AbstractTwo of the four central design problems for FIR filters in the frequency domain are the problems of simultaneous approximation of prescribed magnitude and phase responses and prescribed magnitude and group delay responses, respectively. In the past, these problems have almost always been approached in indirect and approximative ways only. Especially (approximate) solutions of the simpler frequency response approximation problem have served as substitutes for solutions of the magnitude‐phase problem. In this paper, at first a rigorous mathematical formulation of both problems is developed and then, for these problems, the existence of solutions and results on the convergence of the approximation errors are proved. (A method to solve both problems is simultaneously suggested in Görner et al. (Optimization and Engineering 2000; 1:123–154).) Also the improvement, obtained by use of a direct solution of the magnitude‐phase response problem instead of a solution of the frequency response problem, is quantified by computable bounds. In the study, the approximation errors are measured by an arbitrary Lp‐normresp. lp‐norm with 1⩽p⩽∞, and constraints on the filter coefficients are permitted. Copyright © 2001 John Wiley & Sons, Ltd.