We consider the problem of best approximating a given function in $L^2$ of a subset of the unit circle by the trace of an $H^2$ function whose norm on the complementary set is bounded by a prescribed constant. It is known that the solution can be obtained by solving a spectral equation for a certain Toeplitz operator. We show how diagonalization of such an operator ‡ la Rosenblum-Rovnyak allows to estimate the rate of convergence. We also consider the same problem in $L^\infty$ rather than $L^2$ norm, and present its solution that involves some unbounded Toeplitz operator.