Poisson integral and hilbert transformation

Given the transfer function (Formula Presented.) with (Formula Presented.) of a non causal system. Then the operation (Formula Presented.), which cuts off the anti-causal part of the transfer function, is called the Riesz projection. This operation plays a prominent roll in system theory, as soon as the causality of certain system has to be enforced. For example, in estimation and detection problems, the determination of the causal linear filter which minimizes the means square error criterion (the so called Wiener filter) involves the Riesz projection, and also the so called spectral factorization comprises a Riesz projection (cf. Section 10). In Section 6 we will investigate the analytic behavior of the Riesz projection on different Banach spaces in some detail.