On trigonometric form of tchebyshev alternance theorems and on phase iterative method of finding best approximations with weight

The article contains generalization of phase method of finding best approximations for function (in $C[-1,1]$) with weight by use of tchebyshev system of functions, rational functions and trigonometric polynomials. In the article P. L. Tchebyshev's alternance theorems have been attached analytical and constructive trigonometric form of weighted error $r(x)$ representation by means of phase function $\psi(\theta)$: $E\cos(m\theta+\psi(\theta))$, $x=\cos\theta$. Have been formulated iterative methods of finding approximation parameters. Mentioned numerical calculations examples reveal a high effciency of suggested solution method of those formulated extremal problems.