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 Trudy Inst. Mat. i Mekh. UrO RAN, 2005, Volume 11, Number 2, Pages 30–46 (Mi timm187)

Zeros of orthogonal polynomials

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 $1,\sin\tau,\cos\tau,\sin2\tau,\cos2\tau,…$ 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).

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2005, suppl. 2, S30–S48

Bibliographic databases:
UDC: 517.5

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://www.ams.org/mathscinet-getitem?mr=2200220} \zmath{https://zbmath.org/?q=an:1146.42004} \elib{http://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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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. M. Badkov, “Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight”, Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S39–S43
2. V. M. Badkov, “Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces $L^r$ ($r>1$).”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S64–S77
3. V. M. Badkov, “Some properties of Jacobi polynomials orthogonal on a circle”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S49–S58
4. V. M. Badkov, “Estimates of the Lebesgue function of Fourier sums over trigonometric polynomials orthogonal with a weight not belonging to the spaces $L^r$ $(r>1)$”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 21–32
5. V. M. Badkov, “Asymptotic formulae for the zeros of orthogonal polynomials”, Sb. Math., 203:9 (2012), 1231–1243
6. V. M. Badkov, “Asimptoticheskie svoistva nulei ortogonalnykh trigonometricheskikh polinomov polutselykh poryadkov”, Tr. IMM UrO RAN, 19, no. 2, 2013, 54–70
7. A. V. Shilkov, “Chetno-nechetnye kineticheskie uravneniya perenosa chastits. 2: Konechno-analiticheskaya kharakteristicheskaya skhema dlya odnomernykh zadach”, Matem. modelirovanie, 26:7 (2014), 33–53
8. A. V. Shilkov, “Even- and odd-parity kinetic equations of particle transport. 3: Finite analytic scheme on tetrahedra”, Math. Models Comput. Simul., 7:5 (2015), 409–429
9. A. V. Shilkov, “O reshenii lineinykh ellipticheskikh uravnenii vtorogo poryadka”, Matem. modelirovanie, 31:6 (2019), 55–81
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