Sobolev orthogonal polynomials generated by modified Laguerre polynomials and the Cauchy problem for ODE systems

We consider the problem of representing a solution of the Cauchy problem for a system of ordinary differential equations (in general, nonlinear) in the form of a Fourier series in polynomials $ l_{r, k} (x; b) $ $(k = 0,1,\ldots) $, orthonormal by Sobolev with respect to the scalar product $<f,g>=\sum_{\nu=0}^{r-1} f^{(\nu)} (0) g ^ {(\nu)} (0) + \int_ {0} ^ \infty f ^ {(r)} (t) g ^ {(r)} (t) e ^ {-bt} dt $ with $ b> 0 $, generated by the modified Laguerre polynomials $ l_k (x; b) = \sqrt {b} L_k (bx) $ by means of the equalities $ l_ {r, k} (x; b) = \frac {x ^ k} {k!} \, (k = 0,1 , \ldots, r-1) $,   $ l_ {r, r + k} (x; b) = \frac {1} {(r-1)!} \int_ {0} ^ x (xt) ^ {r-1} {l} _ {k} (t; b) dt \, (k = 0,1, \ldots) $. In the infinite-dimensional Hilbert space of $ l_2 ^ m $ $ m $ -dimensional sequences $ C = (c_0, c_1, \ldots) $ for which the norm $ \| C \| = \left (\sum \nolimits_ {j = 0} ^ \infty \sum \nolimits_ {l = 1} ^ {m} (c_j ^ l) ^ 2 \right) ^ \frac12 $, the contracting nonlinear operator $ A: l_2 ^ m \to l_2 ^ m $ is constructed, the fixed point \linebreak $ \hat C = (\hat c_0, \hat c_1, \ldots) $ coincides with the sequence of unknown coefficients of the expansion of the solution of the Cauchy problem in question Fourier series in the system $ l_ {1, k} (x; b) $ $ (k = 0,1, \ldots) $. The corresponding finite-dimensional analogue $ A_N: \mathbb {R} ^ N_m \to \mathbb {R} ^ N_m $ of the operator $ A $ is also constructed, which acts in the finite-dimensional space $ \mathbb {R} ^ N_m $ of matrices $ C $ of dimension $ m \times N $, in which the norm $ \| C \| _N ^ m = \left (\sum \nolimits_ {j = 0} ^ {N-1} \sum \nolimits_ {l = 1} ^ {m} (c_j ^ l) ^ 2 \right ) ^ \frac12 $. The fixed point $ \bar C = (\bar c_0, \bar c_1, \ldots, \bar c_ {N-1}) $ of the operator $ A_N $ is the estimate (approximate value) of the desired point $ \hat C_N = (\hat c_0, \hat c_1, \ldots, \hat c_ {N-1}) $. An estimate of the error $ \| \hat C_N- \bar C_N \| _N ^ m $ is established.