Sobolev-orthonormal system of functions generated by the system of Laguerre functions

We consider the system of functions $\lambda_{r,n}^\alpha(x)$ ($r\in\mathbb{N}$, $n=0, 1, 2, \ldots$), orthonormal with respect to the Sobolev-type inner product $\langle f, g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_{0}^{\infty} f^{(r)}(x)g^{(r)}(x) dx$ and generated by the orthonormal Laguerre functions. The Fourier series in the system $\{\lambda_{r,n}^{\alpha}(x)\}_{k=0}^\infty$ is shown to uniformly converge to the function $f\in W_{L^p}^r$ for $\frac{4}{3}<p<4$, $\alpha\geq0$, $x\in[0, A]$, $0\leq A<\infty$. Recurrence relations are obtained for the system of functions $\lambda_{r,n}^\alpha(x)$. Moreover, we study the asymptotic properties of the functions $\lambda_{1,n}^\alpha(x)$ as $n\rightarrow\infty$ for $0\leq x\leq\omega$, where $\omega$ is a fixed positive real number.