Zeros of orthogonal polynomials in certain discrete Sobolev spaces

In this work discrete Sobolev (pseudo-)inner products of type φ1(p, q) ≔ λp(c)q(c) + ∫ab p′(x)q′(x)dμ(x), where dμ is a quasi-definite Borel measure and λ ≠ 0, and φ2(p, q) ≔ λp(c)q(c) + p(−c)q(−c)) + ∫−aa p′(x)q′(x)dμ(x), where dμ is a symmetric positive Borel measure, c ≠ 0, and λ > 0 are considered. General properties of orthogonal polynomials associated with the above discrete Sobolev inner products and their zeros are studied.