared deviation of \(f\) from the subspace \(R_k(c)\).