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A tool for locating zeros of orthogonal polynomials in Sobolev inner product spaces. (English) Zbl 0792.42010
Summary: In the theory of polynomials orthogonal with respect to an inner product of the form \[ \langle f,g\rangle= \int^ \infty_ 0 f(x)g(x)d\psi(x)+ \sum^ m_{k=1} A_ k f^{(i_ k)}(0) g^{(i_ k)}(0), \] one is confronted with the following situation: for certain values of the parameters, the orthogonal polynomials of degree \(n\) does not have all its zeros inside the support of the distribution function \(d\psi\). This paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial. For the case \(m=2\) it is shown that there are exactly two zeros outside the true interval of orthogonality for \(A_ 1\), \(A_ 2\) large; moreover, it is proved that these zeros are nonreal (complex conjugates) in the case \(i_ 1+ 1= i_ 2\). Also several examples are given.

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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