The objects of the thesis are the spaces of functions being integrable with the weight, Jacobi polynomials, Fourier-Jacobi sums and series, the class of functions whose r-th derivative satisfies a condition of Lipschitz type in the integral metric, and the class of functions with given order of the best approximation. The purposes of the thesis are to find the estimates of an exact order for the generalized Lebesgue constants of Fourier-Jacobi sums in the spaces of functions being integrable with the weight, to prove theorems about approximation by algebraic polynomials in these spaces, and, using that as basis, to obtain the conditions of Fourier-Jacobi sums' convergence in cases when usual Lebesgue constants are unbounded. The methods used in thesis include general methods of solving approximation theory problems, general facts from mathematical analysis, functional analysis, and function theory. The thesis is devoted to the problem of estimates for the generalized Lebesgue constants of Fourier-Jacobi sums in the spaces of functions being integrable with the weight, the problem of approximation by algebraic polynomials in the metric of such spaces with additional weight, and the estimation for deviations of partial sums of Fourier-Jacobi series in these spaces in cases when usual Lebesgue constants are unbounded. We find the estimates of an exact order for the generalized Lebesgue constants of Fourier-Jacobi sums in the spaces of functions being integrable with the weight, in certain cases when usual Lebesgue constants are unbounded. We prove the theorem, strengthening Jackson theorem, about the approximation of functions by algebraic polynomials on [-1;1] in aforementioned spaces with additional weight, for the classes of functions whose r-th derivative satisfies a condition of Lipschitz type in the integral metric. In certain cases, we obtain upper estimates for the deviations of functions from these classes and some classes of functions with given order of the best approximation, from the partial sums of corresponding Fourier-Jacobi sums, in named spaces, and these estimates show that in aforenamed cases the approximation of the functions from aforesaid classes by Fourier-Jacobi sums is of an order not worse than the best, even when usual Lebesgue constants are unbounded. The results presented in the thesis can be used to continue the research of the series by orthogonal polynomials, and the approximation in the spaces of functions being integrable with a weigth.
