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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2005, Volume 11, Number 2, Pages 30–46 (Mi timm187)  

This article is cited in 9 scientific papers (total in 9 papers)

Zeros of orthogonal polynomials

V. M. Badkov
Full-text PDF (309 kB) Citations (9)
References:
Abstract: Let $\{T_{\sigma,n}(\tau)\}_{n=0}^\infty$ be an orthonormal on $[0,2\pi]$, with respect to some measure $d\sigma(\tau)$, system of trigonometric polynomials obtained from the sequence $1,\sin\tau,\cos\tau,\sin2\tau,\cos2\tau,\dots$ by Schmidt's orthogonalization method. A formula is established for the increment, at a point of the unit circle, of the argument of an algebraic polynomial orthogonal on it with respect to measure $d\sigma(\tau)$. Using this formula, for $n>0$, it is proved that zeros of the polynomial $T_{\sigma,n}(\tau)$ are real and simple and that zeros of the linear combinations $aT_{\sigma,2n-1}(\tau)+bT_{\sigma,2n}(\tau)$ and $-bT_{\sigma,2n-1}(\tau)+aT_{\sigma,2n}(\tau)$ alternate if $a^2+b^2>0$. For a wide class of weights with singularities whose orders are defined by finite products of real powers of concave moduli of continuity, it is proved that there exist positive constants $C_1$ and $C_2$, depending only on the weight, such that the distance between neighboring zeros of an orthogonal (with this weight) trigonometric polynomial of order $n$ lies between $C_1n^{-1}$ and $C_2n^{-1}$. In the form of corollaries, we deduce both known and new results on zeros of polynomials orthogonal with respect to a measure on a segment (possibly infinite).
Received: 20.01.2005
Bibliographic databases:
Document Type: Article
UDC: 517.5
Language: Russian
Citation: V. M. Badkov, “Zeros of orthogonal polynomials”, Function theory, Trudy Inst. Mat. i Mekh. UrO RAN, 11, no. 2, 2005, 30–46; Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 2, S30–S48
Citation in format AMSBIB
\Bibitem{Bad05}
\by V.~M.~Badkov
\paper Zeros of orthogonal polynomials
\inbook Function theory
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2005
\vol 11
\issue 2
\pages 30--46
\mathnet{http://mi.mathnet.ru/timm187}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2200220}
\zmath{https://zbmath.org/?q=an:1146.42004}
\elib{https://elibrary.ru/item.asp?id=12040701}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2005
\issue , suppl. 2
\pages S30--S48
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  • https://www.mathnet.ru/eng/timm/v11/i2/p30
  • This publication is cited in the following 9 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Trudy Instituta Matematiki i Mekhaniki UrO RAN
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