Convergence of Szász-Mirakyan-Durrmeyer operators having Laguerre-type weight

TL;DR

The paper extends Szász–Mirakyan–Durrmeyer operators to the unbounded half-line by introducing Szász–Mirakyan–Laguerre–Durrmeyer operators $M_n^{(\alpha,\beta)}$ with a Laguerre-type weight $\omega_{\alpha,\beta}$, enabling weighted $L_p$ convergence on $[0,\infty)$. It derives explicit closed-form moments using confluent hypergeometric functions ${}_1F_1$, establishes a differential recurrence for moments, and provides asymptotic expansions for central moments, capturing the operator’s behavior as $n$ grows. The work further delivers quantitative convergence on compact intervals via modulus of continuity, and Korovkin-type weighted convergence for unbounded functions, along with $L_p$-convergence results on both local and global scales, including kernel representations and eigenstructure considerations. Together, these results offer a comprehensive Laguerre-weighted framework that extends classical approximation theory to unbounded domains with explicit formulas and convergence guarantees.

Abstract

In this paper, we introduce a new family of Szasz-Mirakyan-Durrmeyer operators defined on the half-line [0,\infty), constructed using Laguerre-type kernels. We discuss in detail the algebraic structure and analytical properties of these operators. thoroughly investigated. Explicit closed-form expressions for the moments are derived, along with a differential recurrence relation connecting successive moments. Quantitative estimates on compact intervals are obtained, and Weighted approximation results are provided for unbounded functions. Furthermore, the asymptotic behavior of the central moment is analyzed. We establish both local and global $L_p$-convergence results and identify the eigenfunctions associated with these operators. These findings demonstrate the effectiveness of the proposed generalized operators in extending classical approximation results to the unbounded domain.

Theorems & Definitions (5)

• Theorem 1
• proof
• Theorem 2: Quantitative estimate on compact intervals
• proof
• Theorem 3: Korovkin-type convergence in $C_\varphi([0,\infty))$] Let $n>\beta$ and $\alpha>-1$. Then for every $f\in C_\varphi([0,\infty))$, $\lim_{n\to\infty}\|M_n^{(\alpha,\beta)}f-f\|_\varphi=0.$ We verify the Korovkin test functions $1,t,t^2$. We note that $M_n^{(\alpha,\beta)}1(x)=\sum_{k=0}^\infty\psi_{n,k}(x)=1\qquad(\forall x\ge0).$ Thus $\|M_n^{(\alpha,\beta)}1-1\|_\varphi=0$. Also, $M_n^{(\alpha,\beta)} t(x)=\frac{n x + (\alpha+1)}{\,n-\beta\,}\,.$ Therefore $M_n^{(\alpha,\beta)} t(x) - x = \frac{\beta x + \alpha+1}{n-\beta},$ and dividing by $1+x^2$ gives $\frac{|M_n^{(\alpha,\beta)} t(x)-x|}{1+x^2} \le \frac{1}{n-\beta}\sup_{y\ge0}\frac{|\beta y+\alpha+1|}{1+y^2} = \frac{C_1(\alpha,\beta)}{n-\beta},$ where $C_1(\alpha,\beta)<\infty$. Thus $\|M_n^{(\alpha,\beta)} t-t\|_\varphi\to0$. Recall $M_n^{(\alpha,\beta)} t^2(x)=\frac{n^2x^2 + n x(2\alpha+4) + (\alpha+1)(\alpha+2)}{(n-\beta)^2}.$ Compute the difference $M_n^{(\alpha,\beta)} t^2(x)-x^2= x^2(\frac{n^2}{(n-\beta)^2}-1) + \frac{n x(2\alpha+4)}{(n-\beta)^2} + \frac{(\alpha+1)(\alpha+2)}{(n-\beta)^2}.$ Thus there exists a finite constant $C_2(\alpha,\beta)$ with $\frac{|M_n^{(\alpha,\beta)} t^2(x)-x^2|}{1+x^2}\le \frac{C_2(\alpha,\beta)}{n-\beta},$ for all $x\ge0$. Hence $\|M_n^{(\alpha,\beta)} t^2 - t^2\|_\varphi\to0$. By the weighted Korovkin theorem (positivity, linearity, and approximation of the three test functions), $\|M_n^{(\alpha,\beta)} f - f\|_\varphi\to0$ for every $f\in C_\varphi([0,\infty))$. In this section, we examine the convergence properties of the Szász–Mirakyan–Durrmeyer operators in the $L_p$-metric, both locally and globally on the half-line $[0,\infty)$. The analysis focuses on establishing sufficient conditions under which the operators provide approximation in $L_p$-spaces for $1\leq p < \infty$. Local convergence is studied on compact intervals, while global convergence is obtained through appropriate weight functions ensuring integrability over the unbounded domain. These results extend the classical $L_p$-approximation theory to the present class of operators and demonstrate their robustness in handling functions of varying growth behavior. For all $f \in C(K)$, $\lim_{n \to \infty} \| M_n^{(\alpha,\beta)}[f] - f \|_{C(K)} = 0,$ i.e., $M_n^{(\alpha,\beta)}[f] \to f$ uniformly on every compact subset $K$ of $[0,\infty)$. We follow the classical Korovkin approach by verifying the convergence on the test functions $\{1, t, t^2\}$. From the moment calculations, we have $M_n^{(\alpha,\beta)}[1](x) = 1.$ As $n \to \infty$, we have uniformly on compact sets $M_n^{(\alpha,\beta)}[t](x)=\frac{nx + \alpha + 1}{n - \beta} \to x.$ and $M_n^{(\alpha,\beta)}[t^2](x) =\frac{n^2x^2 + 2n(\alpha+1)x + (\alpha+1)(\alpha+2)}{(n-\beta)^2} \to x^2.$ Since $M_n^{(\alpha,\beta)}$ is a sequence of positive linear operators and the convergence holds for $1, t, t^2$, the classical Korovkin theorem yields $\lim_{n \to \infty} \| M_n^{(\alpha,\beta_n)}[f] - f \|_{C(K)} = 0$ This completes the proof. Under the same conditions as Theorem \ref{['thm:korovkin']}, the operators $M_n^{(\alpha,\beta)}$ approximate continuous functions with exponential weights. That is, for any $f \in C([0,\infty))$ satisfying a growth condition $|f(t)| \leq Me^{At}$ for some $M,A > 0$, we have uniform convergence on compact subsets of $[0,\infty)$. In the study of approximation by positive linear operators on unbounded intervals, convergence may be investigated in different functional settings. In particular, for the operators $M_n^{(\alpha,\beta)}$, it is natural to distinguish between two complementary notions of convergence in the $L_p$-sense. The first concerns the local (or compact) $L_p$-convergence, which ensures that the approximation holds uniformly over every finite subinterval of $[0,\infty)$. The second involves the global $L_p$-convergence in weighted spaces equipped with an exponentially decaying weight, which controls the behavior of functions and operators at infinity. The results presented below address both these aspects, providing a comprehensive analysis of the approximation properties of the Szász--Mirakyan--Laguerre--Durrmeyer operators. Let $\beta\ge0$ be fixed and let $\alpha\in[-\tfrac{1}{2},0]$. For $n>\beta$ and $t\ge0$ define $E_n(t):=\frac{1}{n}(n-\beta)^{\alpha+1}t^{\alpha}\frac{\gamma(\alpha+1,(n-\beta)t)}{\Gamma(\alpha+1)},$ where $\gamma(s,z)=\int_0^z u^{s-1}e^{-u}\,du$ is the lower incomplete Gamma function. Then for every fixed $R>0$ there exists a finite constant $C=C(R,\alpha,\beta)$ such that $\sup_{n>\beta}\sup_{t\in[0,R]} E_n(t)\le C.$ Set $z=(n-\beta)t$ and define $g(z):=\frac{z^{\alpha}\,\gamma(\alpha+1,z)}{\Gamma(\alpha+1)},\qquad z\ge0.$ A simple rearrangement gives $E_n(t)=(1-\frac{\beta}{n})g(z).$ Since $0<1-\beta/n\le1$ for all $n>\beta$, it suffices to show $\sup_{z\ge0} g(z)<\infty$. First consider the behavior as $z\to0^+$. Put $s=\alpha+1\in(\tfrac{1}{2},1]$. The small-argument expansion $\gamma(s,z)=\frac{z^{s}}{s}+o(z^{s})\qquad(z\to0^+)$ yields $g(z)=\frac{z^{\alpha}\gamma(\alpha+1,z)}{\Gamma(\alpha+1)} = \frac{z^{2\alpha+1}}{(\alpha+1)\Gamma(\alpha+1)}+o(z^{2\alpha+1}).$ For $\alpha\ge-1/2$ we have $2\alpha+1\ge0$, so $g(z)$ is finite at $z=0$ (indeed $g(0)=0$ when $\alpha>-1/2$, and $g(0)$ is finite when $\alpha=-1/2$). Next consider the behavior as $z\to\infty$. Since $\gamma(\alpha+1,z)\to\Gamma(\alpha+1)$, we have $g(z)\sim z^{\alpha}\qquad(z\to\infty).$ Because $\alpha\le0$ the right-hand side is bounded as $z\to\infty$ (it tends to $0$ if $\alpha<0$ and to $1$ if $\alpha=0$). Therefore $g$ is bounded at infinity. On $(0,\infty)$ the function $g$ is continuous, hence the finiteness at both endpoints implies $g$ is bounded on $[0,\infty)$. Set $C_0:=\sup_{z\ge0} g(z) <\infty.$ Then for every $n>\beta$ and every $t\in[0,R]$, $E_n(t)=(1-\frac{\beta}{n})g(z)\le g(z)\le C_0,$ so the claimed uniform bound holds with $C(R,\alpha,\beta):=C_0$. Let $1 \le p < \infty$, $-\frac{1}{2}\leq \alpha\leq 0$, and let $\beta \ge 0$ be a fixed real number such that $n > \beta$. Then for every fixed $R > 0$ and every $f \in L_p([0,R])$, $\lim_{n \to \infty} \| M_n^{(\alpha,\beta)}[f] - f \|_{L_p([0,R])} = 0.$ Each $M_n^{(\alpha,\beta)}$ is a positive linear operator. By interchanging sum and integral (justified by Fubini's theorem and non-negativity), we obtain the kernel representation $M_n^{(\alpha,\beta)}[f](x) = \int_0^\infty f(t) K_n(x,t) dt,$ where the kernel is given by $K_n(x,t) = e^{-nx} \sum_{k=0}^{\infty} \frac{(n - \beta)^{k + \alpha + 1}}{\Gamma(k + \alpha + 1)} t^{k+\alpha} e^{-(n - \beta)t} \frac{(nx)^k}{k!}.$ The kernel is non-negative for $x, t \ge 0$. Moreover, for $f \equiv 1$, we have $M_n^{(\alpha,\beta)}[1](x) = 1 \quad \text{for all } x \ge 0,$ which implies that for each fixed $x \ge 0$, $\int_0^\infty K_n(x,t) dt = 1.$ Thus, $K_n(x, \cdot)$ is a probability density on $[0, \infty)$. Using theorem \ref{['thm:korovkin']}, for every $g \in C([0,R])$, $\lim_{n \to \infty} \| M_n^{(\alpha,\beta)}[g] - g \|_{\infty,[0,R]} = 0.$ Let $f \in L_p([0,R])$ and $\varepsilon > 0$. Since $C([0,R])$ is dense in $L_p([0,R])$, there exists $g \in C([0,R])$ such that $\| f - g \|_{L_p([0,R])} < \varepsilon.$ We decompose the error as: $\| M_n^{(\alpha,\beta)}[f] - f \|_{L_p([0,R])} \le \| M_n^{(\alpha,\beta)}[f - g] \|_{L_p([0,R])} + \| M_n^{(\alpha,\beta)}[g] - g \|_{L_p([0,R])} + \| g - f \|_{L_p([0,R])}.$ The third term is less than $\varepsilon$ by construction. The second term satisfies: $\| M_n^{(\alpha,\beta)}[g] - g \|_{L_p([0,R])} \le R^{1/p} \| M_n^{(\alpha,\beta)}[g] - g \|_{\infty,[0,R]} \to 0 \quad \text{as } n \to \infty,$ since uniform convergence implies convergence in $L_p$ on bounded intervals. It remains to bound the first term. We claim there exists a constant $C = C(p,R,\alpha,\beta) > 0$, independent of $n$ for large $n$, such that for all $h \in L_p([0,R])$, $\| M_n^{(\alpha,\beta)}[h] \|_{L_p([0,R])} \le C \| h \|_{L_p([0,R])}.$ To prove this, we use the Schur test zhu2007operator for integral operators . Consider the kernel $K_n(x,t)$ restricted to $x, t \in [0,R]$. We verify: $\sup_{x \in [0,R]} \int_0^R K_n(x,t) dt \le 1$ (since $\int_0^\infty K_n(x,t) dt = 1$ and $K_n \ge 0$).$\sup_{t \in [0,R]} \int_0^R K_n(x,t) dx \le C$ for some constant $C$ independent of $n$. The first condition is immediate. For the second condition, we analyze $I_n(t) = \int_0^R K_n(x,t) dx = \int_0^R e^{-nx} \sum_{k=0}^{\infty} \frac{(n - \beta)^{k + \alpha + 1}}{\Gamma(k + \alpha + 1)} t^{k+\alpha} e^{-(n - \beta)t} \frac{(nx)^k}{k!} dx.$ Interchanging sum and integral (justified by Tonelli's theorem): $I_n(t) = \sum_{k=0}^{\infty} \frac{(n - \beta)^{k + \alpha + 1}}{\Gamma(k + \alpha + 1)} t^{k+\alpha} e^{-(n - \beta)t} \frac{n^k}{k!} \int_0^R x^k e^{-nx} dx.$ Using the bound $\int_0^R x^k e^{-nx} dx \le \int_0^\infty x^k e^{-nx} dx = \frac{k!}{n^{k+1}}$, we obtain: $I_n(t) \le \sum_{k=0}^{\infty} \frac{(n - \beta)^{k + \alpha + 1}}{\Gamma(k + \alpha + 1)} t^{k+\alpha} e^{-(n - \beta)t} \frac{1}{n^{k+1}} n^k = \frac{1}{n} \sum_{k=0}^{\infty} \frac{(n - \beta)^{k + \alpha + 1}}{\Gamma(k + \alpha + 1)} t^{k+\alpha} e^{-(n - \beta)t}.$ Now, observe that the series can be expressed using the confluent hypergeometric function: $\sum_{k=0}^{\infty} \frac{(n - \beta)^{k + \alpha + 1}}{\Gamma(k + \alpha + 1)} t^{k+\alpha} e^{-(n - \beta)t} = (n - \beta)^{\alpha+1} t^\alpha e^{-(n - \beta)t} \sum_{k=0}^{\infty} \frac{[(n - \beta)t]^k}{\Gamma(k + \alpha + 1)}.$ Using the identity $\sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k + \alpha + 1)} = e^z \frac{\gamma(\alpha+1, z)}{\Gamma(\alpha+1)}$, where $\gamma(\alpha+1, z)$ is the lower incomplete Gamma function, we get: $I_n(t) \le \frac{1}{n} (n - \beta)^{\alpha+1} t^\alpha e^{-(n - \beta)t} \cdot e^{(n - \beta)t} \frac{\gamma(\alpha+1, (n - \beta)t)}{\Gamma(\alpha+1)} = \frac{1}{n} (n - \beta)^{\alpha+1} t^\alpha \frac{\gamma(\alpha+1, (n - \beta)t)}{\Gamma(\alpha+1)}.$ Using Lemma \ref{['lem:E_n_bound']}, there exists a constant $C = C(R,\alpha,\beta)$ such that $\sup_{t \in [0,R]} I_n(t) \le C \quad \text{for all } n > \beta.$ Thus, by the Schur test zhu2007operator, the operators $M_n^{(\alpha,\beta)}$ are uniformly bounded on $L_p([0,R])$: $\| M_n^{(\alpha,\beta)}[h] \|_{L_p([0,R])} \le C \| h \|_{L_p([0,R])}.$ Returning to the error decomposition, we have for large $n$: $\| M_n^{(\alpha,\beta)}[f] - f \|_{L_p([0,R])} \le C \varepsilon + o(1) + \varepsilon,$ where $o(1) \to 0$ as $n \to \infty$. Since $\varepsilon > 0$ is arbitrary, we conclude that $\lim_{n \to \infty} \| M_n^{(\alpha,\beta)}[f] - f \|_{L_p([0,R])} = 0.$ We now introduce a weighted $L_p$--space suitable for analyzing the global behavior of the operators $M_n^{(\alpha,\beta)}$. Let $1 \le p < \infty$ and $\gamma \ge 0$. The weighted space $L_p^\gamma([0,\infty))$ consists of all measurable functions $f : [0,\infty) \to \mathbb{R}$ such that $\|f\|_{L_p^\gamma}^p := \int_0^\infty |f(x)|^p e^{\gamma x}\,dx < \infty.$ The factor $e^{\gamma x}$ serves as an exponential weight, emphasizing the behavior of $f$ on the unbounded interval $[0,\infty)$. For $\gamma = 0$, the space $L_p^\gamma([0,\infty))$ coincides with the usual $L_p([0,\infty))$ space. Let $1\le p<\infty$, let $\beta\ge0$ be fixed, and assume $\alpha\in [-\tfrac{1}{2},0].$ Fix $\gamma\ge0$ satisfying the condition $\gamma \le p\beta.$ (If $\beta=0$ this forces $\gamma=0$, i.e. no exponential weight.) Then for all sufficiently large $n>\beta$ the operators $M_n^{(\alpha,\beta)}$ map $L_p^\gamma([0,\infty))$ into itself with a uniform bound $\|M_n^{(\alpha,\beta)}\|_{L_p^\gamma\to L_p^\gamma}\le C,$ where $C=C(\alpha,\beta,p,\gamma)$ is independent of $n$ (for all large $n$). Moreover for every $f\in L_p^\gamma([0,\infty))$, $\lim_{n\to\infty}\|M_n^{(\alpha,\beta)}[f]-f\|_{L_p^\gamma}=0.$ Write the kernel representation $M_n^{(\alpha,\beta)}[f](x)=\int_0^\infty K_n(x,t)\,f(t)\,dt,$ with $K_n(x,t)=e^{-nx}\sum_{k=0}^\infty\frac{(n-\beta)^{k+\alpha+1}}{\Gamma(k+\alpha+1)} t^{k+\alpha}e^{-(n-\beta)t}\frac{(nx)^k}{k!},$ a nonnegative kernel and $\int_0^\infty K_n(x,t)\,dt=1$ for each fixed $x$. We conjugate by the weight as usual. For $h(x)=e^{\gamma x/p}f(x)$ define $\widetilde{M}_n[h](x):=e^{\gamma x/p}M_n[e^{-\gamma\cdot/p}h(\cdot)](x) =\int_0^\infty \widetilde{K}_n(x,t)\,h(t)\,dt,$ with $\widetilde{K}_n(x,t):=e^{\gamma x/p}K_n(x,t)e^{-\gamma t/p}.$ Then $\|M_n[f]\|_{L_p^\gamma}=\|\widetilde{M}_n[h]\|_{L_p}$. Thus it suffices to show $\widetilde{M}_n$ is uniformly bounded on ordinary $L_p([0,\infty))$ and that $\widetilde{M}_n[h]\to h$ in $L_p$ for $h$ in the dense subspace corresponding to continuous compactly supported $f$. We apply the Schur test. It is enough to find constants $A,B$ independent of $n$ (for all large $n$) such that $\sup_{x\ge0}\int_0^\infty \widetilde{K}_n(x,t)\,dt \le A,\qquad \sup_{t\ge0}\int_0^\infty \widetilde{K}_n(x,t)\,dx \le B.$ Then $\|\widetilde{M}_n\|_{L_p\to L_p}\le A^{1/q}B^{1/p}$ with $1/p+1/q=1$. Estimate of the first Schur integral (choice of $A$). For fixed $x$, $\int_0^\infty \widetilde{K}_n(x,t)\,dt= e^{\gamma x/p}\int_0^\infty K_n(x,t)e^{-\gamma t/p}\,dt= e^{\gamma x/p}e^{-nx}\sum_{k=0}^\infty\frac{(n-\beta)^{k+\alpha+1}}{\Gamma(k+\alpha+1)}\frac{(nx)^k}{k!} \int_0^\infty t^{k+\alpha}e^{-(n-\beta+\gamma/p)t}\,dt.$ Evaluating the inner Gamma-integral, $\int_0^\infty t^{k+\alpha}e^{-(n-\beta+\gamma/p)t}\,dt =\frac{\Gamma(k+\alpha+1)}{(n-\beta+\gamma/p)^{k+\alpha+1}},$ and thus (after cancellations) $\int_0^\infty \widetilde{K}_n(x,t)\,dt = (\frac{n-\beta}{\,n-\beta+\gamma/p\,})^{\!\alpha+1} e^{\gamma x/p}e^{-nx}\sum_{k=0}^\infty\frac{(nx)^k}{k!}r_n^k,$ where $r_n:=\dfrac{n-\beta}{\,n-\beta+\gamma/p\,}\in(0,1]$. Summing the exponential series gives $\int_0^\infty \widetilde{K}_n(x,t)\,dt = (\frac{n-\beta}{\,n-\beta+\gamma/p\,})^{\!\alpha+1} \exp\!(\frac{\gamma x}{p}+nx(r_n-1)).$ Note that $r_n-1=-\frac{\gamma/p}{\,n-\beta+\gamma/p\,},$ so the exponent simplifies to $\frac{\gamma x}{p}+nx(r_n-1)=x\cdot\frac{\gamma}{p}\cdot\frac{-\beta+\gamma/p}{\,n-\beta+\gamma/p\,}.$ Under hypothesis (H$_\gamma$), $\gamma\le p\beta$, the factor $-\beta+\gamma/p\le0$, hence the exponential factor is bounded by $1$ for all $x\ge0$. Also $(\tfrac{n-\beta}{n-\beta+\gamma/p})^{\alpha+1}\le1$ because $\alpha+1>0$ and the denominator is at least the numerator. Consequently $\int_0^\infty \widetilde{K}_n(x,t)\,dt \le 1\qquad\text{for all }x\ge0,\; n>\beta,$ and we may take $A=1$. Estimate of the second Schur integral (choice of $B$). We must bound $\int_0^\infty \widetilde{K}_n(x,t)\,dx = e^{-\gamma t/p}\int_0^\infty e^{\gamma x/p}K_n(x,t)\,dx.$ Interchange sum and integral (Tonelli) and compute the $x$-integral termwise. Using the identity (put $u=nx$) $\int_0^\infty e^{\gamma x/p}\frac{(nx)^k}{k!}e^{-nx}\,dx =\frac{1}{n}\int_0^\infty e^{-(1-\gamma/(np))u}\frac{u^k}{k!}\,du =\frac{1}{n}(1-\tfrac{\gamma}{np})^{-(k+1)},$ which is finite for all sufficiently large $n$ provided $\gamma<np$. (For fixed $\gamma$ this holds for all large $n$.) Hence $\int_0^\infty \widetilde{K}_n(x,t)\,dx= e^{-\gamma t/p}\sum_{k=0}^\infty\frac{(n-\beta)^{k+\alpha+1}}{\Gamma(k+\alpha+1)}t^{k+\alpha}e^{-(n-\beta)t}\cdot\frac{1}{n}(1-\tfrac{\gamma}{np})^{-(k+1)}= \frac{1}{n}(n-\beta)^{\alpha+1}t^\alpha e^{-(n-\beta)t}(1-\tfrac{\gamma}{np})^{-1} \sum_{k=0}^\infty\frac{((n-\beta)t(1-\tfrac{\gamma}{np})^{-1})^k}{\Gamma(k+\alpha+1)}.$ As in the local analysis the series is identified with the incomplete Gamma factor and we obtain $\int_0^\infty \widetilde{K}_n(x,t)\,dx \le (1-\tfrac{\gamma}{np})^{-1}\;E_n\!(t(1-\tfrac{\gamma}{np})^{-1}),$ where $E_n(u)=\frac{1}{n}(n-\beta)^{\alpha+1}u^{\alpha}\frac{\gamma(\alpha+1,(n-\beta)u)}{\Gamma(\alpha+1)}.$ By Lemma \ref{['lem:E_n_bound']} (which requires $\alpha\in[-\tfrac{1}{2},0]$) the function $u\mapsto E_n(u)$ is uniformly bounded in $n$ and $u\ge0$. Therefore there exists $C_1=C_1(\alpha,\beta)>0$ such that for all sufficiently large $n$ $\sup_{t\ge0}\int_0^\infty \widetilde{K}_n(x,t)\,dx \le (1-\tfrac{\gamma}{np})^{-1}C_1 \le C,$ with $C$ independent of $n$ (for large $n$). Thus we may take $B=C$. Conclusion of boundedness and convergence. With $A=1$ and the above $B$ we obtain a uniform Schur bound $\|\widetilde{M}_n\|_{L_p\to L_p}\le B^{1/p}$, hence $\|M_n\|_{L_p^\gamma\to L_p^\gamma}\le B^{1/p},$ for all sufficiently large $n$. This proves the uniform operator bound. To prove convergence, fix $f\in L_p^\gamma$. For $R>0$ write $f=f_1+f_2$ with $f_1=f\chi_{[0,R]}$ and $f_2=f\chi_{(R,\infty)}$. Given $\varepsilon>0$ choose $R$ so large that $\|f_2\|_{L_p^\gamma}<\varepsilon$. Then by the uniform operator bound $\|M_n f_2\|_{L_p^\gamma}\le B^{1/p}\varepsilon$. On the compact piece $f_1\in L_p([0,R])$ and Theorem \ref{['thm:local-Lp-fixed-beta']} gives $\|M_n f_1-f_1\|_{L_p^\gamma}\to0$ as $n\to\infty$ (weights are bounded on $[0,R]$). Combining these yields $\limsup_{n\to\infty}\|M_n f-f\|_{L_p^\gamma}\le (1+B^{1/p})\varepsilon,$ and since $\varepsilon>0$ is arbitrary the theorem follows. In this section, we study the spectral properties of the operators $M_n^{(\alpha,\beta)}$, focusing on the characterization of their eigenvalues and associated eigenfunctions.. Let $n>\beta$ and $\alpha>-1$. Define the operator $M_n^{(\alpha,\beta)}$ as defined in \ref{['SMoperators']}. Define the infinite matrix $P=(P_{k,j})_{k,j\ge0}$ by $P_{k,j} =\frac{(n-\beta)^{k+\alpha+1}n^j}{\Gamma(k+\alpha+1)j!}\, \frac{\Gamma(j+k+\alpha+1)}{(2n-\beta)^{\,j+k+\alpha+1}} =(\frac{n-\beta}{2n-\beta})^{k+\alpha+1}(\frac{n}{2n-\beta})^j \frac{(k+\alpha+1)_j}{j!}.$ Then $P$ is nonnegative and row-stochastic, and the operator $M_n^{(\alpha,\beta)}$ and $P$ are related as follows: if $v=(v_j)_{j\ge0}$ is any coefficient vector for which the Poisson series $\Phi_v(x):=\sum_{j\ge0} v_j\,\psi_{n,j}(x)$ converges, then $M_n^{(\alpha,\beta)}[\Phi_v]=\Phi_{Pv}.$ Consequently the following eigenpairs hold for $M_n^{(\alpha,\beta)}$: $\lambda_1=1$ is an eigenvalue with eigenfunction $\phi_1(x)\equiv1$.$\lambda_2=(1-\tfrac{\beta}{n})^{\alpha+1}$ is an eigenvalue with eigenfunction $\phi_2(x)=e^{-\beta x}$. More precisely, the vectors $v^{(1)}=(1,1,1,\dots)$ and $v^{(2)}=(1-\tfrac{\beta}{n})^j_{j\ge0}$ satisfy $Pv^{(1)}=v^{(1)}$ and $Pv^{(2)}=\lambda_2 v^{(2)}$, and lifting these vectors via $\Phi$ yields the stated eigenfunctions of $M_n^{(\alpha,\beta)}$. First, we prove that, one can interchange the sum and the integral in $\int_0^\infty(\sum_{j\ge0} v_j\psi_{n,j}(t))\,t^{k+\alpha}e^{-(n-\beta)t}\,dt =\sum_{j\ge0} v_j\int_0^\infty \psi_{n,j}(t)\,t^{k+\alpha}e^{-(n-\beta)t}\,dt,$ using the Fubini--Tonelli theorem (Folland folland1999real or Royden--Fitzpatrick royden1988real). Two convenient sufficient hypotheses are: (Nonnegative coefficients) If $v_j\ge0$ for all $j$, then the integrand $\sum_{j\ge0} v_j\psi_{n,j}(t)\,t^{k+\alpha}e^{-(n-\beta)t}$ is nonnegative and Tonelli's theorem applies, permitting the interchange.(Bounded coefficients) If $\sup_{j\ge0}|v_j|=:C<\infty$, then for every $t\ge0$ $|\sum_{j\ge0} v_j\psi_{n,j}(t)|\le C\sum_{j\ge0}\psi_{n,j}(t)=C.$ Hence $|\sum_{j\ge0} v_j\psi_{n,j}(t)\,t^{k+\alpha}e^{-(n-\beta)t}| \le C\,t^{k+\alpha}e^{-(n-\beta)t},$ and the right-hand side is integrable on $[0,\infty)$ because $\int_0^\infty t^{k+\alpha}e^{-(n-\beta)t}\,dt<\infty$ (recall $n>\beta$, $\alpha>-1$). Thus by the dominated convergence theorem (or Fubini's theorem for absolutely integrable integrands) we may swap the sum and integral. Expand an arbitrary function $\Phi_v(x)=\sum_{j\ge0}v_j\psi_{n,j}(x)$ (with coefficients such that the series converges). Using above argument and the definition of $M_n^{(\alpha,\beta)}$ we compute $M_n^{(\alpha,\beta)}[\Phi_v](x)=\sum_{k\ge0}\left(\frac{(n-\beta)^{k+\alpha+1}}{\Gamma(k+\alpha+1)} \int_0^\infty(\sum_{j\ge0} v_j\psi_{n,j}(t))\,t^{k+\alpha}e^{-(n-\beta)t}\,dt\right)\psi_{n,k}(x)=\sum_{k\ge0}(\sum_{j\ge0}P_{k,j}v_j)\psi_{n,k}(x) =\Phi_{Pv}(x),$ where $P_{k,j}=\frac{(n-\beta)^{k+\alpha+1}}{\Gamma(k+\alpha+1)}\cdot\frac{n^j}{j!} \int_0^\infty t^{j+k+\alpha} e^{-(2n-\beta)t}\,dt,$ and evaluation of the Gamma integral yields the above formula for $P_{k,j}$. Summing the series for $\sum_j P_{k,j}$ (or invoking $\sum_j\psi_{n,j}(t)=1$) shows each row sums to $1$, hence $P$ is row-stochastic and nonnegative. For the constant eigenfunction, take $v^{(1)}=(1,1,\dots)$. Since each row of $P$ sums to $1$ we have $Pv^{(1)}=v^{(1)}$. Lifting gives $\Phi_{v^{(1)}}(x)=\sum_{j\ge0}\psi_{n,j}(x)=1$ and therefore $M_n^{(\alpha,\beta)}[1]=1$. For the second eigenpair consider the geometric vector $v^{(2)}$ with $v^{(2)}_j=z^j$ and seek $z$ such that $Pv^{(2)}=\lambda v^{(2)}$. A direct summation (using the Pochhammer/binomial series) $\sum_{j\ge0}\frac{(k+\alpha+1)_j}{j!}a^j=(1-a)^{-(k+\alpha+1)},\qquad |a|<1,$ with $a=\dfrac{n z}{2n-\beta}$, yields $(Pv^{(2)})_k=(\frac{n-\beta}{2n-\beta-n z})^{k+\alpha+1}.$ Equating this to $\lambda z^k$ for all $k$ forces $z$ to satisfy $z=\frac{n-\beta}{2n-\beta-n z},$ whose solutions are $z=1$ and $z=1-\dfrac{\beta}{n}$. The $z=1$ case reproduces the constant eigenvector. For $z=1-\beta/n$ the resulting eigenvalue is $\lambda_2=(\frac{n-\beta}{2n-\beta-n z})^{\alpha+1} =(\frac{n-\beta}{n})^{\alpha+1}=(1-\frac{\beta}{n})^{\alpha+1}.$ Lifting $v^{(2)}$ with $z=1-\beta/n$ gives $\Phi_{v^{(2)}}(x)=\sum_{j\ge0}(1-\frac{\beta}{n})^j\psi_{n,j}(x) =e^{-n x}\sum_{j\ge0}\frac{(n x (1-\beta/n))^j}{j!}=e^{-\beta x},$ so $e^{-\beta x}$ is an eigenfunction of $M_n^{(\alpha,\beta)}$ with eigenvalue $\lambda_2$. This completes the proof. The operator $M_n^{(\alpha,\beta)}$ is positive and linear. The constant eigenfunction corresponds to the invariant property of $M_n^{(\alpha,\beta)}$. The second eigenvalue $\lambda_2 = \left(1 - \frac{\beta}{n}\right)^{\alpha + 1}$ satisfies $0 < \lambda_2 < 1$ for $0 < \beta < n$, and therefore the component along $e^{-\beta x}$ decays geometrically under iteration: $(M_n^{(\alpha,\beta)})^{r}[e^{-\beta x}] = \lambda_2^{\,r} e^{-\beta x}, \qquad r \ge 1.$ O. Szasz, Generalization of S. Bernstein's polynomials to the, Journal of research of the National Bureau of Standards 45 (3) (1950) 239.G. Mirakjan, Approximation of continuous functions with the aid of polynomials, in: Dokl. Akad. Nauk SSSR, Vol. 31, 1941, pp. 201--205.P. Sablonnière, Opérateurs de Bernstein--Jacobi et polynômes orthogonaux, publ.D. Lara-Velasco, T. E. Pérez, Bernstein--Jacobi-type operators preserving derivatives, Computational and Applied Mathematics 43 (5) (2024) 277.S. Mazhar, V. Totik, Approximation by modified szász operators, Acta Sci. Math 49 (1-4) (1985) 257--269.V. Gupta, R. P. Agarwal, Convergence estimates in approximation theory, Vol. 13, Springer, 2014.U. Abel, A. M. Acu, M. Heilmann, I. Rasa, Asymptotic properties for a general class of Szász--Mirakjan--Durrmeyer operators, Mathematical Methods in the Applied Sciences (2025).P. G. Patel, J. C. Prajapati, Certain properties of generalized Mittag-Leffler operators, in: Fractional Differential Equations, Elsevier, 2024, pp. 27--42.P. G. Patel, On Durrmeyer variant of Mittag-Leffler operators, Dolomites Research Notes on Approximation 18 (DRNA Volume 18.2) (2025) 39--46.A. B. O. Daalhuis, Confluent hypergeometric functions., NIST handbook of mathematical functions 321 (2010) 349.K. Zhu, Operator theory in function spaces, no. 138, American Mathematical Soc., 2007.G. B. Folland, Real analysis: modern techniques and their applications, John Wiley & Sons, 1999.H. L. Royden, P. Fitzpatrick, Real analysis, Vol. 32, Macmillan New York, 1988.