On real and complex zeros of orthogonal polynomials in a discrete Sobolev space. (English) Zbl 0792.42011
Summary: Let \(\{S_ n(x;c,N)\}\) denote a set of polynomials orthogonal with respect to the discrete Sobolev inner product
\[
\langle f,g\rangle= \int^ \infty_{-\infty} f(x)g(x)d\psi(x)+Nf'(c)g'(c),
\]
where \(N\geq 0\), \(c\in\mathbb{R}\). For \(N=0\), put \(K_ n(x)= S_ n(x;\cdot,0)\). Then \(S_ n(x;c,N)\) has at least \(n-2\) different real zeros; their position with respect to the zeros of \(K_ n\) can be determined using the tangent to the graph of \(y= K_ n(x)\) in \((c,K_ n(c))\). On the other hand, if \(n\geq 3\), then \(c\) can be chosen such that \(S_ n(x;c,N)\) has two complex zeros if \(N\) is sufficiently large.
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
References:
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