Geodesic
Pointwise estimates of polynomials orthogonal on a~circle with respect to a~weight not belonging to the spaces $L^r$ ($r>1$).
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 1, pp. 66-78

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Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle $|z|=1$ with the weight $\varphi(\tau):=h(\tau)|\sin(\tau/2)|^{-1}g(|\sin(\tau/2)|)$ ($\tau\in\mathbb R$), where $g(t)$ is a concave modulus of continuity slowly changing at zero such that $t^{-1}g(t)\in L^1[0,1]$ and $h(\tau)$ is a positive function from the class $C_{2\pi}$ with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point $t=1$ to the greatest zero of a polynomial orthogonal on the segment [-1,1].
Mots-clés : orthogonal polynomials
Keywords: pointwise estimates, the Szegő function. 
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     author = {V. M. Badkov},
     title = {Pointwise estimates of polynomials orthogonal on a~circle with respect to a~weight not belonging to the spaces $L^r$ ($r>1$).},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {66--78},
     publisher = {mathdoc},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2009_15_1_a5/}
}
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V. M. Badkov. Pointwise estimates of polynomials orthogonal on a~circle with respect to a~weight not belonging to the spaces $L^r$ ($r>1$).. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 1, pp. 66-78. http://geodesic.mathdoc.fr/item/TIMM_2009_15_1_a5/