Let N be a natural number greater than 1. Select N uniformly distributed points t k = 2πk/N + u (0 k N − 1), and denote by) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the systemwhere m 2, and denote Ω = {a i } m i=0 . Denote by C r Ω a class of 2π-periodic continuous functions f , where f is r-times differentiable on each segment ∆ i = [a i , a i+1 ] and f (r) is absolutely continuous on ∆ i . In the present article we consider the problem of approximation of functions f ∈ C 2 Ω by the polynomials L n,N (f, x). We show that instead of the estimate |f (x) − L n,N (f, x)| c ln n/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f (x) − L n,N (f, x)| c/n (x ∈ R) which is uniform with respect to n (1 n N/2). Moreover, we found a local estimate |f (x) − L n,N (f, x)| c(ε)/n 2 (|x − a i | ε) which is also uniform with respect to n (1 n N/2). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.
In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials
$\{L_{1,n}(x;b)\}_{n=0}^\infty$, orthonormal with respect to the Sobolev-type inner product
$$
\langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx
$$
and generated by the system of modified Laguerre polynomials $\{L_{n}(x;b)\}_{n=0}^\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss -- Laguerre quadrature formula is used.
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