A numerical algorithm for solving a nonstationary problem of optimal control |
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Authors: | N L Grigorenko D V Kamzolkin L N Luk’yanova |
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Institution: | 1.Institute of Mathematics and Mechanics,Ural Branch of the Russian Academy of Sciences,Yekaterinburg,Russia |
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Abstract: | Let {φ n (α,β) (z)} n=0 ∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)α+1/2(1 + cos τ)β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ n (α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)?α?3/2 and also between the polynomial φ n (α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)?α?1/2. We use these relations to derive an asymptotic formula for φ n (α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ n (α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large. |
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