zbMATH — the first resource for mathematics

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.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
Full Text: DOI
[1] Marcellán, F.; Alfaro, M.; Rezola, M.L., Orthogonal polynomials on Sobolev spaces: old and new directions, J. comput. appl. math., 48, 1-2, 113-131, (1993) · Zbl 0790.42015
[2] Meijer, H.G., Zero distribution of orthogonal polynomials in a certain discrete Sobolev space, J. math. anal. appl., 172, 2, 520-532, (1993) · Zbl 0780.42016
[3] Meijer, H.G., On real and complex zeros of orthogonal polynomials in a discrete Sobolev space, J. comput. appl. math., 49, 179-191, (1993), (this volume) · Zbl 0792.42011
[4] Pólya, G.; Szegő, G., Problems and theorems in analysis II, (1976), Springer New York · Zbl 0311.00002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.