Division problem of moment functionals

Abstract

For a quasi-definite moment functional sigma and nonzero polynomials A(x) and D(x), we define another moment functional tau by the relation D(x)tau = A(x)sigma. In other words, tau is obtained from sigma by a linear spectral transform. We find necessary and sufficient conditions for tau to be quasi-definite when D(x) and A(x) have no nontrivial common factor. When tau is also quasi-definite, we also find a simple representation of orthogonal polynomials relative to tau in terms of orthogonal polynomials relative to sigma. We also give two illustrative examples when sigma is the Laguerre or Jacobi moment functional.