# zbMATH — the first resource for mathematics

On zeros of Sobolev-type orthogonal polynomials. (English) Zbl 0768.33008
Some properties of the zeros of orthogonal polynomials $$Q_ n$$ corresponding with the inner product $\langle f,g\rangle=\int_ I f(x)g(x)d\mu(x)+\lambda f'(c)g'(c)$ are analyzed. In particular various interlacing properties with the zeros of the orthogonal polynomials associated with the measure $$\mu$$ and corresponding kernel polynomials are given and for $$c$$ to the right of the interval $$I$$ it is shown that the largest zero of $$Q_ n$$ is to the right of $$c$$ for $$n$$ large enough. Furthermore it is shown that the zeros of $$Q_ n$$ are an increasing function of $$\lambda$$.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
##### Keywords:
Sobolev inner product; orthogonal polynomials; zeros