Geodesic
On the application of linear methods to polynomial approximation of
Izvestiya. Mathematics, Tome 4 (1970) no. 4, pp. 835-858 
V. K. Dzyadyk. On the application of linear methods to polynomial approximation of. Izvestiya. Mathematics, Tome 4 (1970) no. 4, pp. 835-858. http://geodesic.mathdoc.fr/item/IM2_1970_4_4_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Starting from known linear polynomial operators $U_n(\psi;x)$ which generate good approximations to continuous functions $\psi(x)$, the author proposes a method which for a given right-hand side of the equation \begin{equation} y'=f(x,y) \tag{1} \end{equation} and given initial conditions enables us to construct polynomials $y_n(x)=y_n(U_n;f;x)$ approximating to the unknown solution of the equation (1) with essentially the same precision as these operators $U_n$ would yield if the solution were given. More precisely, it is shown in this paper that $|y(x)-y_n(U_n;f;x)|\leqslant(1+\alpha_n)\cdot C\|y(x)-U_n(y;x)\|$, $C=\operatorname{const}$, $\alpha_n\downarrow0$, and effective upper bounds are placed on the quantities $C$ and $\alpha_n$. The same procedure is used also for the polynomial approximation of the solutions of $k$-th order equations with $k\geqslant2$, systems of equations, Hamrnerstein integral equations and other integral equations.

[1] Bekkenbakh E., Bellman R., Neravenstva, Mir, M., 1965 | MR

[2] Dzyadyk V. K., “O primenenii lineinykh metodov k priblizheniyu polinomami funktsii, yavlyayuschikhsya resheniyami integralnykh uravnenii Fredgolma vtorogo roda”, Ukr. matem. zh., 22:4–5 (1970), 461–480 | Zbl

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