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Symmetric orthogonal polynomials for Sobolev-type inner products. (English) Zbl 0809.42012
In this paper the authors investigate monic orthogonal polynomials $$Q_ n (x)$$ for the inner product $\langle f,g\rangle= \int_{-a}^ a f(x) g(x) d\mu(x)+ \sum_{j=0}^ r M_ j f^{(j)}(0) g^{(j)}(0),$ where $$\mu$$ is a symmetric measure.
Due to the symmetry, it follows that $$Q_{2n}(x)= U_ n(x^ 2)$$ and $$Q_{2n+1}(x)= x V_ n(x^ 2)$$, and it is shown that $$U_ n$$ and $$V_ n$$ are again orthogonal polynomials with a Sobolev-type inner product. The zeros of $$Q_ n(x)$$ are investigated and it is shown that in some cases there may be two complex conjugated zeros, a situation which can never occur for orthogonal polynomial in $$L_ 2(\mu)$$. Finally, the authors also consider a symmetric Sobolev-type inner product with masses at $$\pm c$$.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
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