Best approximation in the energy norm, and the Friedrichs extension of singular differential operators

Dr. Marco Marletta

Department of Mathematics and Computer Science, University of Leicester, England

We consider a formally selfadjoint semi-bounded singular differential operator. In many applications the so-called \bf Friedrichs realization} of the differential operator is of interest. For second order operators there is, for each value of the spectral parameter outside the continuous spectrum, a {\bf principal solution} of the ODE which is asymptotically smaller at the singular endpoint than any other solution. Since the work of Rellich (1952/53) it has been known that it is precisely the principal solutions of the ODE, and no others, which lie in the domain of the Friedrichs realization. Similar results have been proved in special cases for higher order ODEs. In this talk we solve the problem for the general higher- order case, by showing that a sequence of approximations to the principal solutions due to W.T. Reid possesses optimal approximation properties in the energy norm.