Necessary and Sufficient Condition for the Continuity of all Causal Projections with Applications

This paper investigates projections onto the space of causal transfer functions. It gives a complete characterization of the set of functions for which these projections are bounded. This characterization is done in terms of the modulus of continuity of the functions. It is shown that the Riesz projector has the smallest operator norm among all causal projectors, and that the Riesz projector is bounded precisely for those functions for which the modulus of continuity is upper bounded by a regular majorant. Moreover, some consequences of these results for several applications are discussed