Previously, we have provided the properties of the newly defined Bernstein–Chlodowsky–Gadjiev-Type operators that fix the function (e^{-2x} ). Now, we can introduce some approximation properties of these new operators for the different spaces of continuous functions. Additionally, we provide the rate of convergence of (mathcal{G}_{n}^{alpha _{1},beta _{1}} ).

Theorem 1

Let (x>0 ) be fixed and (mathcal{G}_{n}^{alpha _{1},beta _{1}} ), (ngeq 1 ), be the operator defined in (3.3). Then, (mathcal{G}_{n}^{alpha _{1},beta _{1}} ) is a linear positive operator from (C_{*}(mathcal{S}) ) into itself. In addition, (Vert mathcal{G}_{n}^{alpha _{1},beta _{1}} Vert _{ C_{*}( mathcal{S})} =1).

Proof

It can be easily shown that for each (nin mathbb{N} ), (s_{n}(x) ) is an increasing and convex real continuous function satisfying

$$ s_{n} biggl( p_{n}frac{alpha _{1}}{n+beta _{1}} biggr)=p_{n} frac{alpha _{2}}{n+beta _{2}}quad text{and}quad s_{n} biggl( p_{n} frac{n+alpha _{1}}{n+beta _{1}} biggr)=p_{n} frac{n+alpha _{2}}{n+beta _{2}}. $$

As an explicit consequence of equations (3.1) and (3.2), one can conclude that (mathcal{G}_{n}^{alpha _{1},beta _{1}} ) is a positive operator. Additionally, if (fin C_{*}(mathcal{S}) ), one can say that (B_{n,p_{n}}^{alpha , beta }(f)in C_{*}(mathcal{S}) ) resulting from (1.1), which yields (B_{n,p_{n}}^{alpha , beta }(f)in C(mathcal{S}) ). Then, it can be easily seen that (mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)in C(mathcal{S}) ) since (s_{n}(x) ) satisfy the above properties and the relation (3.4). Moreover, it is obvious that (lim_{xto infty }mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)(x)= lim_{xto infty }(f)(x)in mathbb{R} ). As a consequence, (Vert mathcal{G}_{n}^{alpha _{1},beta _{1}} Vert _{C_{*}( mathcal{S})}=Vert mathcal{G}_{n}^{alpha _{1},beta _{1}}(e_{0}) Vert _{infty }=1 ) due to the positivity of each (mathcal{G}_{n}^{alpha _{1},beta _{1}} ). □

Theorem 2

For the same assumptions of Theorem 1, the following expression

$$ mathcal{G}_{n}^{alpha _{1},beta _{1}}bigl(C_{0}(mathcal{S}) bigr)subset C_{0}( mathcal{S}) $$

holds.

Proof

From the direct consequence of Theorem 1 and (lim_{xto infty }mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)(x)= lim_{xto infty }(f)(x)=0 ) whenever (fin C_{0}(mathcal{S}) ), one can easily show the proof of the theorem. □

Theorem 3

For the fixed (n geq 1 ), consider the operators (mathcal{G}_{n}^{alpha _{1},beta _{1}} ) defined by (3.3). Then,

$$ lim_{nrightarrow infty }mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)=fquad textit{uniformly on } mathcal{S} $$

if (fin C_{*}(mathcal{S}) ).

Proof

In an attempt to prove the theorem we need to show that

$$ lim_{nto infty }mathcal{G}_{n}^{alpha _{1},beta _{1}}(f_{mu })=f_{ mu }quad text{uniformly on } mathcal{S}, $$

(4.1)

for every (mu >0 ). In line with this objective, for every (z>0 ), we use the following useful inequality given in [23, Lemma 3.1]

$$ e^{-zvartheta _{n}}-e^{-z}< frac{z_{n}}{2e},quad ngeq 1 , $$

(4.2)

where (vartheta _{n}=frac{1-e^{-z_{n}}}{z_{n}} ) and ((z_{n})_{ngeq 1} ) is a sequence of strictly positive real numbers. Then, by following the similar steps of the proof of [23, Corollary 3.4], we can obtain that

$$begin{aligned}& biglvert mathcal{G}_{n}^{alpha _{1},beta _{1}}(f_{mu }) (x)-(f_{mu }) (x) bigrvert \& quad leq e^{-mu p_{n} alpha _{1}/(n+beta _{1})} biggl[ 1- bigl(1-e^{-mu p_{n}/(n+beta _{1})} bigr) biggl( s_{n}(x) frac{n+beta _{2}}{np_{n}}- frac{alpha _{2}}{n} biggr) biggr]-e^{- mu x}, \& quad = e^{-mu p_{n} alpha _{1}/(n+beta _{1})} e^{nln [ 1- (1-e^{- mu p_{n}/(n+beta _{1})} ) ( s_{n}(x) frac{n+beta _{2}}{np_{n}}-frac{alpha _{2}}{n} ) ]}-e^{-mu x}, \& quad leq e^{-mu p_{n} alpha _{1}/(n+beta _{1})}e^{-ns_{n}(x)[(n+ beta _{2})/np_{n}][1-e^{-mu p_{n}/(n+beta _{1})}] }e^{alpha _{2}[1-e^{- mu p_{n}/(n+beta _{1})}] }-e^{-mu x}, \& quad = e^{-mu p_{n} alpha _{1}/(n+beta _{1})}e^{[alpha _{2} mu p_{n}/(n+ beta _{1})] frac{[1-e^{-mu p_{n}/(n+beta _{1})}]}{mu p_{n}/(n+beta _{1})} }e^{-ns_{n}(x)[(n+ beta _{2})/np_{n}][mu p_{n}/(n+beta _{1})] frac{[1-e^{-mu p_{n}/(n+beta _{1})}]}{mu p_{n}/(n+beta _{1})} } \& qquad {} -e^{-mu x}, \& quad leq e^{-mu p_{n} (alpha _{1}-alpha _{2})/(n+beta _{1})} bigl( e^{-mu s_{n}(x)[(n+beta _{2})/(n+beta _{1})] frac{[1-e^{-mu p_{n}/(n+beta _{1})}]}{mu p_{n}/(n+beta _{1})} }- e^{- mu s_{n}(x)[(n+beta _{2})/(n+beta _{1})]} bigr) end{aligned}$$

since (ln xleq x-1 ), (frac{[1-e^{-mu p_{n}/(n+beta _{1})}]}{mu p_{n}/(n+beta _{1})} leq 1 ) for (2.3) and the inequality (3.6) holds. Then, using (4.2) for

$$ z=-mu s_{n}(x)frac{(n+beta _{2})}{(n+beta _{1})} quad text{and}quad z_{n}= frac{mu p_{n}}{(n+beta _{1})}, $$

we deduce that

$$ biglvert mathcal{G}_{n}^{alpha _{1},beta _{1}}(f_{mu }) (x)-f_{mu }(x) bigrvert leq e^{-mu p_{n} (alpha _{1}-alpha _{2})/(n+beta _{1})} frac{mu p_{n}}{2e(n+beta _{1})} $$

and

$$ biglVert mathcal{G}_{n}^{alpha _{1},beta _{1}}(f_{mu })-f_{mu } bigrVert _{ infty } leq e^{-mu p_{n} (alpha _{1}-alpha _{2})/(n+beta _{1})} frac{mu p_{n}}{2e(n+beta _{1})} $$

(4.3)

for (xin mathcal{S} ) and the proof of (4.1) is completed. Then, relying on the direct result of (4.1) and [14], we can prove the theorem. □

Theorem 4

For the same assumptions of Theorem 3, then

$$ lim_{nrightarrow infty }mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)=f quad textit{uniformly on compact subsets of } mathcal{S} $$

if (fin C_{b}(mathcal{S}) ).

Proof

From the the results provided above, we note that

$$begin{aligned}& vert mathcal{G}_{n}^{alpha _{1},beta _{1}}(e_{0}) (x)-e_{0}(x) =0, \& biglvert mathcal{G}_{n}^{alpha _{1},beta _{1}}(e_{1}) (x)-e_{1}(x) bigrvert leq p_{n}frac{alpha _{1}-alpha _{2}}{n+beta _{2}}+p_{n} frac{alpha _{2}}{n+beta _{1}}+x biggl( N_{n} frac{n+beta _{2}}{n+beta _{1}}-1 biggr), end{aligned}$$

and

$$begin{aligned}& biglvert mathcal{G}_{n}^{alpha _{1},beta _{1}}(e_{2}) (x)-e_{2}(x) bigrvert \& quad leq x^{2} biggl( frac{n-1}{n} biggl( frac{n+beta _{2}}{n+beta _{1}} biggr)^{2}N_{n}^{2}-1 biggr)+p_{n} frac{(2alpha _{1}+1)(n+beta _{2})}{(n+beta _{1})^{2}}N_{n} x+ biggl( p_{n}frac{alpha _{1}}{n+beta _{1}} biggr)^{2}, end{aligned}$$

thereby, (lim_{nto infty }mathcal{G}_{n}^{alpha _{1},beta _{1}}(lbrace e_{0},e_{1},e_{2} rbrace )=lbrace e_{0},e_{1},e_{2} rbrace ) uniformly on compact subsets of (mathcal{S}), due to the fact that (lim_{n rightarrow infty }N_{n}=1 ). As a result, as (lbrace e_{0},e_{1},e_{2} rbrace subset Omega _{2}^{*} ), the consequence follows from [6, Theorem 3.5]. □

In order to estimate the rate of convergence of ((mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)) ) for (ngeq 1 ) to f in Theorem 3, we need to increase our knowledge about the modulus of continuity. In this estimation, we will take advantage of the following definition of the modulus of continuity introduced in [23]:

Definition 1

Let (fin C_{*}(mathcal{S}) ). Then, the modulus of continuity of a function, (omega ^{*} ( f,delta ) ), is defined for (delta geq 0 ) by

$$ omega ^{*} ( f,delta )=sup_{ substack{x,tgeq 0\ vert e^{-x}-e^{-t} vert leq delta }} biglvert f(x)-f(t) bigrvert . $$

(4.4)

In other words, this modulus of continuity can be stated concerning the standard modulus of continuity by

$$ omega ^{*} ( f,delta )=omega ( mathbf{f}, delta ), $$

where (mathbf{f}:C_{*}(mathcal{S}) rightarrow C(mathcal{S}) ) is the continuous function defined by

$$ mathbf{f}(theta )= textstylebegin{cases} f(-ln theta ), & text{if }theta in (0,1], \ 1, & text{if }theta =0 . end{cases} $$

Then, the following theorem would be helpful in order to express the next theorems.

Theorem 5

([23])

If (Q_{n}: C_{*}(mathcal{S}) rightarrow C_{*}(mathcal{S}) ) is a sequence of positive linear operators for (n geq 1 ) with

$$begin{aligned} rho _{n} =& biglVert Q_{n}(e_{0})-e_{0} bigrVert _{infty }, \ xi _{n} =& biglVert Q_{n}(f_{1})-f_{1} bigrVert _{infty }, \ kappa _{n} =& biglVert Q_{n}(f_{2})-f_{2} bigrVert _{infty }, end{aligned}$$

where (rho _{n}, xi _{n}, kappa _{n} rightarrow 0 ) as (nrightarrow infty ), then,

$$ biglVert Q_{n}(f)-f bigrVert _{infty }leq Vert f Vert _{infty }rho _{n}+(2+ rho _{n})omega ^{*} ( f,sqrt{rho _{n}+2xi _{n}+kappa _{n}} ), $$

for (fin C_{*}(mathcal{S}) ).

In this regard, it is clear that there is a close relation between (omega ^{*} ( f,delta ) ) and the particular Korovkin subset chosen for the space (C_{*}(mathcal{S}) ), (see [23]). Then, we can state the following theorem with the help of the above.

Theorem 6

For every (fin C_{*}(mathcal{S}) ) and (ngeq 1 ),

$$ biglVert mathcal{G}_{n}^{alpha _{1},beta _{1}}(f)-f bigrVert _{infty } leq 2omega ^{*} biggl( f,sqrt{e^{-p_{n}(alpha _{1}-alpha _{2})/(n+ beta _{1})} frac{p_{n}}{ e(n+beta _{1})}} biggr), $$

under the same assumptions of Theorem 3.

Proof

It is obvious that, (rho _{n} ) and (kappa _{n} ) equal zero due to their definitions. On the other hand, it is easy to show that

$$ xi _{n}=e^{-p_{n}(alpha _{1}-alpha _{2})/(n+beta _{1})} frac{p_{n}}{2 e(n+beta _{1})}, $$

from (4.3) with (lambda =1 ) for every (ngeq 1 ). Hence, the proof is completed. □

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