This paper discusses linear L-infinity approximation of systems from noisy frequency-domain data. The models are discrete partial sum operators generated by phi-summations including, e.g. the Fejer and the de La Vallee-Poussin summations as special cases. The main contribution of the paper is the derivation of upper bounds on these operators and on the approximation error in terms of the Fourier transform of the window function phi. The results can be used to design the first linear step in the two-step approximate H-infinity identification method with desired approximation and convergence properties.