The approximation method for approximating solutions of linear differential equations by algebraic polynomials

V. K. Dzyadyk

Izvestiya. Mathematics , Tome 8 (1974) no. 4, pp. 937-966

Résumé

In this article we propose an effective method for the approximate solution of linear differential equations with polynomial coefficients, for which the coefficient $a_0(x)$ of the highest derivative is different from zero on the segment being considered. The approximating apparatus for the required solution $y(x)$ is provided by a certain sequence of polynomials $y_n(x)$. We prove that for $a_0(x)=\mathrm{const}$ the polynomials so constructed realize the asymptotically best approximation to the function $y(x)$ in the $L^2$ metric with Chebyshev weight, and that in the general case they have the property that $$ \|y(x)-y_n(x)\|_C\leqslant AE_n(y)_C,\qquad E_n(y)_C=\inf_{c_k}\biggl\|y(x)-\sum_0^nc_kx^k\biggr\|,\quad A=\mathrm{const}. $$