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.