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Zeros of Orthogonal Polynomials Near an Algebraic Singularity of the Measure.
- Source :
-
Constructive Approximation . Jun2018, Vol. 47 Issue 3, p407-435. 29p. - Publication Year :
- 2018
-
Abstract
- In this paper, we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on [-1,1]<inline-graphic></inline-graphic> and behaves like h(x)|x-x0|λdx<inline-graphic></inline-graphic> for some x0∈(-1,1)<inline-graphic></inline-graphic>, where <italic>h</italic>(<italic>x</italic>) is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for λ=0<inline-graphic></inline-graphic>) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function cJλ-12(x)+dJλ+12(x)<inline-graphic></inline-graphic>, where Ja(x)<inline-graphic></inline-graphic> denotes the Bessel function of the first kind and order <italic>a</italic>. Moreover, using Sturm-Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01764276
- Volume :
- 47
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Constructive Approximation
- Publication Type :
- Academic Journal
- Accession number :
- 129344444
- Full Text :
- https://doi.org/10.1007/s00365-017-9411-5