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Sharp convergence rates are provided for interpolation and approximationschemes in the Hardy space \( H^{2} \) that use band-limited data. By meansof new explicit formulae for the spectral decomposition of certain Toeplitz operators,sharp estimates for Carleman and Krein-Nudel'man approximationschemes are derived. In addition, pointwise convergence results are obtained.An illustrative example based on experimental data from a hyperfrequency filter is provided.

Asymptotic Estimates for Interpolation and Constrained Approximation in $ H^{2} $ by Diagonalization of Toeplitz Operators