Properties of positive linear operators connected with squared fundamental functions

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Abstract

The aim of this article is to present a general class of positive linear operators of discrete type related to squared fundamental basis functions. If these operators are expressed by a series, we propose to truncate them by a finite sum while still keeping the property of converging towards the identity operator in a weighted space. A relation between the local smoothness of functions and the local approximation is obtained. In the variant in which the operators are described by a finite sum, we establish the limit of their iterates.

Authors

Octavian Agratini
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

Contraction principle; linear positive operator; Lipschitz function; modulus of smoothness; weighted space

Paper coordinates

O. Agratini, Properties of positive linear operators connected with squared fundamental functions, Numer. Funct. Anal. Optimiz., 45 (2024) no. 2, pp. 103–111, https://doi.org/10.1080/01630563.2024.2316579

PDF

https://www.tandfonline.com/doi/epdf/10.1080/01630563.2024.2316579?needAccess=true

About this paper

Journal

Numerical Functional Analysis and Optimization

Publisher Name

Taylor and Francis Ltd.

DOI

https://doi.org/10.1080/01630563.2024.2316579

Print ISSN

15322467

Online ISSN

01630563

google scholar link

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Properties of positive linear operators connected with squared fundamental functions

Octavian Agratini
Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy,
Str. Fântânele, 57, 400320 Cluj-Napoca, Romania
E-mail: o.agratini@yahoo.com

Abstract

2020 Mathematics Subject Classification: 41A36, 47A10.

Keywords: Linear positive operator, weighted space, modulus of smoothness, contraction principle, Lipschitz function.

1 Introduction

This note falls under the field of Approximation Theory, more precisely it aims at the study of linear approximation processes, the main ingredient being their positivity. We will consider discrete type processes that are often designed as follows

(L_{n} f) (x) = \sum_{k = 0}^{\infty} l_{n, k} (x) f (x_{n, k}), x \in I,

(1)

where $I$ is a certain real interval, the fundamental functions $l_{n, k} : I \to ℝ_{+}$ are continuous on the domain and $f : I \to ℝ$ belongs to a class, say $ℱ (I)$ , which guarantees that the second member in the above formula is well defined. The nodes $x_{n, k}$ , $k \geq 0$ , form a division of the interval $I$ . This construction also includes the case $l_{n, k} = 0$ , $k > n$ , i.e., the sum in (1) is finite.

Recently, has been an increasing interest in the study of some operators associated with those defined in (1) implying the squared fundamental functions. They have the following form

({\tilde{L}}_{n} f) (x) = \frac{1}{\sum_{k = 0}^{\infty} l_{n, k}^{2} (x)} \sum_{k = 0}^{\infty} l_{n, k}^{2} (x) f (x_{n, k})

(2)

under the additional hypothesis $\sum_{k = 0}^{\infty} l_{n, k}^{2} (x) > 0$ for all $x \in I$ . Clearly, both $L_{n}$ and ${\tilde{L}}_{n}$ , $n \geq 1$ , are linear positive operators. We present two examples of ${\tilde{L}}_{n}$ , $n \geq 1$ , type operators which are expressed by a finite sum, respectively by a series.

Example 1.

Choosing

I = [0, 1], l_{n, k} (x) \equiv b_{n, k} (x) = (\binom{n}{k}) x^{k} {(1 - x)}^{n - k}, x_{n, k} = \frac{k}{n},

(3)

$0 \leq k \leq n$ , $L_{n} \equiv B_{n}$ , $n \geq 1$ , become Bernstein operators. The rational Bernstein operators ${({\tilde{L}}_{n})}_{n \geq 1}$ were investigated by Herzog [7]. Among other notable results we mention the papers of Gavrea and Ivan [5], Holhoş [8], Abel and Kushnirevych [2]. Usually, these operators apply to continuous real valued functions on the compact $[0, 1]$ .

Remark 1.

The construction indicated at (2) can be correlated with the theory of probability. To any discrete probability distribution

P = {(p_{λ})}_{λ \geq 0}

we can associate the index of coincidence [6],

I C (P) = \sum_{λ \geq 0} p_{λ}^{2}

which is used in the definition of entropy. For instance, if $P$ is the binomial distribution with parameters $n \in ℕ$ and $x \in [0, 1]$ , its index of coincidence is given by $\sum_{k = 0}^{n} b_{n, k}^{2} (x)$ , see (3).

Example 2.

Set $ℕ_{0} = {0} \cup ℕ$ . Choosing

I = [0, \infty), l_{n, k} (x) \equiv s_{n, k} (x) = e^{- n x} \frac{{(n x)}^{k}}{k!}, x_{n, k} = \frac{k}{n}, k \in ℕ_{0},

$L_{n} \equiv S_{n}$ , $n \geq 1$ , represent the classical Favard-Szász-Mirakjan operators.

In this case $f$ is commonly considered from the Banach lattice $E_{1}$ . We recall, in general the polynomial weighted spaces $E_{m}$ are defined as follows

E_{m} = E_{m} (ℝ_{+}) = {f \in C (ℝ_{+}) : lim_{x \to \infty} w_{m} (x) f (x) is finite}, m \in ℕ,

(4)

$w_{m} (x) = {(1 + x^{2 m})}^{- 1}$ and they are endowed with the norm $∥ \cdot ∥_{m}$ ,

{‖ f ‖}_{m} = sup_{x \geq 0} w_{m} (x) | f (x) | .

These spaces are nested as follows $C_{B} (ℝ_{+}) \subset E_{m} \subset E_{m + 1} \subset C (ℝ_{+})$ , $m \in ℕ$ , where $C_{B} (ℝ_{+})$ indicates the space of continuous and bounded real valued functions defined on $ℝ_{+} = [0, \infty)$ .

Results related to squared fundamental functions $s_{n, k}$ , $k \geq 0$ , can be found in [1, Section 5.1]. Moreover, in Abel’s paper are obtained Voronovskaja theorems for the general class defined by (2).

The purpose of this note is to highlight some properties of ${\tilde{L}}_{n}$ , $n \in ℕ$ , operators. Due to the fact that their expression is very general, we admit that the results cannot be spectacular. This is the reason why we consider additional assumptions on the operators.

The first approach assumed to modify the series into finite sums and to investigate their approximation properties in weighted spaces of continuous functions defined on $ℝ_{+}$ . More exactly, the operators are truncated fading away their tails and sufficient conditions are provided to ensure their convergence to the identity operator. A study for some bivariate classes of discrete operators of type (1) defined following the truncated process is developed, e.g., in [4]. We also establish a relation between the local smoothness of functions and the local approximation.

The second approach involves the operators acting on the space $C ([0, 1])$ and we determine the convergence of their iterates.

2 Results

It should be noted that a property of ${\tilde{L}}_{n}$ , $n \geq 1$ , operators is to reproduce the constants, more precisely ${\tilde{L}}_{n} e_{0} = e_{0}$ , where $e_{0} (x) = 1$ , $x \in I$ .

Such operators are called Markov type operators. The study of the operators defined by (2) requires an estimation of infinite sum which in a certain sense restricts the operators usefulness from computational point of view. In this respect, in order to approximate a function $f$ , it is interesting and useful to consider partial sums of ${\tilde{L}}_{n} f$ which have only finite numbers of terms depending upon $n$ and $x$ . The classes of operators defined by (2) have a very general form. In order to obtain tangible results we will particularize these classes. The first specifications concern the interval $I$ and the network of nodes ${(x_{n, k})}_{k \geq 0}$ . We consider

I = ℝ_{+}, x_{n, k} = \frac{k}{a_{n}} \leq k, k \in ℕ_{0}, with lim_{n \to \infty} a_{n} = \infty,

(5)

where ${(a_{n})}_{n \geq 1}$ is a positive strictly increasing sequence. Set

ψ_{n} (x) = \sum_{k = 0}^{\infty} l_{n, k}^{2} (x), x \geq 0 .

(6)

Also, $τ_{n, r}$ , $r \in ℕ_{0}$ , stands for the $r$ -th central moment of ${\tilde{L}}_{n}$ , i.e.,

τ_{n, r} (x) = {\tilde{L}}_{n} ({(\cdot - x)}^{r}, x) .

Starting from (2), under the additional assumptions (5), we define

({\tilde{L}}_{n, δ} f) (x) = \frac{1}{ψ_{n} (x)} \sum_{k = 0}^{[a_{n} (x + δ (n))]} l_{n, k}^{2} (x) f (\frac{k}{a_{n}}), x \geq 0, f \in ℱ (ℝ_{+}),

(7)

where $δ = {(δ (n))}_{n \geq 1}$ is a certain sequence of positive numbers and $ψ_{n}$ was defined at (6). Here $[α]$ indicates the largest integer not exceeding $α$ . The study of these operators will be developed in $E_{m}$ spaces defined by (4).

Remark 2.

The idea of using a truncation up to the term of rank $[a_{n} (x + δ (n))]$ appeared for several special classes of operators of type (1). Thus, $[n (x + δ)]$ was used both by Lehnoff [9] for Szász-Mirakjan operators and by J. Wang, S. Zhou [11] for Baskakov operators.

In what follows we set $e_{j}$ , $j \in ℕ$ , for the $j$ -th monomial, $e_{j} (t) = t^{j}$ , $t \geq 0$ . The first result represents the answer to the question whether ${({\tilde{L}}_{n, δ})}_{n \geq 1}$ can inherit the property of the approximation process of the original sequence ${({\tilde{L}}_{n})}_{n \geq 1}$ . We mention that the conditions required by us ensure only sufficiency.

Theorem 1.

Let ${\tilde{L}}_{n}$ , $n \in ℕ$ , be defined by (2) subject to conditions (5). Let ${\tilde{L}}_{n, δ}$ , $n \in ℕ$ , be defined by (7). We suppose that $lim_{n \to \infty} \sqrt{a_{n}} δ (n) = \infty$ and the central moment $τ_{n, 2 m}$ satisfies

τ_{n, 2 m} (x) \leq C a_{n}^{- m},

(8)

where $C$ is a constant depending only on $m$ . Let $f \in E_{m} \cap ℱ (ℝ_{+})$ . If ${({\tilde{L}}_{n} f)}_{n \geq 1}$ converges pointwise to $f$ , then ${({\tilde{L}}_{n, δ} f)}_{n \geq 1}$ enjoys the same property.

Proof.

Since $\min_{λ \geq 0} (λ^{2 m} + {(1 - λ)}^{2 m}) = 2^{1 - 2 m}$ , $m \in ℕ$ , the following elementary inequality

t^{2 m} \leq 2^{2 m - 1} (x^{2 m} + {(t - x)}^{2 m}), t \geq 0, x \geq 0, m \in ℕ,

(9)

holds. On the other hand, for every $f \in E_{m}$ , the positive constants $a_{f}$ , $b_{f}$ exist such that $| f | \leq a_{f} + b_{f} e_{2 m}$ and, consequently, by using (9) we get

| f (t) | \leq g_{m} (x) + 2^{2 m - 1} b_{f} {(t - x)}^{2 m},

where $g_{m} = a_{f} + 2^{2 m - 1} b_{f} e_{2 m}$ . This implies

| f (\frac{k}{a_{n}}) | \leq g_{m} (x) + 2^{2 m - 1} b_{f} {(\frac{k}{a_{n}} - x)}^{2 m}, k \in ℕ_{0}, x \geq 0 .

(10)

Because $x$ , $δ (n)$ , $a_{n}$ are positive quantities, if $k \geq [a_{n} (x + δ (n))] + 1$ , then

{k \in ℕ_{0} : k \geq [a_{n} (x + δ (n))] + 1} \subset {k \in ℕ_{0} : | \frac{k}{a_{n}} - x | > δ (n)} := I_{n, x, δ} .

(11)

Setting $R_{n} := {\tilde{L}}_{n} - {\tilde{L}}_{n, δ}$ , we get

| ({\tilde{L}}_{n, δ} f) (x) - f (x) | \leq | (R_{n} f) (x) | + | ({\tilde{L}}_{n} f) (x) - f (x) | .

(12)

Taking into account both (11) and (10), we can write

$\| (R_{n} f) (x) \|$	$\leq \sum_{k \in I_{n, x, δ}} \frac{l_{n, k}^{2} (x)}{ψ_{n} (x)} f (\frac{k}{a_{n}})$
	$\leq \sum_{k \in I_{n, x, δ}} \frac{l_{n, k}^{2} (x)}{ψ_{n} (x)} g_{m} (x) + 2^{2 m - 1} b_{f} \sum_{k \in I_{n, k, δ}} \frac{l_{n, k}^{2} (x)}{ψ_{n} (x)} {(\frac{k}{a_{n}} - x)}^{2 m}$
	$:= \sum_{1} + \sum_{2} .$	(13)

Concerning the sums $\sum_{1}$ and $\sum_{2}$ we have

	$\sum_{1}$	$\leq g_{m} (x) δ^{- 2 m} (n) \sum_{k = 0}^{\infty} \frac{l_{n, k}^{2} (x)}{ψ_{n} (x)} {(\frac{k}{a_{n}} - x)}^{2 m}$
		$= g_{m} (x) δ^{- 2 m} (n) τ_{n, 2 m} (x),$

respectively

\sum_{2} \leq 2^{2 m - 1} b_{f} τ_{n, 2 m} (x), x \geq 0 .

Returning to (2) and using both (8) and our hypotheses

lim_{n \to \infty} \sqrt{a_{n}} δ (n) = \infty,

we obtain

| (R_{n} f) (x) | \leq (g_{m} (x) δ^{- 2 m} (n) + 2^{2 m - 1} b_{f}) \frac{C}{a_{n}^{m}} = o (1) (n \to \infty) .

Using (12), the above relation along with the hypothesis of the theorem completes the proof of the statement. ∎

Next, our result concerns the operators defined by (2) in the version that the sum is finite. This time a compact interval $I$ will be chosen. Since any compact $[a, b]$ is isomorphic to $[0, 1]$ , in what follows we use only the compact $[0, 1]$ . For this sequence is of interest to investigate the sequences of iterates ${({\tilde{L}}_{n}^{j})}_{n \geq 1}$ , where ${\tilde{L}}_{n}^{0} = I$ , the identity operator of the space $C ([0, 1])$ and

{\tilde{L}}_{n}^{1} = {\tilde{L}}_{n}, {\tilde{L}}_{n}^{j} = {\tilde{L}}_{n} ({\tilde{L}}_{n}^{j - 1}), j > 1,

assuming that $j$ does not depend on $n$ . In our study we maintain the general net $Δ_{n}$ on $[0, 1]$ ,

Δ_{n} = (0 = x_{n, 0} < x_{n, 1} < \dots < x_{n, n} = 1) .

Our operators have therefore the following form

({\tilde{L}}_{n} f) (x) = \sum_{k = 0}^{n} \frac{l_{n, k}^{2} (x)}{\sum_{k = 0}^{n} l_{n, k}^{2} (x)} f (x_{n, k}),

(14)

$x \in [0, 1]$ , $f \in C ([0, 1])$ . In addition to the fulfilled condition ${\tilde{L}}_{n} e_{0} = e_{0}$ , we add the following working hypothesis

({\tilde{L}}_{n} e_{1}) (x) = x, x \in [0, 1] .

(15)

Consequently, the degree of exactness of ${\tilde{L}}_{n}$ , $n \geq 1$ , has the value 1, which means that they preserve the affine functions. In other words, for any real constant $c$ , $e_{0} + c e_{1}$ is a fixed point for ${\tilde{L}}_{n}$ , $n \geq 1$ .

Remark 3.

By virtue of the classical result obtained by Shisha and Mond [10], regarding the error of approximation one has

| ({\tilde{L}}_{n} f) (x) - f (x) | \leq 2 ω_{1} (f, δ_{n} (x)), x \in [0, 1],

(16)

where

δ_{n} (x) = {(\frac{\sum_{k = 0}^{n} l_{n, k}^{2} (x) x_{n, k}^{2}}{\sum_{k = 0}^{n} l_{n, k}^{2} (x)} - x^{2})}^{1 / 2}

(17)

and $ω_{1} (f, \cdot)$ represents the modulus of continuity of $f$ defined by

ω_{1} (f, δ) = sup_{\begin{matrix} x_{1}, x_{2} \in [0, 1] \\ | x_{1} - x_{2} | \leq δ \end{matrix}} | f (x_{1}) - f (x_{2}) |, δ \geq 0 .

For any function $h \in C ([0, 1])$ , $ω_{1}$ enjoys the property $lim_{δ \to 0^{+}} ω (f, δ) = 0$ . Consequently, if we have $lim_{n \to \infty} δ_{n} (x) = 0$ , see (17), then

lim_{n \to \infty} {\tilde{L}}_{n} f = f uniformly on [0, 1],

for any $f \in C ([0, 1])$ .

Considering that ${\tilde{L}}_{n}$ , $n \geq 1$ , are interpolating operators at the ends of the interval $[0, 1]$ , we define

Γ_{α, β} = {f \in C ([0, 1]) : f (0) = α, f (1) = β}, (α, β) \in ℝ \times ℝ .

The system ${(Γ_{α, β})}_{(α, β) \in ℝ \times ℝ}$ makes up a partition of the space $C ([0, 1])$ and the interpolation operators of our operators guarantees that $Γ_{α, β}$ is an invariant subset of every operator ${\tilde{L}}_{n}$ . Assume that

λ_{n} = \min_{x \in [0, 1]} (l_{n, 0}^{2} (x) + l_{n, n}^{2} (x)) > 0 .

(18)

According to [3, Theorem 4] one has ${{\tilde{L}}_{n} |}_{Γ_{α, β}} : Γ_{α, β} \to Γ_{α, β}$ is a contraction for every $(α, β) \in ℝ \times ℝ$ and $n \geq 1$ . In addition

‖ {\tilde{L}}_{n} f - {\tilde{L}}_{n} g ‖ \leq (1 - λ_{n}) ‖ f - g ‖, f, g \in Γ_{α, β},

where $∥ \cdot ∥$ stands for the uniform norm assigns to the space $C ([0, 1])$ .

The above results imply the following statement.

Theorem 2.

Let ${\tilde{L}}_{n}$ be defined by (14) such that (15) and (18) hold. For each $n \in ℕ$ , the iterates sequence ${({\tilde{L}}_{n}^{j})}_{j \geq 1}$ satisfies

lim_{j \to \infty} ({\tilde{L}}_{n}^{j} f) (x) = f (0) + (f (1) - f (0)) x uniformly on [0, 1]

(19)

for every $f \in C ([0, 1])$ .

Notice that the affine function $f^{*} = f (0) + (f (1) - f (0)) e_{1}$ is a fixed point in $Γ_{f (0), f (1)}$ of any operator ${\tilde{L}}_{n}$ and thus (19) is nothing else then the conclusion of Banach’s fixed point theorem applied in the mentioned space $Γ_{f (0), f (1)}$ regarding the limit of the sequence of successive approximations of the contraction ${{\tilde{L}}_{n} |}_{X_{f (0), f (1)}}$ .

Knowing that $(C ([0, 1]), d)$ is a complete metric space,

d (f, g) = ‖ f - g ‖,

we establish upper limits of approximation error between ${\tilde{L}}_{n}^{j} f$ and $f^{*}$ .

Theorem 3.

Let ${\tilde{L}}_{n}$ be defined by (14) such that (15) and (18) take place. For each $n \in ℕ$ , the iterates of ${\tilde{L}}_{n}$ satisfy the following relation

‖ {\tilde{L}}_{n}^{j} f - f^{*} ‖ \leq \frac{2 {(1 - λ_{n})}^{j}}{λ_{n}} sup_{x \in [0, 1]} ω_{1} (f, δ_{n} (x)), f \in C ([0, 1]), j \geq 1,

where $δ_{n} (x)$ is given by (17).

Proof.

Let $n \in ℕ$ and $f \in C ([0, 1])$ be arbitrarily chosen. We use the known inequality describing the speed of convergence

d (x_{j}, x^{*}) \leq \frac{q^{j}}{1 - q} d (x_{1}, x_{0}), j \geq 1,

(20)

valid in any complete metric space, where $x_{j} = Λ (x_{j - 1})$ , $j \geq 1$ , $Λ$ is a contraction and $q \in (0, 1)$ is a Lipschitz constant of $Λ$ . In our case we assign the following values

x_{0} = f, x_{j} = {\tilde{L}}_{n}^{j} f, x^{*} = f^{*} .

Then (20) reads as follows

‖ {\tilde{L}}_{n}^{j} f - f^{*} ‖ \leq \frac{{(1 - λ_{n})}^{j}}{λ_{n}} ‖ {\tilde{L}}_{n} f - f ‖ .

By using (16) one gets

‖ {\tilde{L}}_{n} f - f ‖ \leq 2 sup_{x \in [0, 1]} ω_{1} (f, δ_{n} (x)) .

Combining the previous two inequalities, the statement of the theorem is proved. ∎

In the final part, we expand the domain of the definition of the functions on which the operators from (14) act, returning to the interval $I = ℝ_{+}$ . Our aim is to prove a relation between the local smoothness of functions and the local approximation. We recall that a continuous function $f : ℝ_{+} \to ℝ$ is locally $L i p α$ on $E$ ( $0 < α \leq 1$ , $E \subset ℝ_{+}$ ) if it satisfies the condition

| f (x) - f (y) | \leq M_{f} {| x - y |}^{α}, (x, y) \in ℝ_{+} \times E,

(21)

where $M_{f}$ is a constant depending only on $f$ .

Also, the distance between $x \in ℝ_{+}$ and $E$ denoted by $d (x, E)$ is given by

d (x, E) = inf {| x - y | : y \in E} .

Theorem 4.

Let ${\tilde{L}}_{n}$ , $n \in ℕ$ , be defined by (14) where $x \in ℝ_{+}$ and $f \in C (ℝ_{+})$ . If $f$ is locally $L i p α$ on $E \subset ℝ_{+}$ , then

| ({\tilde{L}}_{n} f) (x) - f (x) | \leq M_{f} (τ_{n, 2}^{α / 2} (x) + 2 d^{α} (x, E)), x \geq 0

takes place.

Proof.

In the relation $1 / r + 1 / s = 1$ ( $r > 0$ , $s > 0$ ) that characterizes Hölder’s inequality, choosing $r = 2 α^{- 1}$ and knowing that ${\tilde{L}}_{n} e_{0} = e_{0}$ , we get

{\tilde{L}}_{n} (h^{α}; x) \leq {({\tilde{L}}_{n} (h^{2}; x))}^{α / 2}, x \geq 0,

where $α \in (0, 1]$ and $h \geq 0$ . Taking $h = | e_{1} - x e_{0} |$ , we can write

{\tilde{L}}_{n} ({| e_{1} - x e_{0} |}^{α}; x) \leq τ_{n, 2}^{α / 2} (x), x \geq 0 .

(22)

By using the continuity of $f$ , it is obvious that (21) holds for any $x \geq 0$ and $y \in \bar{E}$ , $\bar{E}$ being the closure of the set $E$ in $ℝ_{+}$ . Let $(x, x_{0}) \in ℝ_{+} \times \bar{E}$ be such that $d (x, E) = | x - x_{0} |$ . In the following obvious inequality

| f - f (x) | \leq | f - f (x_{0}) | + | f (x_{0}) - f (x) | e_{0},

by applying the linear and positive operators ${\tilde{L}}_{n}$ , and using (21) we have

	$\| ({\tilde{L}}_{n} f) (x) - f (x) \|$	$= \| {\tilde{L}}_{n} (f - f (x) e_{0}; x) \| \leq {\tilde{L}}_{n} (\| f - f (x) e_{0} \|; x)$
		$\leq {\tilde{L}}_{n} (\| f - f (x_{0}) \|; x) + {\tilde{L}}_{n} (\| f (x_{0}) - f (x) \| e_{0}; x)$
		$\leq {\tilde{L}}_{n} (M_{f} {\| e_{1} - x_{0} e_{0} \|}^{α}; x) + M_{f} {\| x - x_{0} \|}^{α}$
		$\leq M_{f} ({\tilde{L}}_{n} ({\| e_{1} - x e_{0} \|}^{α}; x) + {\tilde{L}}_{n} ({\| x - x_{0} \|}^{α} e_{0}; x) + {\| x - x_{0} \|}^{α}) .$

We also took into account the inequality ${(A + B)}^{α} \leq A^{α} + B^{α}$ ( $A \geq 0$ , $B \geq 0$ , $0 < α \leq 1$ ) and the monotonicity of ${\tilde{L}}_{n}$ . Based on (22) the conclusion of our theorem is completely motivated. ∎

Disclosure statement. No potential conflict of interest was reported by the author.

Funding. Not applicable.

ORCID: 0000-0002-2406-4274

References

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[2] Abel, U., Kushnirevych, V. (2019). Voronovskaja type theorems for positive linear operators related to squared Bernstein polynomials, Positivity, 23, 697-710.
[3] Agratini, O., Rus, I.A. (2003). Iterates of a class of discrete operators via contraction principle, Comment. Math. Univ. Carolinae, 44, no. 3, 555-563.
[4] Agratini, O., Tarabie, S., Trîmbiţaş, R. (2014). Approximation of bivariate function by truncated classes of operators, In: Proceedings of the Third International Conference on Modelling and Development of Intelligent Systems, Sibiu 2013 (Dana Simian, Ed.), pp. 11-19, Lucian Blaga University Press, Sibiu.
[5] Gavrea, I., Ivan, M. (2017). On a new sequence of positive linear operators related to squared Bernstein polynomials, Positivity, 21, 911-917.
[6] Harremoes, P., Topsoe, F. (2001). Inequalities between entropy and index of coincidence derived from information diagrams, IEEE Transactions on Information Theory, 47, no. 7, 2944-2960.
[7] Herzog, F. (2009). Heuristische Untersuchungen der Konwergenzrate spazieller linearer Approximationsprozesse, Diploma Thesis, Friedberg.
[8] Holhoş, A. (2019). Voronovskaya theorem for a sequence of positive linear operators related to squared Bernstein polynomials, Positivity, 23, 571-580.
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[10] Shisha, O., Mond, B. (1968). The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA, 60, 1196-1200.
[11] Wang, J., Zhou, S. (2000). On the convergence of modified Baskakov operators, Bull. Inst. Math. Academia Sinica, 28, no. 2, 117-123.

[1] Herzog, F. (2009). Heuristische untersuchungen der konwergenzrate spazieller linearerapproximationsprozesse. Diploma Thesis, Friedberg.
[2] Gavrea, I., Ivan, M. (2017). On a new sequence of positive linear operators related tosquared Bernstein polynomials.Positivity21:911–917.
[3] Holhos, A. (2019). Voronovskaya theorem for a sequence of positive linear operatorsrelatedtosquaredBernsteinpolynomials.Positivity23:571–580.
[4] Abel, U., Kushnirevych, V. (2019). Voronovskaja type theorems for positive linear opera-tors related tosquare d Bernstein polynomials. Positivity23:697–710.
[5] Harremoes, P., Topsoe, F. (2001). Inequalities between entropy and index of coincidence derived from in formation diagrams.IEEE Trans. Inf. Theory47(7):2944–2960.
[6] Abel, U. (2020). Voronovskaja type theorems for positive linear operators related to squared fundamental functions.In:Draganov,B.,Ivanov,K.,Nikolov, G.,Uluchev,
R.,eds. Constructive Theory of Functions, Sozopol 2019. Sofia: Prof. Marin Drinov PublishingHouse of BAS, pp. 1–21.
[7] Agratini, O., Tarabie, S., Trîmbitas, R. (2014). Approximation of bivariate function bytruncated classes of operators, In: Simian, D., ed.Proceedings of the Third International Conference on Modelling and Development of Intelligent Systems, Sibiu 2013. Sibiu:Lucian Blaga University Press, pp. 11–19.
[8] Lehnhoff, H.-G. (1984). On a modified Szász-Mirakjan operator.J. Approx. Theory42:278–282.
[9] Wang, J., Zhou, S. (2000). On the convergence of modified Baskakov operators. Bull. Inst. Math. Academia Sinica 28(2):117–123.
[10] Shisha, O., Mond, B. (1968). The degree of convergence of linear positive operators.Proc.Nat. Acad. Sci. USA60:1196–1200.
[11] Agratini, O., Rus, I. A. (2003). Iterates of a class of discrete operators via contraction principle. Comment. Math. Univ. Carolinae 44(3):555–563.

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	$\| ({\tilde{L}}_{n} f) (x) - f (x) \|$	$= \| {\tilde{L}}_{n} (f - f (x) e_{0}; x) \| \leq {\tilde{L}}_{n} (\| f - f (x) e_{0} \|; x)$
		$\leq {\tilde{L}}_{n} (\| f - f (x_{0}) \|; x) + {\tilde{L}}_{n} (\| f (x_{0}) - f (x) \| e_{0}; x)$
		$\leq {\tilde{L}}_{n} (M_{f} {\| e_{1} - x_{0} e_{0} \|}^{α}; x) + M_{f} {\| x - x_{0} \|}^{α}$
		$\leq M_{f} ({\tilde{L}}_{n} ({\| e_{1} - x e_{0} \|}^{α}; x) + {\tilde{L}}_{n} ({\| x - x_{0} \|}^{α} e_{0}; x) + {\| x - x_{0} \|}^{α}) .$

Properties of positive linear operators connected with squared fundamental functions

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Properties of positive linear operators connected with squared fundamental functions

Abstract

1 Introduction

Example 1.

Remark 1.

Example 2.

2 Results

Remark 2.

Theorem 1.

Proof.

Remark 3.

Theorem 2.

Theorem 3.

Proof.

Theorem 4.

Proof.

References

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