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 Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, Issue 13, Pages 35–44 (Mi vyuru66)

Mathematical Modelling

Optimization of a Polyharmonic Impulse

V. N. Ermolenkoa, V. A. Kostinb, D. V. Kostinb, Yu. I. Sapronovb

a Ingineering company «Vibroinnovation» (Voronezh, Russian Federation)
b Voronezh State University (Voronezh, Russian Federation)

Abstract: In theory and practice of building some technical devices, it is necessary to optimize trigonometric polynomials. In this article, we provide optimization of a trigonometric polynomial (polyharmonic impulse) $f(t):=\sum\limits_{k=1}^n f_k\cos(kt)$ with the asymmetry coefficient $k := \frac{f_{max}}{|f_{min}|}$, $f_{max} := \max\limits_t f(t,\lambda)$, $f_{min} := \min\limits_t f(t,\lambda)$. We have calculated optimal values of main amplitudes. The basis of the analysis represented in the article is the idea of the “minimal Maxwell stratum” by which we understand the subset of polynomials of a fixed degree with maximal possible number of minima under condition that all these minima are located at the same level. Polynomial $f(t)$ is then called maxwellian. The starting point of the present study was an experimentally obtained optimal set of coefficients $f_k$ for arbitrary $n$. Later, we proved uniqueness of the optimal polynomial with maximal number of minima on interval $[0,\pi]$ and derived general formula of a maxwellian polynomial of degree $n$, which was related to Fejer kernel with the asymmetry coefficient $n$. Thus, a natural hypothesis arose that Fejer kernel should define the optimal polynomial. The present paper provides justification of this hypothesis.

Keywords: polyharmonic impulse, trigonometric polynom, asymmetry coefficient, optimization, Maxwell stratum, orthogonal polynomials.

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UDC: 517.9+621.67
MSC: 90C30, 90C90

Citation: V. N. Ermolenko, V. A. Kostin, D. V. Kostin, Yu. I. Sapronov, “Optimization of a Polyharmonic Impulse”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 13, 35–44

Citation in format AMSBIB
\Bibitem{ErmKosKos12} \by V.~N.~Ermolenko, V.~A.~Kostin, D.~V.~Kostin, Yu.~I.~Sapronov \paper Optimization of a Polyharmonic Impulse \jour Vestnik YuUrGU. Ser. Mat. Model. Progr. \yr 2012 \issue 13 \pages 35--44 \mathnet{http://mi.mathnet.ru/vyuru66}