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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.

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