Optimization of a polyhamonic 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 n polynomial (polyharmonic impulse) f (t) := Σ fk cos(kt) with the asymmetry coefficient k=1 k :=f max / |fmin|, f max : f (t, λ), f min := min t f (t, λ). We have calculated optimal t 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, π] 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.