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Estimates for Jacobi-Sobolev type orthogonal polynomials. (English) Zbl 0888.33006
Let the Sobolev-type inner product \(\langle f,g\rangle=\int_\mathbb{R} fgd \mu_0+ \int_\mathbb{R} f'g'd \mu_1\) with \(\mu_0= w+M \delta_c\), \(\mu_1= N\delta_c\) where \(w\) is the Jacobi weight, \(\delta_c\) denotes a Dirac measure supported at the point \(c\), \(c\) is either 1 or \(-1\) and \(M,N\geq 0\). We obtain estimates and asymptotic properties on \([-1,1]\) for the polynomials orthonormal with respect to \(\langle .,.\rangle\) (the so-called Jacobi-Sobolev type polynomials) and we compare these polynomials with Jacobi orthonormal polynomials; as a consequence, a result about the convergence acceleration to \(c\) of the zeros is given. We also find some bounds and estimates for the kernels associated with the Jacobi-Sobolev type polynomials; in particular, the analogue of the generalized Szegő extremum problem concerning Christoffel functions is deduced.
Reviewer: M.Alfaro

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
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