Study of ordinary differential equations in the space of hyperfunctions and its applications to orthogonal polynomials

Abstract

As a kind of generalized functions, hyperfunctions are, in some sense, more naturally adapted to the study of partial differential equations than L. Schwartz``s distributions. For example, the microlocal analysis of differential operators, which is now an indispensable tool in the development of the theory of differential equations, comes very naturally from the hyperfunctional viewpoint. We first extend the celebrated theorem bt H. Komatsu on the index of ordinary differential operators with analytic coefficients on the space of hyperfunctions to the index theorem of the microdifferential operators on the space of microfunctions. In particular, we prove the surjectivity of the microdifferential operators on the space of microfunctions. The latter together with the index theorem gives an elegant formula for the number of linearly independent solutions of homogeneous microdifferential equations. Secondly, we apply the theory of hyperfunctions to the study of orthogonal polynomials. Use of hyperfunction theory in the study of orthogonal polynomials not only answers many fundamental questions arising in the field of orthogonal polynomials but also opens a new way of investigating them. In particular, via the Fourier hyperfunction, we unify the real and complex orthogonality of Tchebychev ploynomial sets under a single theory, and find a net complex orthogonality of Laguerre polynomials, and answer to the open question, raised by E. Grosswald, of characterizing orthogonal polynomials having the Bessel type orthogonality. Finally, we construct a bounded variation measure in a closed form with respect to which Bessel polynomials are orthogonal. It resolves the long standing open problem of finding a real orthogonalizing weight of bounded variation for Bessel polynomials, which was first raised by, H. L. Krall and O.Frink and remained enigmatic for the last forty years.