THE KLEIN-GORDON EQUATION, THE HILBERT TRANSFORM AND GAUSS-TYPE MAPS: H-infinity APPROXIMATION

The impetus to this work is the need to characterize when the system {phi(m)(1), phi(n)(2)} where m, n = 0, 1, 2, horizontal ellipsis ., is complete in the weak-star topology of H-infinity on the unit disk (or the half-plane). Here, phi(1) and phi(2) are two atomic inner functions, of the form phi(1)(z)=exp(lambda(1)z+1/z-1) and phi(2)(z)=exp(lambda(2)z-1/z+1), where lambda(1), lambda(2) are positive reals. Our main result asserts that the system of non-negative integral powers {phi(m)(1), phi(n)(2)} is weak-star dense in H-infinity of the unit disk if and only if lambda(1)lambda(2) <= pi(2). In earlier work in the L-infinity setting on the unit circle all the integer powers were considered, and the corresponding result was obtained (Hedenmalm and Montes-Rodriguez, 2011). The approach was to first transfer the completeness problem to the real line via the Cayley transform, and to then connect with the dynamics of Gauss-type transformations on the interval [-1, 1]. Indeed, the nonexistence of nontrivial finite absolutely continuous invariant measures for the Gauss-type map was the key ingredient of the analysis. Moreover, it was shown that the answer to the completeness problem has striking consequences for the Klein-Gordon equation. Here, the analysis is much more subtle as a result of the required finer topology. To appreciate the difference, we observe that the standard quotient space L-1/H-1 used as the predual of H-infinity is not appropriate for our purposes. Instead we model the predual in the real way, as L-1 plus the Hilbert transform of L-1, in analogy with the decomposition of BMO. The next step is to analyze carefully the iterates of the transfer operator applied to the Hilbert kernel. The approach involves a splitting of the Hilbert kernel which is induced by the transfer operator. The careful analysis of this splitting involves detours to the Hurwitz zeta function as well as to the theory of totally positive matrices.