Markov's formula for numerical integration and its application in orthogonal expansions

Some properties of Chebyshev's series are discussed. These series are used as the basis for numerical analytical methods of solving Cauchy problems for systems of ordinary differential equations. Particular attention has been given to the calculation of Chebyshev's coefficients with the aid of numerical integration. A Markov quadrature formula with a single node and a weight function that corresponds to the orthogonal system of Chebyshev's polynomial of the first kind is derived. Properties of partial sums of Chebyshev's series with coefficients obtained by Markov's formula are described.