Kolomoitsev Y. Approximation of function by polynomials in the space Lp, 0<p<1

The dissertation is devoted to problems which are concerned with approximation of functions by polynomials in the space Lp, 0<p<1. The main results of the dissertation are summarized as follows: the criterion of completeness for trigonometric system with gaps in the the space Lp, 0<p<1, is obtained; it is proved that in the space Lp, 0<p<1, there exist only trivial Fourier multipliers; it is proved that linear differential operators are not comparable in the Lp-metric 0<p<1; the full description for a class of functions that have the maximal order of decrease to zero of the module of smoothness is obtained; the extension of a function in the space Lp, 0<p<1, with preservation the decrease order of the module of smoothness is presented; new results on two-sided approximation of functions by trigonometric polynomials in Lp, 0<p<1, are given; the analogues of Jackson type theorem in space Lp, 0<p<1, for trigonometric system with gaps is obtained; a new property of a constant of the best approximation offunction from the space Lp, 0<p<1, is obtained.