A family of singular semi-classical functionals

The distributional equation of a semi-classical linear functional allows us an efficient study of other characterizations and properties of the semi-classical OPS/functionals. In [5] an extensive survey of this approach has been presented. A particular case of semi-classical OPS/functionals are the classical ones. For the distributional equation of a classical functional a regularity condition holds. For compatibility reasons in [5] it is assumed that this condition holds also for all semi-classical functionals. Here we give a counterexample, and we show that the regularity condition does not hold in general for semi-classical functionals.