Sobolev orthogonal polynomials and spectral differential equations

Abstract:

We find necessary and sufficient conditions for a spectral differential equation \[ {L_N}[y](x) = \sum \limits _{i = 1}^N {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)} \] to have Sobolev orthogonal polynomials of solutions, which are orthogonal relative to the Sobolev (pseudo-) inner product \[ \phi (p,q) = \int _\mathbb {R}^{} {pqd\mu + \int _\mathbb {R}^{} {p’q’dv,} } \] where $d\mu$ and $dv$ are signed Borel measures having finite moments. This result generalizes a result by H. L. Krall, which handles the case when $dv = 0$.