Optimization of a Polyharmonic Impulse

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.