Zeros of orthogonal polynomials in certain discrete Sobolev spaces

In this work discrete Sobolev (pseudo-)inner products of type phi(1)(p,q):=lambda p(c)q(c)+integral(a)(b)p'(x)q'(x)d mu(x), where d mu is a quasi-definite Borel measure and lambda not equal 0, and phi(2)(p,q):=lambda(p(c)q(c)+p(-c)q(-c))+integral(-a)(a) p'(x)q'(x)d mu(x), where d mu is a symmetric positive Borel measure, c not equal 0, and lambda>0 are considered. General properties of orthogonal polynomials associated with the above discrete Sobolev inner products and their zeros are studied.