Further approximations of Durrmeyer modification of Szasz-Mirakjan operators

The main purpose of this paper is to determine the approximations of Durrmeyer modification of Szasz-Mirakjan operators, defined by Mishra et al. (Boll. Unione Mat. Ital. (2016) 8(4):297-305). We estimate the order of approximation of the operators for the functions belonging to the different spaces. Here, the rate of convergence of the said operators is established by means of the function with derivative of the bounded variation. At last, the graphical analysis is discussed to support the approximation results of the operators.


Introduction
In 1944, Mirakjan [8] and 1950, Szász [16] introduced operators on unbounded interval [0, ∞), known as Szász-Mirakjan operators defined by An integral modification of the above operators (1) can be seen in [2] to estimate the approximation results for the integrable function. The important properties including global results, local results, simultaneous approximation, convergence properties, etc. have been studied with the above operators and their modifications in various studies (see [1,3,[11][12][13]). One of them, an interesting modification was the Durrmeyer modification of the Szász-Mirakjan operators and is written as: seen in [7]. Also, another modification into Stancu variant appeared in [9] of the above operators (2) and related properties like density, direct results as well as Voronovskaya type theorem are studied. Many approximation results are also discussed in [14,20]. A natural generalization is carried out for the above operators (2) in [10] by Mishra et al. for the study of simultaneous approximation, like where s un,j (x) = e −unx (unx) j j! by considering the sequence u n is strictly increasing of positive real number as well as u n → ∞ as n → ∞ with u 1 = 1.
Our main motive is to study the approximation properties of the proposed operators (3) for the functions from different spaces. The important properties of the above proposed operators (3) are studied by authors which can also be applied to the operators defined by (2).
In order to study the operators (3), we divide the paper into sections. Section second contains preliminary results, which are used to prove the main theorems. Section third deals with the approximation properties of the operators for the function belongs to the different spaces of functions classes. In section fourth, the rate of convergence of the operators is estimated for the functions with derivative of bounded variation. At last, we present the graphical and numerical representation for the operators in order to show the convergence of the operators.

Preliminary
This section contains the basic properties of the defined operators (3). In order to prove approximations properties, we need basic lemmas.

Lemma 1.
For all x ≥ 0 and n ∈ N, we have Proof. We can easily proof the above parts of the lemma, so we omit the proof.
Lemma 2. Consider the function g is integrable, continuous, bounded on given interval [0, ∞), then the central moments can be obtained as: where m = 0, 1, 2, . . .. So for m = 0, 1, we get the the central moments as follows: in general, we have this lead us to Lemma 3. Let the function g be the continuous and bounded on [0, ∞) endowed with supremum norm g(x) = sup x≥0 |g| then, we have Remark 1. For second order central moment, it can be written as where ζ 2 n (x) = x + 1 un .

Approximation properties
Consider C B [0, ∞) be the space of all continuous and bounded function defined on [0, ∞), endowed with supremum norm g = sup x≥0 |g(x)|, also let for any δ > 0 Also a relation can be seen for which there exists a positive constant M such that: where ω 2 (g, √ δ) is second order modulus of smoothness for the function g ∈ C B [0, ∞), which is defined by: also usual modulus of continuity can be defined for the function g ∈ C B [0, ∞) as follows: Theorem 1. Consider g ∈ C B [0, ∞) and for all x ≥ 0 then there exists a positive constant C such that where δ n =B * n ((t − x) 2 ; x) + 1 u 2 n and γ n =B * n ((t − x); x).
Proof. Here, we consider the auxiliary operators as follows: Let f ∈ E, x ≥ 0 then using Taylor's formula, we get Applying the operatorsB * n on the both sides to the above expression, it yields: Here, the following inequalities are as: and 1 un +x By considering the above inequalities (18,19) and with the help of (17), we obtaiñ = δ n f .
Also, |S * n (g; x)| ≤ g . Using this property, we get using (20) and with the help of modulus of continuity, we obtain Taking the infimum for all f ∈ E on the right hand side and by relation (11), we get Thus, the proof is completed. Now, we estimate the approximation of the defined operators (3), by new type of Lipschitz maximal function with order s ∈ (0, 1], defined by Lenze [6] as Using definition of Lipschitz maximal function, we have a theorem. Proof. By equation (22), we can write Using, Hölder's inequality with j = 2 s , l = 2 2−s , one can get Next theorem is based on modified Lipschitz type spaces [15] and this spaces is defined by and m 1 , m 2 are the fixed numbers and M > 0 is a constant.
Theorem 3. For g ∈ Lip m 1 ,m 2 M (s) and 0 < s ≤ 1, an inequality holds: Proof. We have s ∈ (0, 1] and in order to prove the above theorem, we discuss the cases on s. Case 2. for s ∈ (0, 1) then using Hölder inequality with p = 2 s , q = 2 2−s , we get This complete the proof.
Theorem 4. For the function g which is continuous and bounded on [0, ∞), the convergence of the operators can be obtained as: uniformly on any compact interval of [0, ∞).
Proof. Using Bohman-Korovkin theorem, we can get our required result. Since lim x) → x 2 and hence the proposed operators B * n (g; x) converge uniformly to the function g(x) on any compact interval of [0, ∞).

Rate of convergence by means of the function with derivative of bounded variation
This section consists the rate of convergence by means of the function with derivative of bounded variation. Let DBV [0, ∞) be the set of all class of function having derivative of bounded variation on every compact interval of [0, ∞). The following representation for the function g ∈ DBV [0, ∞), is as follows: where h(t) is a function with derivative of bounded variation on any compact interval of [0, ∞). For investigation of the convergence of the above operators (3) to the function with derivative of bounded variation, we rewrite (3) as follows: where Such type of properties have been studied by researchers using various operators (see [4,5,[17][18][19]).

Lemma 4.
For sufficiently large value of n and for all x ≥ 0, we have Proof. Using the Lemma 2 and since the value of n is sufficiently large, so we have Similarly, we can prove other inequality.
Theorem 5. Let g ∈ DBV [0, ∞), then for all x ≥ 0, an upper bound of the operators to the function can be as: where be an auxiliary operator and V b a g(x) denotes the total variation of the function g(x) on [a, b].
Proof. Since, B * n (1; x) = 1 and hence, one can write Now, for g ∈ DBV [0, ∞), we can write as: And then, one can show Using (25), we can get Using (9), we get: Here, where Here, we consider y = x − x √ un then by the above equality, one can write Using Lemma 4 for solving second term by substituting t = x − x u , we get Hence, To solve P 2 , we reform P 2 and integrating by parts, we have On substituting t = x 1 + 1 β , we obtain Using the value of P 1 , P 2 in (32), we obtain Put the above value from (37) in (31), we obtain required result

Graphical and numerical analysis of the operators
In this section, we study the graphical representation and numerical analysis of the operators to the function.  ) represent green, red and black colors respectively in the given Figure 1. One can observe that as the value of n is increased, the error of the operators to the function is going to be least. We can say that the approach of the operators to the function is good for the large value of n.
But for the same function, if we move towards the truncation type error, we can observe by Figure 2, the approximation is not better throughout the interval [0, 2.5]. Here we consider the u n = n = 15, 35, 50 and j = 15, 35, 50, using these values, the truncation is determined. So one can observe that at a some stage, its going good but not at all. Now, we determine the convergence of the operators to the function by considering the different sequences for the operators and then we see that the variation of the convergence to the function is changed.  Table 1 at different points of x, which is going to be better as the value of n is increased.  x ↓, u n = n → at n=10 at n=50 at n=100 at n=200 at n=250 at n=500 at n=1000 0.    x ↓, u n = n  Table 3 at different points of x. By observing, we can see, the function's curve almost overlapped by the curves of the operators.  Example 5. At the same time for the same function g(x) = x 2 e 2x , 0 ≤ x ≤ 2.5, we can observe by the given Figure 6 that the accuracy of the convergence for the operators (3) is better when u n = n 2 is taken rather than when we choose the sequences u n = n and u n = n 3 2 for the same operators (3).

Remark:
After observing by all the Figures (1)- (6) and Tables (1)- (3), we can conclude that the better approximation can be obtained by choosing the appropriate sequence for the operators (3) and in addition, will get good approximation by the operators (3) for the large value of n of the positive and real sequence.
Conclusion: The approximation properties have been determined for the functions belonging to different spaces and moreover the rate of the convergence of the operators has been discussed. To validate the approximation results, the graphical representation and numerical analysis have been studied.