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Gadzhimirzaev R. M. On the Uniform Convergence of the Fourier Series by the System of Polynomials Generated by the System of Laguerre Polynomials. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2020, vol. 20, iss. 4, pp. 416-423. DOI: 10.18500/1816-9791-2020-20-4-416-423, EDN: FGZZSJ

# On the Uniform Convergence of the Fourier Series by the System of Polynomials Generated by the System of Laguerre Polynomials

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and L^{p}_{w} be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by W^{r}_{Lpw} the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space L^{p}_{w}. In the case when p = 2 we introduce in the space W^{r}_{L2w} an inner product of Sobolev-type, which makes it a Hilbert space. Further, by l^{α}_{r,n}(x), where n = r, r + 1, ..., we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions l^{α}_{r,n}(x) = xn / n! , where n = 0, 1, r − 1, form a complete and orthonormal system in the space W^{r}_{L2w}. In this paper, the problem of uniform convergence on any segment [0,A] of the Fourier series by this system of polynomials to functions from the Sobolev space W^{r}_{Lpw} is considered. Earlier, uniform convergence was established for the case p = 2. In this paper, it is proved that uniform convergence of the Fourier series takes place for p > 2 and does not occur for 1 ≤ p < 2. The proof of convergence is based on the fact that W^{r}_{Lpw} ⊂ W^{r}_{L2w} for p > 2. The divergence of the Fourier series by the example of the function ecx using the asymptotic behavior of the Laguerre polynomials is established.

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