Approximate factorization for one class of second-order matrix functions

Approximate factorization of Holder matrix function on a simple smooth closed contour has been defined. Components for one class of the $2\times2$ matrix function have been approximated by polynomials in $z$ and $1/z$. Sufficient conditions have been found for zero equality of the partial indices of the approximating matrix function. Factorization of the approximating matrix function has been constructed in an explicit form.