Optimization of a Polyhamonic Impulse

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.