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On orthogonal polynomials of Sobolev type: Algebraic properties and zeros. (English) Zbl 0764.33003
The authors study orthogonal polynomials with respect to a special Sobolev inner product given by $\langle f,g\rangle=\int_ I f(x)g(x)d\mu(x)+Mf(c)g(c)+Nf'(c)g'(c),$ with $$c\in\mathbb{R}$$ and $$M,N\geq 0$$. The emphasis is on algebraic properties, such as a representation of the Sobolev orthogonal polynomials $$Q_ n(x)$$ in terms of orthogonal polynomials related to the measure $$\mu$$ and polynomial modifications of it. A five-term recurrence relation for $$Q_ n(x)$$ is given and formulas for the reproducing kernels are derived. A large section deals with various properties of the zeros of the Sobolev orthogonal polynomials. Particular interest is given to the symmetric case, i.e., the case when both the interval $$I$$ and the measure $$\mu$$ are symmetric. The authors also examine differential properties of the polynomials $$Q_ n(x)$$ for the Sobolev modification. Some examples are worked out, e.g., the cases when the polynomials $$P_ n(x)$$ are Gegenbauer polynomials and Poisson- Charlier polynomials.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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