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Trudy Inst. Mat. i Mekh. 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

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
Received: 20.01.2005

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
\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
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2005
\issue , suppl. 2
\pages S30--S48

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    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  mathnet  crossref  isi  elib
    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  mathnet  crossref  mathscinet  isi  elib
    3. V. M. Badkov, “Some properties of Jacobi polynomials orthogonal on a circle”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S49–S58  mathnet  crossref  isi  elib
    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  mathnet  crossref  isi  elib
    5. V. M. Badkov, “Asymptotic formulae for the zeros of orthogonal polynomials”, Sb. Math., 203:9 (2012), 1231–1243  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. V. M. Badkov, “Asimptoticheskie svoistva nulei ortogonalnykh trigonometricheskikh polinomov polutselykh poryadkov”, Tr. IMM UrO RAN, 19, no. 2, 2013, 54–70  mathnet  mathscinet  elib
    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  mathnet  mathscinet  zmath  elib
    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  mathnet  crossref  elib
    9. A. V. Shilkov, “O reshenii lineinykh ellipticheskikh uravnenii vtorogo poryadka”, Matem. modelirovanie, 31:6 (2019), 55–81  mathnet  crossref  elib
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