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, $1\le p<\infty$, 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 $W^r_{L^p_w}$ 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 $W^r_{L^2_w}$ an inner product of Sobolev-type, which makes it a Hilbert space. Further, by $l_{r,n}^\alpha(x)$, where $n = r, r + 1, \dots$, we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions $l_{r,n}^\alpha(x)=\frac{x^n}{n!}$, where $n=0, 1, r-1$, form a complete and orthonormal system in the space $W^r_{L^2_w}$. 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 $W^r_{L^p_w}$ 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 $1\le p<2$. The proof of convergence is based on the fact that $W^r_{L^p_w}\subset W^r_{L^2_w}$ for $p>2$. The divergence of the Fourier series by the example of the function $e^{cx}$ using the asymptotic behavior of the Laguerre polynomials is established.