Extraction of Several Harmonics from Trigonometric Polynomials. Fejér-Type Inequalities

Given a trigonometric polynomial $T_n(t)=\sum _{k=1}^n\tau _k(t)$, $\tau _k(t):=a_k\cos kt+b_k\sin kt$, we consider the problem of extracting the sum of harmonics $\sum \tau _{\mu _s}(t)$ of prescribed orders $\mu _s$ by the method of amplitude and phase transformations. Such transformations map the polynomials $T_n(t)$ into similar ones using two simple operations: the multiplication by a real constant $X$ and the shift by a real phase $\lambda $, i.e., $T_n(t)\mapsto XT_n(t-\lambda )$. We represent the sum of harmonics as a sum of such polynomials and then use this representation to obtain sharp Fejér-type estimates.