On the behavior of causal projections with applications

This paper investigates the behavior of causal projections. Such projections play an important role in filter design, system approximation and representation, spectral factorization, Wiener filter design, robust estimation, and in many more applications. The paper gives a complete characterization of function spaces for which the causal projections are continuous and 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, in the second part of the paper, consequences of these results for the construction of bases, for robust approximations, for the behavior of numerical algorithms, and for filter design applications are discussed in detail.