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

Abstract

Given a trigonometric polynomial $${T_n}(t) = \sum\nolimits_{k = 1}^n {{\tau _k}\left( t \right),{\tau _k}\left( t \right): = {a_k}\cos kt + {b_k}\sin kt}$$ we consider the problem of extracting the sum of harmonics $$\sum \tau_{\mu_s}(t)$$ prescribed orders µs by the method of amplitude and phase transformations. Such transformations map the polynomials Tn(t) into similar ones using two simple operations: the multiplication by a real constant X and the shift by a real phase λ, i.e., Tn(t) → XTn(t — λ). We represent the sum of harmonics as a sum of such polynomials and then use this representation to obtain sharp Fejer-type estimates.