Limit Theorems for Iterates of the Szász–Mirakyan Operator in Probabilistic View

Abstract

The Szász–Mirakyan operator is known as a positive linear operator which uniformly approximates a certain class of continuous functions on the half line. The purpose of the present paper is to find out limiting behaviors of the iterates of the Szász–Mirakyan operator in a probabilistic point of view. We show that the iterates of the Szász–Mirakyan operator uniformly converge to a continuous semigroup generated by a second-order degenerate differential operator. A probabilistic interpretation of the convergence in terms of a discrete Markov chain constructed from the iterates and a limiting diffusion process on the half line is captured as well.

Appendices

Appendix A. Trotter’s Approximation Theorem and Its Rate of Convergence

Trotter’s approximation theorem provides a sufficient condition that the iterates of a bounded linear operator acting on a Banach space converge to a $C_0$-semigroup. We give the statement of Trotter’s approximation theorem when the iteration of a linear operator enjoys the contraction property.

Proposition A.1

(cf.  [26, Theorem 5.1], [22, Theorem 2.13]) Let $(\mathcal {U}, \Vert \cdot \Vert _{\mathcal {U}})$ be a Banach space. Suppose that a sequence $\{T_n\}_{n=1}^\infty$ of bounded linear operators on $\mathcal {U}$ satisfies that $\Vert T_n\Vert \le 1$ for $n \in \mathbb {N}$. Put $\mathcal {A}_n:=n(T_n-I), \, n \in \mathbb {N}.$ We define a linear operator $\mathcal {A}$ by the closure of the limit of $\mathcal {A}_n$. If the domain $D(\mathcal {A})$ is dense in $\mathcal {U}$ and the range of $\lambda - \mathcal {A}$ is dense in $\mathcal {U}$ for some $\lambda >0$, then there exists a $C_0$-semigroup $({\mathsf {T}}_t)_{t \ge 0}$ acting on $\mathcal {U}$ satisfying $\Vert {\mathsf {T}}_t\Vert \le 1, \, t \ge 0$, and

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert T_n^{[nt]}f - {\mathsf {T}}_tf\Vert _{\mathcal {U}}=0, \qquad t \ge 0, \, f \in \mathcal {U}. \end{aligned}$$

(A.1)

On the other hand, as is easily seen, Proposition A.1 does not imply any quantitative estimates of (A.1). Campiti and Tacelli established in [9] a refinement of Proposition A.1 by giving the rate of convergence of (A.1).

Proposition A.2

(cf.  [9, Theorem 1.1], see also [10]) Suppose that $\mathcal {U}$, $\{T_n\}_{n=1}^\infty$, $\{\mathcal {A}_n\}_{n=1}^\infty$ and $({\mathsf {T}}_t)_{t \ge 0}$ are as in Proposition  A.1. Let $\mathcal {D}$ be a dense subspace of $\mathcal {U}$. We assume that

$$\begin{aligned} \Vert \mathcal {A}_n f\Vert _{\mathcal {U}} \le \varphi _n(f), \qquad \Vert \mathcal {A}_n f - \mathcal {A}f\Vert _{\mathcal {U}} \le \psi _n(f), \qquad f \in \mathcal {D}, \end{aligned}$$

where $\varphi _n, \psi _n : \mathcal {D} \rightarrow [0, \infty )$ are semi-norms with $\psi _n(f) \rightarrow 0$ as $n \rightarrow \infty$ for $f \in \mathcal {D}$. Then, for $t \ge 0$ and $f \in \{g \in \mathcal {D} \, | \, {\mathsf {T}}_t g \in \mathcal {D}, \, t \ge 0\}$, we have

$$\begin{aligned} \Vert T_n^{[nt]}f - {\mathsf {T}}_tf\Vert _{\mathcal {U}} =\sqrt{\frac{t}{n}}\varphi _n(f)+\frac{1}{n}\varphi _n(f) +\int _0^t \psi _n({\mathsf {T}}_sf) \, \mathrm {d}s, \qquad n \in \mathbb {N}. \end{aligned}$$

(A.2)

Indeed, we have used Proposition A.2 in order to deduce the rate of convergence of the iterates of the Szász–Mirakyan operator (see Theorem 1.5).

Appendix B. Korovkin-Type Theorems

It is well-known that the Korovkin-type theorems provide a quite powerful sufficient condition to deduce that a sequence of positive linear operators acting on some function spaces converges strongly to the identity operator. Korovkin’s first theorem, which was first discovered by Korovkin himself in [21], is stated as follows:

Proposition B.1

(Korovkin’s first theorem) Let $e_0(x) \equiv 1$, $e_1(x)=x$ and $e_2(x)=x^2$ for $x \in [0, 1]$. If a sequence $\{L_n\}_{n=1}^\infty$ of positive linear operators acting on C([0, 1]) satisfies

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert L_ne_i - e_i\Vert _\infty =0, \qquad i=0, 1, 2, \end{aligned}$$

then, it holds that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert L_nf - f\Vert _\infty =0, \qquad f \in C([0, 1]). \end{aligned}$$

So far, a number of generalizations of Proposition B.1 have been established in various settings. We refer to [1] and [2] for good surveys of this topic. In the sequel, $(\mathfrak {X}, \Vert \cdot \Vert _{\mathfrak {X}})$ is used in order to represent Banach spaces of some continuous functions on a locally compact space $E \subset \mathbb {R}$. We give a general formulation of the Korovkin-type theorem.

Definition B.2

(Korovkin set) A subset $\mathcal {K}$ of $\mathfrak {X}$ is called a Korovkin set of $\mathfrak {X}$ if for every sequence $\{L_n\}_{n=1}^\infty$ of positive linear operators acting on $\mathfrak {X}$ satisfying $\sup _{n \in \mathbb {N}}\Vert L_n\Vert <\infty$ and

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert L_nf - f\Vert _{\mathfrak {X}}=0, \qquad f \in \mathcal {K}, \end{aligned}$$

then, it holds that

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert L_nf - f\Vert _{\mathfrak {X}}=0, \qquad f \in \mathfrak {X}. \end{aligned}$$

Namely, Korovkin-type results aim to find out which functions form Korovkin sets of a given function space. Note that, in other words, Korovkin’s first theorem asserts that the set $\{e_0, e_1, e_2\} \subset C([0, 1])$ is a Korovkin set of C([0, 1]) . We next give several examples of Korovkin sets of $C_\infty ([0, \infty ))$.

Proposition B.3

(cf.  [1, Corollary 6.7]) Suppose that $0<\lambda _1<\lambda _2<\lambda _3$. Then, $\{f_{\lambda _1}, f_{\lambda _2}, f_{\lambda _3}\}$ is a Korovkin set of $C_\infty ([0, \infty ))$, where $f_{\lambda _k}(x):=\exp (-\lambda _k x)$ for $k=1, 2, 3$.

Moreover, we obtain an easy way to find out Korovkin sets of $C_\infty ^w([0, \infty ))$ when the weight function w is supposed to be bounded.

Proposition B.4

(cf.  [1, Proposition 6.16]) Let $w \in C_b([0, \infty ))$. Then, every Korovkin set of $C_\infty ([0, \infty ))$ is also a Korovkin set of $C_\infty ^w([0, \infty ))$. In particular, the set $\{f_{\lambda _1}, f_{\lambda _2}, f_{\lambda _3}\}$ as in Proposition  B.3 is a Korovkin set of $C_\infty ^w(E[0, \infty )$.