On Robinson's energy delay theorem (Q2010652)

In engineering, since outer functions from \(H^2\) satisfy some requested physical conditions for filtering processes, those outer functions play the role of minimum-phase functions. A useful theorem, especially for applications in geology, is the Robinson's energy delay theorem on minimum-phase functions, which says that if the modulus of two functions \(f\) and \(g\) from \(H^2\) are equal almost everywhere on the unit circle, and \(f\) is a minimum-phase function, then \(\sum_{n=0}^N|a_n|^2\ge\sum_{n=0}^N|b_n|^2\), where \(a_n\) and \(b_n\) are the Taylor coefficients of \(f\) and \(g\), respectively. A physical interpretation for this inequality was given by Robinson ``that among all filters with the same gain, the outher filter makes the energy built-up as large as possible, and it does so for every positive time''. In this paper a nice short proof for Robinson's delay theorem is given using the classical theory of \(H^2\)-functions. Also a proof for the corresponding matrix case of this theorem is given. Since the case when for some values of \(N\) the inequality becomes an equality is important as energy conservation, in this paper this case is analyzed and the result describes when this is possible, helping for a characterization to the case of having equality for infinitely many values of \(N\).