  We first give a characterization for Mathieu subspaces of univariate
polynomial algebras over fields in terms of their radicals. We then deduce that
for some classes of classical univariate orthogonal polynomials the Image
Conjecture is true. We also prove two special cases of the one-dimensional
Image Conjecture for univariate polynomial algebras $A[t]$ over commutative
$\Bbb Q$-algebras $A$.
