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This article is cited in 2 scientific papers (total in 2 papers) Approximation properties of de la Vallée Poussin means of partial Fourier series in Meixner–Sobolev polynomials R. M. Gadzhimirzaev Daghestan Federal Research Centre of the Russian Academy of Sciences, Makhachkala, Russia
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    § 1. Introduction1.1. Discrete orthogonal Meixner polynomialsLet $\alpha$ and $q$ be arbitrary real numbers, $q\neq0$. The classical Meixner polynomials $M_n^\alpha(x,q)$ can be defined using Rodrigues’s formula (see [1] or [2]) by
$$
\begin{equation*}
M_n^\alpha(x,q)=\frac{q^{-n}}{n!\,\rho(x)}\Delta^n\{\rho(x)x^{[n]}\};
\end{equation*}
\notag
$$
here,
$$
\begin{equation*}
x^{[n]}=x(x-1)\dotsb(x-n+1), \ \ \Delta f(x)=f(x+1)-f(x)\ \ \text{and} \ \ \rho(x)=q^x\frac{\Gamma(x+\alpha+1)}{\Gamma(x+1)}.
\end{equation*}
\notag
$$
These polynomials were first studied by Meixner [3]. The interest in these polynomials stems, in particular, from their applications to problems in the theory of spectral estimation [4], optimum parameter location [5], approximate integration [6] and so on. For $0<q<1$ and $\alpha>-1$ the polynomials $M_n^\alpha(x,q)$ are orthogonal on the uniform grid $\Omega=\{0, 1, \dots\}$ with respect to the weight $\rho(x)$:
$$
\begin{equation*}
\sum_{x\in\Omega}M_n^\alpha(x,q)M_m^\alpha(x,q)\rho(x)=(1-q)^{-\alpha-1}h_n^{\alpha,q}\delta_{nm}, \qquad h_n^{\alpha,q}=\frac{\Gamma(n+\alpha+1)}{\Gamma(n+1)}q^{-n}.
\end{equation*}
\notag
$$
For various algebraic properties of these polynomials $M_n^\alpha(x,q)$, see [1] and [2]. Asymptotic properties of the Meixner polynomials and of their generalizations have been studied extensively (see [7]–[11] and the references cited there). These studies depend on various methods based on explicit formulae, recurrence relations, links to other classical orthogonal polynomials, and also on the machinery of vector equilibrium problems in the theory of logarithmic potential, which was developed by Gonchar and Rakhmanov (see [12]–[14]). In the present paper we consider the polynomials $M_{n,N}^\alpha(x)$ obtained from the $M_n^\alpha(x,q)$ when $x$ is changed to $Nx$, $N>0$, and $e^{-1/N}$ is substituted in for $q$. Various properties of these polynomials are recalled in the concluding section, § 5. 1.2. Discrete Sobolev orthogonal polynomialsFor more than three decades, the study of polynomial systems orthogonal with respect to the Sobolev inner product has been of great interest. This is partly due to the fact that the Sobolev inner products and the corresponding orthogonal systems (as well as their differential analogues) play an important role in many problems in the theory of functions, quantum mechanics, mathematical physics, numerical analysis and so on [15]–[17]. The Fourier series with respect to these polynomials have important properties for applications, lacking in Fourier series with respect to classical orthogonal systems. For example, for an approximate solution of boundary-value problems, where the behaviour of an approximate solution must be controlled at one or several points, the machinery of Fourier series with respect to Sobolev polynomials is more natural than that of Fourier series in classical orthogonal polynomials, [18]–[20]. In the literature systems of Sobolev orthogonal polynomials are constructed in various ways, which differ by the choice of inner products. We recall some inner products related to Meixner polynomials. In [21] and [22] the Sobolev inner product of the form
$$
\begin{equation*}
\langle f,g\rangle_S=\sum_{x=0}^{\infty}f(x)g(x)\rho(x)+Mf(0)g(0)+N\Delta f(0)\Delta g(0),
\end{equation*}
\notag
$$
where $M,N\geqslant 0$, was considered. In [23] and [24] the more general inner product
$$
\begin{equation*}
\langle f,g\rangle_S=\sum_{x=0}^{\infty}f(x)g(x)\rho(x)+\lambda\sum_{x=0}^{\infty}\Delta f(x)\Delta g(x)\rho(x),
\end{equation*}
\notag
$$
where $\lambda\geqslant0$, was used. For other types of Sobolev inner products related to Meixner polynomials, see [25] and [26]. These studies are mainly concerned with the distribution of zeros of the Meixner–Sobolev polynomials, and with their algebraic, asymptotic and difference properties. The Sobolev inner product
$$
\begin{equation}
\langle f,g\rangle_S=\sum_{k=0}^{r-1}\Delta^kf(0)\Delta^kg(0)+\sum_{x=0}^\infty\Delta^rf(x)\Delta^rg(x)\rho(x)
\end{equation}
\tag{1.1}
$$
was studied by Sharapudinov and his students [27]–[29], who proposed a method of the construction of systems of polynomials orthonormal with respect to the inner product (1.1). Let us describe the essence of this method. Let $1\leqslant p<\infty$, $l_w^p(\Omega)$ be the space of discrete functions $f$ defined on the grid $\Omega = \{0, 1, \dots\}$ and such that $\|f\|_{l_{w}^p(\Omega)}^p{=}\sum_{x\in\Omega}|f(x)|^pw(x) < \infty$. Next, let $W^r_{l_{w}^p(\Omega)}$ be the space of discrete functions $f$ defined on the grid $\Omega$ such that $\lim_{x\to+\infty}|f(x)|w(x)=0$ and $\Delta^rf\in l_{w}^p(\Omega)$. We denote the system of polynomials orthonormal in the usual sense on the grid $\Omega$ with respect to a weight $w(x)$ by $\{\varphi_n(x)\}$. For $r\in\mathbb{N}$ we consider the new system $\{\varphi_{r,n}(x)\}$ generated by $\{\varphi_n(x)\}$ via the formulae
$$
\begin{equation*}
\varphi_{r,n}(x)=\frac{x^{[n]}}{n!}, \qquad n=0,\dots,r-1,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\varphi_{r,n}(x)= \frac{1}{(r-1)!}\sum_{t=0}^{x-r}(x-1-t)^{[r-1]}\varphi_{n-r}(t), \qquad x\geqslant r, \quad n\geqslant r.
\end{equation*}
\notag
$$
We refer to the same studies for various properties of this system. In particular, they showed that
$$
\begin{equation*}
\Delta^\nu \varphi_{r,n}(x)= \begin{cases} \varphi_{r-\nu,n-\nu}(x),& 0\leqslant\nu\leqslant r-1, \ r\leqslant n, \\ \varphi_{n-r}(x),& \nu=r\leqslant n, \\ \varphi_{r-\nu,n-\nu}(x),& \nu\leqslant n<r, \\ 0,& n<\nu\leqslant r. \end{cases}
\end{equation*}
\notag
$$
In addition, the following result was established. Theorem A. The system of polynomials $\{\varphi_{r,n}(x)\}$ is complete in the space $W^r_{l_w^2(\Omega)}$. The Fourier series of a function $f\in W^r_{l_w^2(\Omega)}$ in the system $\{\varphi_{r,n}(x)\}$ has the form
$$
\begin{equation*}
f(x)=\sum_{k=0}^{r-1}\Delta^kf(0)\frac{x^{[k]}}{k!}+\sum_{k=r}^\infty c_{r,k}(f)\varphi_{r,k}(x), \qquad x\in\Omega,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
c_{r,k}(f)=\sum_{t\in\Omega}\Delta^r f(t)\varphi_{k-r}(t)w(t), \qquad k\geqslant r.
\end{equation*}
\notag
$$
In this paper we consider the system of polynomials
$$
\begin{equation}
\begin{gathered} \, \notag m_{n,N}^{\alpha,r}(x)=\frac{(Nx)^{[n]}}{n!}, \qquad n=0,\dots,r-1, \\ m_{n,N}^{\alpha,r}(x)=\frac{1}{(r-1)!}\sum_{t\in \Omega_\delta^x}(Nx-1-Nt)^{[r-1]}m_{n-r,N}^\alpha(t), \qquad x\geqslant r\delta, \quad n\geqslant r, \end{gathered}
\end{equation}
\tag{1.2}
$$
where
$$
\begin{equation*}
\Omega_\delta^x=\{0, \delta, \dots, x-r\delta\}, \ \ \delta=\frac 1N,\quad\text{and} \quad m_{n,N}^\alpha(x)=\frac{1}{\sqrt{h_{n}^\alpha}}\, M_{n,N}^\alpha(x), \ \ h_n^\alpha=h_n^{\alpha,e^{-\delta}}.
\end{equation*}
\notag
$$
Some known properties of this system are recalled in the next subsection. 1.3. Some properties of the Meixner–Sobolev polynomialsTo begin with, we note that $m_{n,N}^{\alpha,r}(x)$, which is defined in (1.2), is an algebraic polynomial of degree $n$. Indeed, from (1.2) and (5.2) we have
$$
\begin{equation}
m_{n,N}^{\alpha,r}(x)=\frac{1}{\sqrt{h_{n-r}^\alpha}} \binom{n-r+\alpha}{n-r}\sum_{k=0}^{n-r}\frac{(n-r)^{[k]}(1-e^\delta)^k}{(\alpha+1)_kk!}P_{k+r}(x),
\end{equation}
\tag{1.3}
$$
where
$$
\begin{equation*}
P_{k+r}(x)=\frac{1}{(r-1)!}\sum_{t\in \Omega_\delta^x}(Nx-1-Nt)^{[r-1]}(Nx)^{[k]}.
\end{equation*}
\notag
$$
The discrete analogue of Taylor’s formula for the function $d(x)=(Nx)^{[k+r]}$ is as follows:
$$
\begin{equation*}
\begin{aligned} \, d(x) &=\sum_{k=0}^{r-1}\Delta_\delta^kd(0)\frac{(Nx)^{[k]}}{k!}+\frac{1}{(r-1)!} \sum_{t\in \Omega_\delta^x}(Nx-1-Nt)^{[r-1]}\Delta^r_\delta d(x) \\ &=\sum_{k=0}^{r-1}\Delta_\delta^kd(0)\frac{(Nx)^{[k]}}{k!}+(k+r)^{[r]}P_{k+r}(x)=(k+r)^{[r]}P_{k+r}(x), \end{aligned}
\end{equation*}
\notag
$$
where $\Delta_\delta f(x)=f(x+\delta)-f(x)$. Hence
$$
\begin{equation*}
P_{k+r}(x)=\frac{d(x)}{(k+r)^{[r]}}=\frac{(Nx)^{[k+r]}}{(k+r)^{[r]}}.
\end{equation*}
\notag
$$
As a result, from (1.3) we obtain
$$
\begin{equation*}
m_{n,N}^{\alpha,r}(x)=\frac{1}{\sqrt{h_{n-r}^\alpha}} \binom{n-r+\alpha}{n-r}\sum_{k=0}^{n-r}\frac{(n-r)^{[k]}(1-e^\delta)^k}{(\alpha+1)_k}\frac{(Nx)^{[k+r]}}{(k+r)!}.
\end{equation*}
\notag
$$
For other properties of the system of polynomials $\{m_{n,N}^{\alpha,r}(x)\}$, see [30]. In particular, $\bullet$ if $n\geqslant r$ and $x\in\{0, \delta, \dots, (r-1)\delta\}$, then $m_{n,N}^{\alpha,r}(x)=0$; $\bullet$ the $\nu$th finite difference satisfies
$$
\begin{equation*}
\Delta_\delta^\nu m_{n,N}^{\alpha,r}(x)= \begin{cases} m_{n-\nu,N}^{\alpha,r-\nu}(x),& 0\leqslant\nu\leqslant r-1,\ r\leqslant n, \\ m_{n-r,N}^{\alpha}(x),& \nu=r\leqslant n, \\ m_{n-\nu,N}^{\alpha,r-\nu}(x),& \nu\leqslant n<r, \\ 0,& n<\nu\leqslant r; \end{cases}
\end{equation*}
\notag
$$
$\bullet$ the polynomial $m_{n,N}^{\alpha,r}(x)$ obeys the equality
$$
\begin{equation}
m_{n,N}^{\alpha,r}(x)= \frac{1}{(1-e^\delta)^r}\frac{1}{\sqrt{h_{n}^{\alpha}}}\biggl[M_{n,N}^{\alpha-r}(x) -\sum_{k=0}^{r-1}\Delta_\delta^kM_{n,N}^{\alpha-r}(0)\frac{(Nx)^{[k]}}{k!}\biggr],
\end{equation}
\tag{1.4}
$$
where
$$
\begin{equation*}
\Delta^k_\delta M_{n,N}^{\alpha-r}(0)=(1-e^\delta)^k\frac{\Gamma(n+\alpha-r+1)}{(n-k)!\,\Gamma(\alpha-r+k+1)};
\end{equation*}
\notag
$$
note that $\Delta^k_\delta M_{n,N}^{\alpha-r}(0)=0$ for $\alpha=0$; $\bullet$ the system $\{m_{n,N}^{\alpha,r}(x)\}$ is complete in the space $W^r_{l^2_{\rho_N}(\Omega_\delta)}$ and orthonormal with respect to the Sobolev inner product
$$
\begin{equation*}
\langle f,g\rangle_S=\sum_{k=0}^{r-1}\Delta^k_\delta f(0)\Delta^k_\delta g(0)+\sum_{x\in \Omega_\delta}\Delta^r_\delta f(x)\Delta^r_\delta g(x)\rho_N(x),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\Omega_\delta=\{0, \delta, 2\delta, \dots\}\quad\text{and} \quad \rho_N(x)=e^{-x}\frac{\Gamma(Nx+\alpha+1)}{\Gamma(Nx+1)}(1-e^{-\delta})^{\alpha+1}.
\end{equation*}
\notag
$$
The Fourier series of a function $f\in W^r_{l^2_{\rho_N}(\Omega_\delta)}$ with respect to this system has the form
$$
\begin{equation}
f(x)\sim \sum_{k=0}^{r-1}\Delta_\delta^kf(0)\frac{(Nx)^{[k]}}{k!}+\sum_{k=r}^\infty c^\alpha_{r,k}(f)m^{\alpha,r}_{k,N}(x),
\end{equation}
\tag{1.5}
$$
where
$$
\begin{equation}
c^\alpha_{r,k}(f)=\sum_{t\in\Omega_\delta}\Delta_\delta^r f(t)m^\alpha_{k-r,N}(t)\rho_N(t), \qquad k\geqslant r.
\end{equation}
\tag{1.6}
$$
The problem of the convergence of the Fourier series (1.5) to a function $f\in W^r_{l^p_{\rho_N}(\Omega_\delta)}$ for $1\leqslant p<\infty$ was considered in [30]. Theorem B. Let $\alpha>-1$ and $1\leqslant p<\infty$. Let $f\in W^r_{l^p_{\rho_N}(\Omega_\delta)}$. Then for $p\geqslant2$ the series (1.5) converges pointwise to $f$ on $\Omega_\delta$. If $1\leqslant p<2$, then there exist a grid $\Omega_\delta$ and a function $f\in W^r_{l^p_{\rho_N}(\Omega_\delta)}$ whose Fourier series diverges at some point $x_0\in \Omega_\delta$. In what follows we denote by $S_{n+r,N}^{\alpha,r}(f,x)$ a partial sum of the series (1.5):
$$
\begin{equation}
S_{n+r,N}^{\alpha,r}(f,x)=\sum_{k=0}^{r-1}\Delta_\delta^kf(0)\frac{(Nx)^{[k]}}{k!} +\sum_{k=r}^{n+r} c^\alpha_{r,k}(f)m^{\alpha,r}_{k,N}(x).
\end{equation}
\tag{1.7}
$$
The following important properties of the sums $S_{n+r,N}^{\alpha,r}(f,x)$ can be derived from (1.7):
Consider the case $\alpha=0$. From (1.4) and (5.9) we obtain
$$
\begin{equation*}
m_{n+r,N}^{0,r}(x)=\frac{(Nx)^{[r]}}{\sqrt{(n+r)^{[r]}}}\, m_{n,N}^r(x-r\delta).
\end{equation*}
\notag
$$
The partial sum (1.7) can be written as
$$
\begin{equation*}
S_{n+r,N}^{0,r}(f,x)=\sum_{k=0}^{r-1}\Delta_\delta^kf(0)\frac{(Nx)^{[k]}}{k!}+(Nx)^{[r]}\sum_{k=r}^{n+r} c^0_{r,k}(f)\frac{m^{r}_{k-r,N}(x-r\delta)}{\sqrt{k^{[r]}}}.
\end{equation*}
\notag
$$
In addition, by (5.15) we have
$$
\begin{equation}
\begin{aligned} \, \notag &\Delta_\delta^l S_{n+r,N}^{0,r}(f,x) \\ \notag &\qquad =\sum_{k=0}^{r-l-1}\Delta_\delta^{k+l}f(0)\frac{(Nx)^{[k]}}{k!} +(Nx)^{[r-l]}\sum_{k=r}^{n+r} c^0_{r,k}(f)\frac{m^{r-l}_{k-r,N}(x-(r-l)\delta)}{\sqrt{(k-l)^{[r-l]}}} \\ \notag &\qquad =\sum_{k=0}^{r-l-1}\Delta_\delta^{k+l}f(0)\frac{(Nx)^{[k]}}{k!} +(Nx)^{[r-l]}\sum_{k=r}^{n+r} c^0_{r-l,k-l}(\Delta_\delta^lf)\frac{m^{r-l}_{k-r,N}(x-(r-l)\delta)}{\sqrt{(k-l)^{[r-l]}}} \\ &\qquad =S_{n+r-l,N}^{0,r-l}(\Delta_\delta^lf,x). \end{aligned}
\end{equation}
\tag{1.9}
$$
The approximation properties of the partial sums $\Delta_\delta^lS_{n+r,N}^{0,r}(f,x)$ were considered in [30]. In particular, it was shown that
$$
\begin{equation*}
e^{-x/2}x^{-(r-l)/2+1/4}|\Delta_\delta^lf(x)-\Delta_\delta^lS_{n+r,N}^{0,r}(f,x)| \leqslant E_{n+r-l}^{r-l}(\Delta_\delta^lf,\delta)(1+\lambda_{n,N}^{r-l}(x)),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{gathered} \, \notag \lambda_{n,N}^{r}(x)\leqslant c(r,\lambda) \begin{cases} \log(n+1),& x\in\biggl[r\delta, \dfrac{\nu}{2}\biggr], \\ \log(n+1)+\biggl(\dfrac{\nu}{\nu^{1/3}+|x-\nu|}\biggr)^{1/4},& x\in\biggl(\dfrac{\nu}{2}, \dfrac{3\nu}{2}\biggr], \\ n^{-r/2+7/4}x^{r/2+1/4}e^{-x/4}, & x\in\biggl(\dfrac{3\nu}{2}, \infty\biggr), \end{cases} \\ \notag \nu=\nu_n(r)=4n+2r+2, \\ E_{n+r}^r(f,\delta)=\inf_{q_{n+r}}\sup_{x\in\Omega_{r,\delta}} e^{-x/2}x^{-r/2+1/4}|f(x)-q_{n+r}(x)|, \end{gathered}
\end{equation}
\tag{1.10}
$$
and the infimum is taken over all algebraic polynomials $q_{n+r}(x)$ of degree $n+r$ such that
$$
\begin{equation*}
\Delta_\delta^i f(0)=\Delta_\delta^i q_{n+r}(0), \quad i=0,\dots, r-1,\quad\text{where } \Omega_{r,\delta}=\{r\delta, (r+1)\delta, \dots\}.
\end{equation*}
\notag
$$
Here and in what follows, by $c(\alpha)$, $c(\alpha, \lambda)$ and $c(a,b,\alpha,\lambda)$ we denote positive numbers depending only on the parameters and maybe different at different places. 1.4. The main resultsIn this paper we study the approximation properties of the de la Vallée Poussin means
$$
\begin{equation}
\mathcal{V}_{n+m+r}(f,x)=\frac{1}{m+1}\sum_{k=n}^{n+m}S_{k+r,N}^{0,r}(f,x)
\end{equation}
\tag{1.11}
$$
and of their finite differences. An important role in this problem is played by the behaviour of the corresponding Lebesgue function $\Lambda_{n,m}^{\alpha}(x)$. The following theorem is one of our main results. Theorem 1. Let $\alpha\geqslant0$, $0<a\leqslant b$, $an\leqslant m\leqslant bn$ and $x\in[0,\infty)$. Then In the proof of the next theorem we employ Lebesgue’s inequality for the means $\mathcal{V}_{n+m+r}(f,x)$ and an estimate for $\Lambda_{n,m}^{\alpha}(x)$. Theorem 2. Let $f\in W_{l_{\omega}^2(\Omega_\delta)}^r$ and $\omega(x)=e^{-x}(1-e^{-\delta})$, and let $a$ and $b$ be fixed real numbers, $0<a\leqslant b$. Then for $an\leqslant m\leqslant bn$ and $0\leqslant l\leqslant r-1$, 1.5. The structure of the paperSection 1 contains five subsections. Some properties of the classical Meixner polynomials are collected in § 1.1. General facts on polynomials orthogonal with respect to Sobolev inner products are collected in § 1.2. Some known properties of the Meixner–Sobolev polynomials are recalled in § 1.3. The main results of the paper are formulated in § 1.4. This subsection explains the structure of the paper. In § 2 we present a representation and estimates for Fejér kernels of the form
$$
\begin{equation*}
\mathcal{K}_{n,N}^{\alpha}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j=0}^{k}m_{j,N}^\alpha(x)m_{j,N}^\alpha(y).
\end{equation*}
\notag
$$
Theorems 1 and 2 are proved in §§ 3 and 4, respectively. In § 5, the concluding section, we collect some properties of the modified Meixner polynomials.
§ 2. Representation and estimates for Fejér kernelsA representation and estimates for the Fejér kernels
$$
\begin{equation*}
K_{n,1}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j=0}^{k}\frac{j!}{\Gamma(j+\alpha+1)}L_j^\alpha(x)L_j^\alpha(y)
\end{equation*}
\notag
$$
were obtained in [31], in the study of the boundedness (in $L^p$-norms) of the Fejér means of Fourier–Laguerre sums. In this section we obtain similar estimates for Fejér kernels in the case of Meixner polynomials $M_{n,N}^\alpha(x)$. Consider the Fejér-type kernels
$$
\begin{equation}
\mathcal{K}_{n,N}^{\alpha}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}K_{k,N}^\alpha(x,y),
\end{equation}
\tag{2.1}
$$
where $K_{n,N}^\alpha(x,y)$ is defined by (5.10) (see § 5). The following result holds. Lemma 1. Let $x\neq y$. Then Proof. In view of (5.10), (5.4) and (5.5) we can write (2.1) as
$$
\begin{equation*}
\begin{aligned} \, &2(x-y)n\mathcal{K}_{n,N}^{\alpha}(x,y) \\ &\ \ =\frac{\delta}{(e^{\delta}-1)e^{(n-1)\delta}}\frac{n!}{\Gamma(n+\alpha)}\bigl(M_{n-1,N}^\alpha(x)M_{n,N}^\alpha(y) -M_{n,N}^\alpha(x)M_{n-1,N}^\alpha(y)\bigr) \\ &\ \ \qquad +\frac{\delta}{(e^{\delta}-1)}\sum_{k=1}^{n-1}\frac{1}{e^{(n-1)\delta}}\frac{k!}{\Gamma(k+\alpha)} \bigl(M_{k-1,N}^\alpha(x)M_{k,N}^\alpha(y)-M_{k,N}^\alpha(x)M_{k-1,N}^\alpha(y)\bigr) \\ &\ \ \qquad +\frac{e^{-\delta}-1}{(e^{\delta}-1)}(y-x)\sum_{k=1}^{n-1}\frac{1}{e^{(k-1)\delta}} \frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x)M_{k,N}^\alpha(y) \\ &\ \ \qquad +\frac{\delta}{(e^{\delta}-1)\Gamma(\alpha+1)}\bigl(M_{0,N}^\alpha(x)M_{1,N}^\alpha(y)-M_{1,N}^\alpha(x)M_{0,N}^\alpha(y)\bigr) \\ &\ \ \qquad +\frac{1}{e^{\delta}-1}\sum_{k=1}^{n-1}\frac{k!}{e^{(k-1)\delta}}\frac{yM_{k,N}^\alpha(x)\Delta_\delta M_{k,N}^\alpha(y-\delta)-xM_{k,N}^\alpha(y)\Delta_\delta M_{k,N}^\alpha(x-\delta)}{\Gamma(k+\alpha+1)}. \end{aligned}
\end{equation*}
\notag
$$
The first two terms on the right give in combination $n(x- y)\mathcal{K}_{n,N}^{\alpha}(x,y)$. Next,
$$
\begin{equation*}
\begin{aligned} \, &\frac{\delta}{(e^{\delta}-1)\Gamma(\alpha+1)}\bigl(M_{0,N}^\alpha(x)M_{1,N}^\alpha(y)-M_{1,N}^\alpha(x)M_{0,N}^\alpha(y)\bigr) \\ &\qquad =\frac{x-y}{\Gamma(\alpha+1)}=\frac{x-y}{\Gamma(\alpha+1)}M_{0,N}^\alpha(x)M_{0,N}^\alpha(y), \end{aligned}
\end{equation*}
\notag
$$
and so the third and fourth terms give in combination $(x-y)K_{n-1,N}^{\alpha}(x,y)$. Thus, we have
$$
\begin{equation}
\begin{aligned} \, \notag &(x-y)n\mathcal{K}_{n,N}^{\alpha}(x,y)=(x-y)K_{n-1,N}^{\alpha}(x,y) \\ &\qquad +\frac{1}{e^{\delta}-1}\sum_{k=1}^{n-1}\frac{k!}{e^{(k-1)\delta}}\frac{yM_{k,N}^\alpha(x)\Delta_\delta M_{k,N}^\alpha(y-\delta)-xM_{k,N}^\alpha(y)\Delta_\delta M_{k,N}^\alpha(x-\delta)}{\Gamma(k+\alpha+1)}. \end{aligned}
\end{equation}
\tag{2.3}
$$
Now let us transform the expression under the sum sign. From the recurrence formula
$$
\begin{equation}
\begin{aligned} \, \notag &(k(e^{-\delta}+1)+e^{-\delta}(\alpha+1)+(e^{-\delta}-1)Nx)M_{k,N}^\alpha(x) \\ &\qquad =(k+1)e^{-\delta}M_{k+1,N}^\alpha(x)+(k+\alpha)M_{k-1,N}^\alpha(x) \end{aligned}
\end{equation}
\tag{2.4}
$$
and since ${\Delta_\delta(f(t-\delta)g(t-\delta))=f(t)\Delta_\delta g(t-\delta)+g(t-\delta)\Delta_\delta f(t-\delta)}$, we have
$$
\begin{equation*}
\begin{aligned} \, &\Delta_\delta\bigl[(k(e^{-\delta}+1)+e^{-\delta}(\alpha+1)+(e^{-\delta}-1)N(y-\delta))M_{k,N}^\alpha(y-\delta)\bigr] \\ &\qquad =(k+1)e^{-\delta}\Delta_\delta M_{k+1,N}^\alpha(y-\delta)+(k+\alpha)\Delta_\delta M_{k-1,N}^\alpha(y-\delta) \\ &\qquad =(k(e^{-\delta}+1)+e^{-\delta}(\alpha+1)+(e^{-\delta}-1)Ny)\Delta_\delta M_{k,N}^\alpha(y-\delta) \\ &\qquad\qquad+(e^{-\delta}-1)M_{k,N}^\alpha(y-\delta). \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{aligned} \, \notag &(k(e^{-\delta}+1)+e^{-\delta}(\alpha+1)+(e^{-\delta}-1)Ny)\Delta_\delta M_{k,N}^\alpha(y-\delta) \\ \notag &\qquad =(k+1)e^{-\delta}\Delta_\delta M_{k+1,N}^\alpha(y-\delta)+(k+\alpha)\Delta_\delta M_{k-1,N}^\alpha(y-\delta) \\ &\qquad\qquad+ (1-e^{-\delta})M_{k,N}^\alpha(y-\delta). \end{aligned}
\end{equation}
\tag{2.5}
$$
Now we multiply (2.4) by
$$
\begin{equation*}
\frac{1}{e^{(k-1)\delta}}\frac{k!}{\Gamma(k+\alpha+1)}\Delta_\delta M_{k,N}^\alpha(y-\delta),
\end{equation*}
\notag
$$
(2.5) by
$$
\begin{equation*}
\frac{1}{e^{(k-1)\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x),
\end{equation*}
\notag
$$
and subtract the second equality from the first. As a result,
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{(e^{\delta}-1)N(x-y)}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x)\Delta_\delta M_{k,N}^\alpha(y-\delta) \\ &\qquad=\Delta_{\delta_y}\mathcal{M}_{k,N}^\alpha(x,y-\delta) -\Delta_{\delta_y}\mathcal{M}_{k-1,N}^\alpha(x,y-\delta) \notag \\ &\qquad\qquad+\frac{e^{\delta}-1}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x) M_{k,N}^\alpha(y-\delta), \end{aligned}
\end{equation}
\tag{2.6}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathcal{M}_{k,N}^\alpha(\tau,t)=\frac{(k+1)!}{e^{k\delta}\Gamma(k+\alpha+1)} \bigl[M_{k,N}^\alpha(\tau)M_{k+1,N}^\alpha(t)-M_{k+1,N}^\alpha(\tau) M_{k,N}^\alpha(t)\bigr], \\ \Delta_{\delta_y}f(x,y)=f(x, y+\delta)-f(x, y). \end{gathered}
\end{equation*}
\notag
$$
A similar analysis shows that
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{(e^{\delta}-1)N(x-y)}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(y)\Delta_\delta M_{k,N}^\alpha(x-\delta) \\ \notag &\qquad=\Delta_{\delta_x}\mathcal{M}_{k,N}^\alpha(x-\delta,y) -\Delta_{\delta_x}\mathcal{M}_{k-1,N}^\alpha(x-\delta,y) \\ &\qquad\qquad-\frac{e^{\delta}-1}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x-\delta) M_{k,N}^\alpha(y). \end{aligned}
\end{equation}
\tag{2.7}
$$
From (2.6) and (2.7) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\frac{y}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x)\Delta_\delta M_{k,N}^\alpha(y-\delta) \\ &\qquad =\frac{y}{(e^\delta-1)N(x-y)}\bigl[ \Delta_{\delta_y}\mathcal{M}_{k,N}^\alpha(x,y-\delta)-\Delta_{\delta_y}\mathcal{M}_{k-1,N}^\alpha(x,y-\delta)\bigr] \\ &\qquad\qquad +\frac{y}{N(x-y)}\frac{1}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x) M_{k,N}^\alpha(y-\delta), \\ &\frac{x}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(y)\Delta_\delta M_{k,N}^\alpha(x-\delta) \\ &\qquad =\frac{x}{(e^\delta-1)N(x-y)}\bigl[ \Delta_{\delta_x}\mathcal{M}_{k,N}^\alpha(x-\delta,y)-\Delta_{\delta_x}\mathcal{M}_{k-1,N}^\alpha(x-\delta,y)\bigr] \\ &\qquad\qquad -\frac{x}{N(x-y)}\frac{1}{e^{k\delta}}\frac{k!}{\Gamma(k+\alpha+1)}M_{k,N}^\alpha(x-\delta) M_{k,N}^\alpha(y). \end{aligned}
\end{equation*}
\notag
$$
In view of these equalities we can write (2.3) as
$$
\begin{equation*}
\begin{aligned} \, &(x-y)n\mathcal{K}_{n,N}^{\alpha}(x,y)=(x-y)K_{n-1,N}^{\alpha}(x,y) \\ &\qquad +\frac{e^\delta y}{(e^\delta-1)^2N(x-y)}\bigl[\Delta_{\delta_y}\mathcal{M}_{n-1,N}^\alpha(x,y-\delta)-\Delta_{\delta_y}\mathcal{M}_{0,N}^\alpha(x,y-\delta)\bigr] \\ &\qquad -\frac{e^\delta x}{(e^\delta-1)^2N(x-y)}\bigl[\Delta_{\delta_x}\mathcal{M}_{n-1,N}^\alpha(x-\delta,y) -\Delta_{\delta_x}\mathcal{M}_{0,N}^\alpha(x-\delta,y)\bigr] \\ &\qquad +\frac{e^\delta}{(e^\delta-1)N(x-y)}\biggl[xK_{n-1,N}^{\alpha}(x-\delta,y)+yK_{n-1,N}^{\alpha}(x,y-\delta)- \frac{x+y}{\Gamma(\alpha+1)}\biggr]. \end{aligned}
\end{equation*}
\notag
$$
Next, we have
$$
\begin{equation*}
\Delta_{\delta_y}\mathcal{M}_{0,N}^\alpha(x,y-\delta)=-\frac{e^\delta-1}{\Gamma(\alpha+1)}\quad\text{and} \quad \Delta_{\delta_x}\mathcal{M}_{0,N}^\alpha(x-\delta,y)=\frac{e^\delta-1}{\Gamma(\alpha+1)},
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\begin{aligned} \, &(x-y)n\mathcal{K}_{n,N}^{\alpha}(x,y)=(x-y)K_{n-1,N}^{\alpha}(x,y) \\ &\qquad +\frac{e^\delta}{(e^\delta-1)N(x-y)}\bigl[x K_{n-1,N}^{\alpha}(x-\delta,y)+y K_{n-1,N}^{\alpha}(x,y-\delta)\bigr] \\ &\qquad +\frac{e^\delta}{(e^\delta-1)^2N(x-y)}\bigl[y\Delta_{\delta_y}\mathcal{M}_{n-1,N}^\alpha(x,y-\delta)- x\Delta_{\delta_x}\mathcal{M}_{n-1,N}^\alpha(x-\delta,y)\bigr]. \end{aligned}
\end{equation*}
\notag
$$
Using (5.6) and (5.10), we find that
$$
\begin{equation}
(x-y)K_{n-1,N}^{\alpha}(x,y)=a_n^{\alpha,\delta} \bigl[M_{n,N}^{\alpha}(x)M_{n,N}^{\alpha-1}(y)-M_{n,N}^{\alpha-1}(x)M_{n,N}^{\alpha}(y)\bigr],
\end{equation}
\tag{2.8}
$$
where
$$
\begin{equation*}
a_n^{\alpha,\delta}=\frac{\delta n!}{(e^\delta-1)e^{(n-1)\delta}\Gamma(n+\alpha)}.
\end{equation*}
\notag
$$
Next, an appeal to (5.4), (5.7) and (5.6) shows that
$$
\begin{equation*}
\begin{aligned} \, &\frac{e^\delta y}{(e^\delta-1)^2N(x-y)}\Delta_{\delta_y}\mathcal{M}_{n-1,N}^\alpha(x,y-\delta) \\ &\qquad =a_n^{\alpha,\delta}\frac{y}{x-y}\bigl[M_{n,N}^{\alpha}(x)\bigl(M_{n,N}^{\alpha-1}(y) -M_{n,N}^{\alpha}(y-\delta)\bigr) \\ &\qquad\qquad+ M_{n,N}^{\alpha-1}(x)\bigl(M_{n,N}^{\alpha+1}(y-\delta)-M_{n,N}^{\alpha}(y)\bigr)\bigr], \\ &\frac{e^\delta x}{(e^\delta-1)^2N(x-y)}\Delta_{\delta_x}\mathcal{M}_{n-1,N}^\alpha(x-\delta,y) \\ &\qquad =a_n^{\alpha,\delta}\frac{x}{x-y}\bigl[M_{n,N}^{\alpha}(y)\bigl(M_{n,N}^{\alpha}(x-\delta) -M_{n,N}^{\alpha-1}(x)\bigr) \\ &\qquad\qquad+ M_{n,N}^{\alpha-1}(y)\bigl(M_{n,N}^{\alpha}(x)-M_{n,N}^{\alpha+1}(x-\delta)\bigr)\bigr]. \end{aligned}
\end{equation*}
\notag
$$
Now (2.2) follows from (5.6), (5.8) and (2.8).
This proves Lemma 1. From equality (2.2) and the weighted estimate (5.14) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &e^{-(x+y)/2}|\mathcal{K}_{n,N}^\alpha(x,y)|\leqslant c(\alpha,\lambda) \frac{1}{n^\alpha(x-y)^2}\biggl\{(x+y)A_{n}^{\alpha}(x)A_{n}^{\alpha}(y) \\ \notag &\quad +\frac{x}{n}A_{n-1}^{\alpha+1}(x)A_{n}^{\alpha}(y)\biggl[\frac{x}{|x-y|}+1\biggr]+ \frac{y}{n}A_{n}^{\alpha}(x)A_{n-1}^{\alpha+1}(y)\biggl[\frac{y}{|x-y|}+1\biggr] \\ &\quad +\frac{xy}{n}\biggl(\frac{1}{|x-y|}+1\biggr)\bigl[A_{n}^{\alpha}(x)A_{n-1}^{\alpha+1}(y)+A_{n-1}^{\alpha+1}(x)A_{n}^{\alpha}(y)\bigr]+ \frac{2xy}{n}A_{n-1}^{\alpha+1}(x)A_{n-1}^{\alpha+1}(y)\biggr\}. \end{aligned}
\end{equation}
\tag{2.9}
$$
Next, by (5.12), for the function $A_n^\alpha(x)$ we have
$$
\begin{equation}
A_{n-1}^{\alpha+1}(x)\leqslant c(\alpha)nA_{n}^{\alpha}(x), \qquad 0\leqslant x\leqslant \frac{1}{\theta_n},
\end{equation}
\tag{2.10}
$$
$$
\begin{equation}
A_{n-1}^{\alpha+1}(x)\leqslant c(\alpha)\sqrt{\frac{n}{x}}A_{n}^{\alpha}(x), \qquad \frac{1}{\theta_n}\leqslant x\leqslant \frac{\theta_n}{2},
\end{equation}
\tag{2.11}
$$
and
$$
\begin{equation}
A_{n-1}^{\alpha+1}(x)\leqslant c(\alpha)A_{n}^{\alpha}(x), \qquad \frac{\theta_n}{2}<x<\infty.
\end{equation}
\tag{2.12}
$$
From (2.9)–(2.12) and (5.14) it is easy to find estimates for the kernel $\mathcal{K}_{n,N}^\alpha(x,y)$ which depend on the position of the variables $x$ and $y$ on the half-axis $[0,\infty)$. Namely, the following results hold. Estimate (2.13) follows from (2.9)–(2.11) since $|x-y|>\frac{1}{2}\max(x,y)$ for $x>2y$ $(y>2x)$. Lemma 4. If $x\in[{2}/{\theta_n},{\theta_n}/{2}]$, $y\in[{1}/{\theta_n},{3\theta_n}/{4}]$ and $|x-y|\geqslant\sqrt{{x}/{\theta_n}}$, then Proof. From (2.9) and (2.11) we obtain
$$
\begin{equation*}
\begin{aligned} \, &e^{-(x+y)/2}|\mathcal{K}_{n,N}^\alpha(x,y)|\leqslant c(\alpha,\lambda) \frac{(xy)^{-\alpha/2-1/4}}{\sqrt{n}(x-y)^2}\biggl[x+y+\sqrt{\frac{x}{n}}\biggl(\frac{x}{|x-y|}+1\biggr) \\ &\qquad +\sqrt{\frac{y}{n}}\biggl(\frac{y}{|x-y|}+1\biggr) +\frac{xy}{n}\biggl(\frac{1}{|x-y|}+1\biggr)\biggl(\sqrt{\frac{n}{x}}+\sqrt{\frac{n}{y}}\biggr)+2\sqrt{xy}\biggr]. \end{aligned}
\end{equation*}
\notag
$$
Now estimate (2.14) follows from the condition $|x-y|\geqslant\sqrt{{x}/{\theta_n}}$. § 3. Estimate for the Lebesgue functionConsider the sum
$$
\begin{equation}
\Lambda_{n,m}^{\alpha}(x)=x^{\alpha/2+1/4}\delta\sum_{t\in\Omega_{\delta}}t^{\alpha/2-1/4} e^{-(x+t)/2}|V_{n+m,N}^\alpha(x,t)|,
\end{equation}
\tag{3.1}
$$
where
$$
\begin{equation*}
V_{n+m,N}^\alpha(x,y)=\frac{1}{m+1}\sum_{k=n}^{n+m}K_{k,N}^\alpha(x,y).
\end{equation*}
\notag
$$
Let us estimate the behaviour of $\Lambda_{n,m}^{\alpha}(x)$ for $\alpha\geqslant0$. Let $x\in[0,{2}/{\theta_n}]$, $\theta_n=4n+2\alpha+2$. Then
$$
\begin{equation}
\Lambda_{n,m}^{\alpha}(x)=x^{\alpha/2+1/4}\delta \biggl[\sum_{t\in B_1}+\sum_{t\in B_2}\biggr]t^{\alpha/2-1/4} e^{-(x+t)/2}|V_{n+m,N}^\alpha(x,t)|=I_1+I_2,
\end{equation}
\tag{3.2}
$$
where $B_1=[0,4/\theta_n]\cap\Omega_{\delta}$ and $B_2=(4/\theta_n,\infty)\cap\Omega_{\delta}$. Let us estimate $I_1$. From (5.13) and (5.12) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag I_1 &\leqslant x^{\alpha/2+1/4}\delta\sum_{t\in B_1} t^{\alpha/2-1/4}\frac{e^{-(x+t)/2}}{m+1} \sum_{k=n}^{n+m}\sum_{j=0}^{k}m_{j,N}^\alpha(x)m_{j,N}^\alpha(t) \\ \notag &\leqslant c(\alpha,\lambda)\frac{x^{\alpha/2+1/4}}{m+1}\delta\sum_{t\in B_1} t^{\alpha/2-1/4}\sum_{k=n}^{n+m}\sum_{j=0}^{k}\theta_j^\alpha \\ &\leqslant c(\alpha,\lambda,a,b)\frac{x^{\alpha/2+1/4}\theta_{n+m}^{\alpha+2}}{m+1} \biggl(\frac{4}{\theta_n}\biggr)^{\alpha/2+3/4} \leqslant c(\alpha,\lambda,a,b). \end{aligned}
\end{equation}
\tag{3.3}
$$
Now we estimate $I_2$. We note first that
$$
\begin{equation}
\begin{aligned} \, \notag V_{n+m,N}^\alpha(x,t) &=\frac{1}{m+1}\sum_{k=n}^{n+m}K_{k,N}^\alpha(x,t) \\ &=\frac{n+m+1}{m+1}\mathcal K_{n+m+1,N}^\alpha(x,t)-\frac{n}{m+1}\mathcal K_{n,N}^\alpha(x,t). \end{aligned}
\end{equation}
\tag{3.4}
$$
As a result,
$$
\begin{equation}
\begin{aligned} \, \notag I_2 &\leqslant c(b) x^{\alpha/2+1/4}\delta\sum_{t\in B_2}t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n+m+1,N}^\alpha(x,t)| \\ &\qquad +c(a)x^{\alpha/2+1/4}\delta\sum_{t\in B_2}t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n,N}^\alpha(x,t)|=I_{21}+I_{22}. \end{aligned}
\end{equation}
\tag{3.5}
$$
We estimate $I_{21}$ ($I_{22}$ is estimated similarly). Let $B_3=(4/\theta_n,\theta_{n+m+1}/2)\cap\Omega_{\delta}$. Then
$$
\begin{equation}
I_{21}=c(b) x^{\alpha/2+1/4}\delta \biggl[\sum_{t\in B_3}+\sum_{t\in B_2\setminus B_3}\biggr]t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n+m+1,N}^\alpha(x,t)|= I_{21}^1+I_{21}^2.
\end{equation}
\tag{3.6}
$$
From Lemmas 2 and 3 and equality (5.12) we obtain
$$
\begin{equation}
\nonumber I_{21}^1 \leqslant c(\alpha,\lambda,b)x^{\alpha/2+1/4}\theta_{n+m+1}^{-\alpha}A_{n+m+1}^\alpha(x) \delta\sum_{t\in B_3}\frac{t^{\alpha/2-1/4}A_{n+m+1}^\alpha(t)}{\max(x,t)}
\end{equation}
\notag
$$
$$
\begin{equation}
\leqslant c(\alpha,\lambda,a,b)x^{\alpha/2+1/4}\theta_{n+m+1}^{\alpha/2-1/4} \delta\sum_{t\in B_3}t^{-3/2}\leqslant c(\alpha,\lambda,a,b),
\end{equation}
\tag{3.7}
$$
$$
\begin{equation}
\nonumber I_{21}^2 \leqslant c(\alpha,\lambda,a,b)x^{\alpha/2+1/4}\theta_{n+m+1}^{-\alpha-1}A_{n+m+1}^\alpha(x) \delta\sum_{t\in B_2\setminus B_3}t^{\alpha/2-1/4}A_{n+m+1}^\alpha(t)
\end{equation}
\notag
$$
$$
\begin{equation}
\leqslant c(\alpha,\lambda,a,b)\theta_{n+m+1}^{-\alpha/2-5/4} \biggl(\frac{\theta_{n+m+1}^{\alpha/2+3/4}}{\theta_{n+m+1}^{1/3}}+e^{-3n/4}\biggr)= \frac{c(\alpha,\lambda,a,b)}{\theta_{n+m+1}^{5/6}}.
\end{equation}
\tag{3.8}
$$
Now an appeal to (3.2)–(3.8) shows that
$$
\begin{equation*}
\Lambda_{n,m}^{\alpha}(x)\leqslant c(\alpha,\lambda,a,b), \qquad x\in\biggl[0,\frac{2}{\theta_n}\biggr].
\end{equation*}
\notag
$$
Now let $x\in[2/\theta_{n}, \theta_{n+m+1}/2]$. Using equality (3.4), we have
$$
\begin{equation*}
\begin{aligned} \, \Lambda_{n,m}^{\alpha}(x) &\leqslant c(b) x^{\alpha/2+1/4}\delta\sum_{t\in \Omega_\delta}t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n+m+1,N}^\alpha(x,t)| \\ &\qquad +c(a)x^{\alpha/2+1/4}\delta\sum_{t\in \Omega_\delta}t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n,N}^\alpha(x,t)|=J_{1}+J_{2}. \end{aligned}
\end{equation*}
\notag
$$
Let us estimate $J_{1}$ alone. To this end we set
$$
\begin{equation*}
\begin{gathered} \, D_1=\biggl[0,x-\sqrt{\frac{x}{\theta_{n+m+1}}}\biggr]\cap\Omega_\delta, \qquad D_2=\biggl(x-\sqrt{\frac{x}{\theta_{n+m+1}}},x+\sqrt{\frac{x}{\theta_{n+m+1}}}\biggr)\cap\Omega_\delta, \\ D_3=\biggl(x+\sqrt{\frac{x}{\theta_{n+m+1}}},\frac{3\theta_{n+m+1}}4\biggr]\cap\Omega_\delta\quad\text{and} \quad D_4=\biggl(\frac{3\theta_{n+m+1}}4,\infty\biggr)\cap\Omega_\delta. \end{gathered}
\end{equation*}
\notag
$$
In view of this notation, we can write $J_1$ as a sum: where
$$
\begin{equation*}
J_{1i}=c(b) x^{\alpha/2+1/4}\delta\sum_{t\in D_i}t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n+m+1,N}^\alpha(x,t)|, \qquad i=1,2,3,4.
\end{equation*}
\notag
$$
We estimate $J_{12}$. By Lemma 7 we have
$$
\begin{equation}
\begin{aligned} \, \notag J_{12} &\leqslant\frac{c(a,b)x^{\alpha/2+1/4}}{n+m+1}\,\delta\sum_{t\in D_2}t^{\alpha/2-1/4} e^{-(x+t)/2}K_{0,N}^\alpha(x,t) \\ \notag &\qquad +\frac{c(a,b) x^{\alpha/2+1/4}}{n+m+1}\,\delta \\ \notag &\qquad\qquad\times\sum_{t\in D_2}t^{\alpha/2-1/4} e^{-(x+t)/2}\sum_{k=1}^{n+m}(K_{k,N}^\alpha(x,x))^{1/2}(K_{k,N}^\alpha(t,t))^{1/2} \\ \notag &\leqslant c(\alpha, a,b)\frac{x^{\alpha/2+1/4}}{n+m+1}\, e^{-x}x^{\alpha/2+3/4} \\ \notag &\qquad +\frac{c(\alpha,\lambda,a,b) x^{\alpha/2+1/4}}{n+m+1}\,\delta \\ \notag &\qquad\qquad\times\sum_{t\in D_2}t^{\alpha/2-1/4} \sum_{k=1}^{n+m}k^{1-\alpha}\theta_k^{\alpha/2-1/4}x^{-\alpha/2-1/4} \theta_k^{\alpha/2-1/4}t^{-\alpha/2-1/4} \\ &\leqslant \frac{c(\alpha, a,b)}{n+m+1}+c(\alpha,\lambda,a,b)(n+m)^{1/2}x^{-1/2}\sqrt{\frac{x}{\theta_{n+m+1}}}\leqslant c(\alpha,\lambda,a,b). \end{aligned}
\end{equation}
\tag{3.10}
$$
To estimate $J_{13}$ we set
$$
\begin{equation*}
D_3^1=\biggl(x+\sqrt{\frac{x}{\theta_{n+m+1}}},2x\biggr]\cap\Omega_\delta\quad\text{and} \quad D_3^2=\biggl(2x,\frac{3\theta_{n+m+1}}4\biggr]\cap\Omega_\delta.
\end{equation*}
\notag
$$
Note that $t-x\geqslant\sqrt{x/\theta_{n+m+1}}$ for $x\in[2/\theta_{n},\theta_{n+m+1}/2]$ and $t\in D_3$. Now, by Lemma 4
$$
\begin{equation}
\begin{aligned} \, \notag J_{13} &\leqslant \frac{c(\alpha,\lambda,a,b)}{\sqrt{n+m+1}}\, \delta\sum_{t\in D_3}\frac{t^{-1/2}(x+t)}{(x-t)^2} \\ \notag &\leqslant \frac{c(\alpha,\lambda,a,b)}{\sqrt{n+m+1}}\sqrt{x}\, \delta\sum_{t\in D_3^1}\frac{1}{(t-x)^2} +\frac{c(\alpha,\lambda,a,b)}{\sqrt{n+m+1}}\, \delta\sum_{t\in D_3^2}\frac{\sqrt{t}}{(t-x)^2} \\ &\leqslant \frac{c(\alpha,\lambda,a,b)}{\sqrt{n+m+1}}\biggl(\sqrt{x} \sqrt{\frac{\theta_{n+m+1}}{x}}+c(\alpha)\delta\sum_{t\in D_3^2}t^{-3/2}\biggr)\leqslant c(\alpha,\lambda,a,b). \end{aligned}
\end{equation}
\tag{3.11}
$$
A similar argument shows that Next we estimate $J_{14}$. Setting
$$
\begin{equation*}
D_4^1=\biggl(\frac{3\theta_{n+m+1}}4,\frac{3\theta_{n+m+1}}2\biggr]\cap\Omega_\delta, \qquad D_4^2=\biggl(\frac{3\theta_{n+m+1}}2,\infty\biggr)\cap\Omega_\delta,
\end{equation*}
\notag
$$
it follows from Lemma 2 that
$$
\begin{equation}
\begin{aligned} \, \notag J_{14} &\leqslant c(\alpha,\lambda,a,b) x^{\alpha/2+1/4}\theta_{n+m+1}^{-(\alpha+1)}A_{n+m+1}^\alpha(x)\delta\sum_{t\in D_4}t^{\alpha/2-1/4} A_{n+m+1}^\alpha(t) \\ &\leqslant c(\alpha,\lambda,a,b)\theta_{n+m+1}^{-\alpha/2-5/4}\delta\biggl(\sum_{t\in D_4^1}\frac{t^{\alpha/2-1/4}} {\theta_{n+m+1}^{1/3}}+\sum_{t\in D_4^2}t^{\alpha/2-1/4}e^{-t/4}\biggr)\leqslant \frac{c(\alpha,\lambda,a,b)}{\theta_{n+m+1}^{5/6}}. \end{aligned}
\end{equation}
\tag{3.13}
$$
A similar estimate holds for $J_2$, and therefore
$$
\begin{equation*}
\Lambda_{n,m}^{\alpha}(x)\leqslant c(\alpha,\lambda,a,b), \qquad x\in\biggl[\frac{2}{\theta_{n}}, \frac{\theta_{n+m+1}}{2}\biggr].
\end{equation*}
\notag
$$
Consider the case $x\in(\theta_{n+m+1}/2, 3\theta_{n+m+1}/2]$. For $J_1$ we have where
$$
\begin{equation*}
\begin{gathered} \, H_i=c(a,b) x^{\alpha/2+1/4}\delta\sum_{t\in G_i}t^{\alpha/2-1/4} e^{-(x+t)/2}|\mathcal K_{n+m+1,N}^\alpha(x,t)|, \qquad i=1,2,3,4,5, \\ G_1=\biggl[0,x-\frac{\theta_{n+m+1}}4\biggr]\cap\Omega_\delta, \qquad G_2=\biggl(x-\frac{\theta_{n+m+1}}4,x-1\biggr)\cap\Omega_\delta, \\ G_3=[x-1,x+1]\cap\Omega_\delta, \qquad G_4=\biggl(x+1,x+\frac{\theta_{n+m+1}}4\biggr)\cap\Omega_\delta \\ \text{and}\quad G_5=\biggl[x+\frac{\theta_{n+m+1}}4,\infty\biggr)\cap\Omega_\delta. \end{gathered}
\end{equation*}
\notag
$$
For $H_3$ estimate (3.10) holds. As concerns, $H_1$ and $H_5$, by Lemma 2 and equality (5.12) we have
$$
\begin{equation*}
\begin{aligned} \, H_1 &\leqslant \frac{c(\alpha,\lambda,a,b)x^{\alpha/2+1/4}} {\theta_{n+m+1}^{\alpha+5/4}(\theta_{n+m+1}^{1/3}+ |x-\theta_{n+m+1}|)^{1/4}}\delta\sum_{t\in G_1}t^{\alpha/2-1/4}A_{n+m+1}^\alpha(t) \\ &\leqslant \frac{c(\alpha,\lambda,a,b)}{\theta_{n+m+1}^{\alpha/2+1}\theta_{n+m+1}^{1/12}} \bigl(\theta_{n+m+1}^{\alpha/2-3/4}+\theta_{n+m+1}^{\alpha/2+1/4}\bigr)\leqslant \frac{c(\alpha,\lambda,a,b)}{\theta_{n+m+1}^{5/6}} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, H_5 &\leqslant \frac{c(\alpha,\lambda,a,b)}{\theta_{n+m+1}^{\alpha/2+13/12}} \delta\sum_{t\in G_5}t^{\alpha/2-1/4}A_{n+m+1}^\alpha(t) \\ &\leqslant \frac{c(\alpha,\lambda,a,b)}{n^{\alpha/2+13/12}} \biggl(\frac{1}{\theta_{n+m+1}^{1/3}}\theta_{n+m+1}^{\alpha/2+3/4}+ \frac{1}{e^{3n/4}}\biggr) \leqslant \frac{c(\alpha,\lambda,a,b)}{\theta_{n+m+1}^{2/3}}. \end{aligned}
\end{equation*}
\notag
$$
In view of Lemma 5, for $H_2$ and $H_4$ we have
$$
\begin{equation*}
\begin{aligned} \, H_2 &\leqslant c(\alpha,\lambda,a,b)x^{\alpha/2+1/4}\delta\sum_{t\in G_2}t^{\alpha/2-1/4} \frac{\theta_{n+m+1}^{-\alpha}}{(x-t)^{3/2}} \\ &\leqslant c(\alpha,\lambda,a,b)\delta\sum_{t\in G_2}(x-t)^{-3/2}\leqslant c(\alpha,\lambda,a,b) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
H_4\leqslant c(\alpha,\lambda,a,b)\delta\sum_{t\in G_4}(t-x)^{-3/2}\leqslant c(\alpha,\lambda,a,b).
\end{equation*}
\notag
$$
From the estimates for $H_i$ and equality (3.14) we obtain Therefore,
$$
\begin{equation*}
\Lambda_{n,m}^\alpha(x)\leqslant c(\alpha,\lambda,a,b), \qquad x\in\biggl(\frac{\theta_{n+m+1}}{2},\frac{3\theta_{n+m+1}}{2}\biggr].
\end{equation*}
\notag
$$
Let $x\in(3\theta_{n+m+1}/2,\infty)$. In this case
$$
\begin{equation*}
\Lambda_{n,m}^{\alpha}(x)\leqslant \frac{x^{\alpha/2+1/4}}{m+1}\, \delta\sum_{t\in \Omega_\delta}t^{\alpha/2-1/4} e^{-(x+t)/2}\sum_{k=n}^{n+m}(K_{k,N}^\alpha(x,x))^{1/2} (K_{k,N}^\alpha(t,t))^{1/2}.
\end{equation*}
\notag
$$
By Lemma 7
$$
\begin{equation*}
\Lambda_{n,m}^{\alpha}(x)\leqslant c(\alpha,\lambda)\frac{x^{\alpha/2+1/4}}{m+1}\, e^{-x/4} \delta\sum_{t\in \Omega_\delta}t^{\alpha/2-1/4}e^{-t/2}\sum_{k=n}^{n+m}k^{1/2-\alpha/2} (K_{k,N}^\alpha(t,t))^{1/2}.
\end{equation*}
\notag
$$
Setting
$$
\begin{equation*}
\begin{gathered} \, E_1=\biggl[0,\frac{1}{\theta_{n}}\biggr]\cap\Omega_\delta, \qquad E_2=\biggl(\frac1{\theta_{n}},\frac{\theta_{n+m}}2\biggr]\cap\Omega_\delta, \\ E_3=\biggl(\frac{\theta_{n+m}}2,\frac{3\theta_{n+m}}2\biggr]\cap\Omega_\delta\quad\text{and} \quad E_4=\biggl(\frac{3\theta_{n+m}}2,\infty\biggr)\cap\Omega_\delta, \end{gathered}
\end{equation*}
\notag
$$
we have
$$
\begin{equation}
\Lambda_{n,m}^{\alpha}(x)\leqslant c(\alpha,\lambda)\frac{x^{\alpha/2+1/4}}{m+1}e^{-x/4}(W_1+W_2+W_3+W_4),
\end{equation}
\tag{3.15}
$$
where
$$
\begin{equation*}
\begin{aligned} \, W_1 &\leqslant c(\alpha,\lambda)\delta\sum_{t\in E_1}t^{\alpha/2-1/4} \sum_{k=n}^{n+m}k^{1-\alpha}\theta_k^\alpha\leqslant c(\alpha,\lambda)\frac{(n+m)^2}{\theta_n^{\alpha/2+3/4}}\leqslant \frac{c(\alpha,\lambda,a,b)}{n^{\alpha/2-5/4}}, \\ W_2 &\leqslant c(\alpha,\lambda)\delta\sum_{t\in E_2} t^{\alpha/2-1/4}\sum_{k=n}^{n+m}k^{1-\alpha}\theta_k^{\alpha/2-1/4} t^{-\alpha/2-1/4}\leqslant \frac{c(\alpha,\lambda,a,b)}{n^{\alpha/2-9/4}}, \\ W_3 &\leqslant c(\alpha,\lambda)\delta\sum_{t\in E_3} t^{\alpha/2-1/4}\sum_{k=n}^{n+m}k^{1/2-\alpha/2}k^{-\alpha/2}\leqslant \frac{c(\alpha,\lambda,a,b)}{n^{\alpha/2-9/4}} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
W_4\leqslant c(\alpha,\lambda)\delta\sum_{t\in E_4} t^{\alpha/2-1/4}\sum_{k=n}^{n+m}k^{1-\alpha}e^{-t/4}\leqslant \frac{c(\alpha,\lambda,a,b)}{n^{\alpha-2}}e^{-3n/4}.
\end{equation*}
\notag
$$
Now from (3.15) and the estimates for the $W_i$ $(i=1,2,3,4)$ we obtain
$$
\begin{equation*}
\Lambda_{n,m}^{\alpha}(x)\leqslant c(\alpha,\lambda,a,b), \qquad x\in \biggl(\frac{3\theta_{n+m+1}}{2},\infty\biggr).
\end{equation*}
\notag
$$
§ 4. Approximation properties of de la Vallée Poussin means for $\alpha=0$Let $q_{n+r}(x)$ be an algebraic polynomial of degree $n+r$ such that $\Delta_\delta^i f(0)=\Delta_\delta^i q_{n+r}(0)$, $i=0,\dots, r-1$. From (1.8) and (1.11) we obtain
$$
\begin{equation*}
f(x)-\mathcal V_{n+m+r}(f,x)=f(x)-q_{n+r}(x)+\mathcal V_{n+m+r}(q_{n+r}-f,x).
\end{equation*}
\notag
$$
Since
$$
\begin{equation*}
\sum_{k=0}^{r-1}\Delta_\delta^k(q_{n+r}(0)-f(0))\frac{(Nx)^{[k]}}{k!}=0,
\end{equation*}
\notag
$$
it follows from (1.6) that
$$
\begin{equation*}
\begin{aligned} \, &\mathcal V_{n+m+r}(q_{n+r}-f,x) =\frac{1}{m+1}\sum_{k=n}^{m+n}S_{k+r,N}^{0,r}(q_{n+r}-f,x) \\ &\qquad=\frac{(Nx)^{[r]}}{m+1}\sum_{k=n}^{n+m}\sum_{j=0}^{k}\frac{m^{r}_{j,N}(x-r\delta)}{\sqrt{(j+r)^{[r]}}} \sum_{t\in\Omega_{\delta}}\Delta_\delta^r(q_{n+r}(t)-f(t))e^{-t}(1-e^{-\delta})m_{j,N}^0(t). \end{aligned}
\end{equation*}
\notag
$$
Applying the Abel transform to the inner sum (and using equalities (5.1) and (5.9)), we get that
$$
\begin{equation*}
\begin{aligned} \, &\sum_{t\in\Omega_{\delta}}\Delta_\delta^r(q_{n+r}(t)-f(t))e^{-t}(1-e^{-\delta})m_{j,N}^0(t) \\ &\qquad =(-1)^r(1-e^{-\delta})\sum_{t\in\Omega_{\delta}}(q_{n+r}(t+r\delta)-f(t+r\delta)) \frac{\sqrt{e^{j\delta}}}{j!}\, \Delta_\delta^{j+r}\biggl\{\frac{\Gamma(Nt+1)e^{-t}}{\Gamma(Nt-j+1)}\biggr\} \\ &\qquad =(-1)^r\frac{1-e^{-\delta}}{\sqrt{e^{j\delta}}}\frac{(j+r)!}{j!}\sum_{t\in\Omega_{r,\delta}} (q_{n+r}(t)-f(t))\frac{\Gamma(Nt-r+1)}{\Gamma(Nt+1)}\, e^{-t}M_{j+r,N}^{-r}(t) \\ &\qquad =\frac{(e^{\delta}-1)^{r+1}}{e^\delta}\sum_{t\in\Omega_{r,\delta}}(q_{n+r}(t)-f(t))e^{-t}\sqrt{(j+r)^{[r]}} \, m_{j,N}^{r}(t-r\delta). \end{aligned}
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation*}
\begin{aligned} \, &\mathcal V_{n+m+r}(q_{n+r}-f,x) \\ &\qquad =\frac{(e^{\delta}-1)^{r+1}}{e^\delta}(Nx)^{[r]}\sum_{t\in\Omega_{r,\delta}}(q_{n+r}(t)-f(t))e^{-y}V_{n+m}^r(x-r\delta,t-r\delta). \end{aligned}
\end{equation*}
\notag
$$
Thus, for $x\in\Omega_{r,\delta}$ we have
$$
\begin{equation*}
\begin{aligned} \, &e^{-x/2}x^{-r/2+1/4}\bigl|f(x)-\mathcal V_{n+m+r}(f,x)\bigr|\leqslant E_{n+r}^r(f,\delta) \\ &\qquad +c(r)E_{n+r}^r(f,\delta)x^{r/2+1/4}\delta\sum_{t\in\Omega_{r,\delta}} t^{r/2-1/4}e^{-(x+t)/2}|V_{n+m}^r(x-r\delta,t-r\delta)|. \end{aligned}
\end{equation*}
\notag
$$
An appeal to (3.1) shows that
$$
\begin{equation*}
e^{-x/2}x^{-r/2+1/4}\bigl|f(x)-\mathcal V_{n+m+r}(f,x)\bigr| \leqslant \bigl(1+c(r)\Lambda_{n,m}^r(x)\bigr)E_{n+r}^r(f,\delta).
\end{equation*}
\notag
$$
Now by Theorem 1 we have the estimate
$$
\begin{equation}
e^{-x/2}x^{-r/2+1/4}\bigl|f(x)-\mathcal V_{n+m+r}(f,x)\bigr|\leqslant c(r,\lambda,a,b)E_{n+r}^r(f,\delta).
\end{equation}
\tag{4.1}
$$
Next, from (1.9) we obtain
$$
\begin{equation*}
\Delta_\delta^{l}\mathcal V_{n+m+r}(f,x)=\mathcal V_{n+m+r-l}(\Delta_\delta^{l}f,x), \qquad l=0,\dots,r-1.
\end{equation*}
\notag
$$
As a result, (1.12) follows from this equality and (4.1).
§ 5. Appendix. Some properties of Meixner polynomialsIn this section we recall some properties of the polynomials $M_{n,N}^{\alpha}(x)$ from [2]: $\bullet$ Rodrigues’s formula
$$
\begin{equation}
M_{n,N}^{\alpha}(x)=\frac{\Gamma(Nx+1)e^{n\delta+x}}{n!\,\Gamma(Nx+\alpha+1)} \Delta^n_\delta\biggl\{\frac{\Gamma(Nx+\alpha+1)} {\Gamma(Nx-n+1)}\, e^{-x}\biggr\};
\end{equation}
\tag{5.1}
$$
$\bullet$ the explicit representation
$$
\begin{equation}
M_{n,N}^\alpha(x)=\binom{n+\alpha}{n}\sum_{k=0}^n\frac{n^{[k]}(Nx)^{[k]}}{(\alpha+1)_kk!}(1-e^\delta)^k;
\end{equation}
\tag{5.2}
$$
$\bullet$ the orthogonality relation
$$
\begin{equation*}
\sum_{x\in\Omega_\delta}M_{n,N}^\alpha(x)M_{m,N}^\alpha(x)\rho_N(x)=h_n^\alpha\delta_{nm}, \qquad \alpha>-1;
\end{equation*}
\notag
$$
$\bullet$ the recurrence relation
$$
\begin{equation}
\begin{aligned} \, \notag M_{n+1,N}^\alpha(x) &=\frac{n(e^{-\delta}+1)+e^{-\delta}(\alpha+1)+(e^{-\delta}-1)Nx}{(n+1)e^{-\delta}}M_{n,N}^\alpha(x) \\ &\qquad -\frac{n+\alpha}{(n+1)e^{-\delta}}M_{n-1,N}^\alpha(x), \end{aligned}
\end{equation}
\tag{5.3}
$$
where $M_{0,N}^\alpha(x)=1$ and $M_{1,N}^\alpha(x)=(1-e^\delta) Nx+\alpha+1$; $\bullet$ the fundamental equalities
$$
\begin{equation}
\Delta_\delta^r M_{n,N}^{\alpha}(x)=(1-e^{\delta})^rM_{n-r,N}^{\alpha+r}(x),
\end{equation}
\tag{5.4}
$$
$$
\begin{equation}
(1-e^{\delta})NxM_{n,N}^{\alpha+1}(x-\delta)=(n+\alpha+1)M_{n,N}^{\alpha}(x)-(n+1)M_{n+1,N}^{\alpha}(x),
\end{equation}
\tag{5.5}
$$
$$
\begin{equation}
M_{n-1,N}^{\alpha}(x)=M_{n,N}^\alpha(x)-M_{n,N}^{\alpha-1}(x),
\end{equation}
\tag{5.6}
$$
$$
\begin{equation}
e^\delta M_{n-2,N}^{\alpha}(x-\delta)=M_{n-1,N}^\alpha(x-\delta)-M_{n-1,N}^{\alpha-1}(x),
\end{equation}
\tag{5.7}
$$
$$
\begin{equation}
M_{n,N}^{\alpha-1}(x)=\frac{\alpha}{n+\alpha}M_{n,N}^\alpha(x) -\frac{(e^\delta-1)Nx}{n+\alpha}M_{n-1,N}^{\alpha+1}(x-\delta)
\end{equation}
\tag{5.8}
$$
and
$$
\begin{equation}
M^{-l}_{n,N}(x)=\frac{(n-l)!}{n!}(e^\delta-1)^l(-Nx)_lM_{n-l,N}^l(x-l\delta), \qquad 1\leqslant l\leqslant n;
\end{equation}
\tag{5.9}
$$
$\bullet$ the Christoffel–Darboux formula
$$
\begin{equation}
\begin{aligned} \, \notag &K_{n,N}^{\alpha}(x,y) =\sum_{k=0}^n m_{k,N}^{\alpha}(x)m_{k,N}^{\alpha}(y) \\ &\quad=\frac{\delta}{(e^{\delta}-1)e^{n\delta}}\frac{(n+1)!}{\Gamma(n+\alpha+1)} \frac{M_{n,N}^\alpha(x)M_{n+1,N}^\alpha(y)-M_{n+1,N}^\alpha(x)M_{n,N}^\alpha(y)}{x-y}. \end{aligned}
\end{equation}
\tag{5.10}
$$
For some asymptotic properties of the polynomials $m_{n,N}^\alpha(x)$, see [2], p. 197. In particular, the following asymptotic formula holds for $\alpha>-1$ and $x\in[0,\infty)$:
$$
\begin{equation}
m_{n,N}^\alpha(x)=l_n^\alpha(x)+\upsilon_{n,N}^\alpha(x),
\end{equation}
\tag{5.11}
$$
where $l_n^\alpha(x)$ is the normalized Laguerre polynomial of degree $n$. For the remainder term $\upsilon_{n,N}^\alpha(x)$, for $n/N\leqslant \lambda$ we have the estimate
$$
\begin{equation*}
e^{-x/2}|\upsilon_{n,N}^\alpha(x)|\leqslant c(\alpha,\lambda)A_n^\alpha(x)\sqrt{\frac{n}{N}}n^{-\alpha/2},
\end{equation*}
\notag
$$
where
$$
\begin{equation}
A_n^\alpha(x)=\begin{cases} \theta_n^{\alpha},& 0\leqslant x\leqslant \dfrac{1}{\theta_n}, \\ \theta_n^{\alpha/2-1/4}x^{-\alpha/2-1/4},& \dfrac{1}{\theta_n}<x\leqslant \dfrac{\theta_n}{2}, \\ \bigl[\theta_n(\theta_n^{1/3}+|x-\theta_n|)\bigr]^{-1/4},& \dfrac{\theta_n}{2}<x\leqslant\dfrac{3\theta_n}{2}, \\ e^{-x/4}, & \dfrac{3\theta_n}{2}<x<\infty, \end{cases}
\end{equation}
\tag{5.12}
$$
and $\theta_n=4n+2\alpha+2$. The weighed estimates
$$
\begin{equation}
e^{-x/2}\bigl|m_{n,N}^\alpha(x\pm s\delta)\bigr|\leqslant c(\alpha,\lambda,s)\theta_n^{-\alpha/2}A_n^\alpha(x)
\end{equation}
\tag{5.13}
$$
and
$$
\begin{equation}
e^{-x/2}\bigl|M_{n,N}^\alpha(x\pm s\delta)\bigr|\leqslant c(\alpha,\lambda,s)A_n^\alpha(x),
\end{equation}
\tag{5.14}
$$
where $s\geqslant0$, were obtained in [2] on the basis of an estimate for Laguerre polynomials and the asymptotic formula (5.11). The following results hold. Lemma 6 [30]. Let $0\leqslant l$ be an integer, and let $r\in\mathbb{N}$, $l\leqslant r$. Then Lemma 7 [32]. Let $-1<\alpha\in\mathbb{R}$, $\theta_n=4n+2\alpha+2$, $\lambda>0$ and $N=1/\delta$, $0<\delta\leqslant1$. Then, for $1\leqslant n\leqslant \lambda N$, AcknowledgementThe author is grateful to the referee for their useful comments and suggestions. 
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