Sobolev inner products and orthogonal polynomials of Sobolev type (Q688139)

This is a short survey of some results on orthogonal polynomials with a Sobolev inner product of the form \[ (f,g)=\int F^ T AG, \] where \(F^ T=[f(x),f'(x),\ldots, f^{(r)} (x)]\), \(G^ T=[g(x)\), \(g'(x),\ldots,g^{(r)}(x)]\) and \(A\) is an \((r+1) \times(r+1)\) matrix of measures on the real line. Some explicitly computable cases are given and some properties of the case where all the measure in the matrix \(A\) are Dirac measures, except for the measure \(A_{0,0}\), which gives Sobolev type orthogonal polynomials.