In this paper we use orthogonal system of Jacobi's polynomials as a tool for study the operators of fractional integration and differentiation in the Riemann-Liouville sense on the compact. This approach has some advantages and alow us to reformulate well-known results of fractional calculus in the new quantity. We consider several modification of Jacobi's polynomials what give us opportunity to study invariant property of operator. As shown by us in this direction is that the operator of fractional integration acting in weighted Lebesgue spaces of summable with square functions has a sequence of including invariant subspaces. The proved theorem on acting of fractional integration operator formulated in terms of Legendre's coefficients is of particular interest. Finely we obtain the sufficient condition in terms of Legendre's coefficients for representation of function by fractional integral. 查看全文>>