On Approximation by Linear Combinations of Modified Summation Operators of Integral Type in Orlicz Spaces

In this paper, the authors introduce the Orlicz spaces corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combinations of modified summation operators of integral type in the Orlicz spaces.


Introduction and Main Results
Throughout this paper, we use C to denote an absolute constant independent of anything, which may be not necessarily the same in different cases.
There are many types of integral operators (see, for example, [1][2][3][4][5][6][7] and closely related references therein).In the paper [8], Ueki provided a characterization for the boundedness and compactness of the Li-Stević type integral operators from the weighted Bergman space L p a (dA α ) to the β-Zygmund space Z β .Later, Li and Ma [9] investigated the boundedness and compactness of products of composition operators and integral type operators from Zygmund-type spaces to Q k spaces.Recently, the equivalent characterizations for the boundedness and compactness of several integral type operators from F(p, q, s) space to α-Bloch-Orlicz and β-Zygmund-Orlicz spaces were developed in [10].Gupta and Yadav [11] estimated the approximation by complex summation integral type operator in compact disks, where p n,k (z) = ( n k )z k (1 − z) n−k .In [8][9][10][11], some approximating properties of integral type operators in complex spaces were established.Vural, Altin, and Yüksel [12] provided weighted approximation and obtained a rate of convergence of Schurer's generalization of the q-Hybrid summation operators of integral type .
In [13], Govil and Gupta considered the simultaneous approximation for the Stancu-type generalization of certain summation operators of integral type by hypergeometric series.Srivastava and Gupta [14] introduced and investigated a new sequence of linear positive operators which included some well-known operators as its special cases and obtained an estimate on the rate of convergence by means of the decomposition technique for functions of bounded variation.In [15], Gupta, Mohapatra, and Finta studied the mixed summation operators of integral type and obtained the rate of point-wise convergence, where In [12][13][14][15], some approximating properties of integral type operators were discussed in C[0, ∞), which is a special case of the Orlicz space.For f ∈ L * Φ [0, ∞), the modified summation operators of integral type B n ( f , x) are defined in [16] as where b n,k (x) = (n+k)!(k−1)!n!
Recently, Han and Wu [16] obtained the following direct, inverse, and equivalent theorems of modified summation operators of integral type in Orlicz spaces.
In recent years, since the Orlicz spaces are more general than the classical L p spaces, which is composed of measurable functions f (x) such that | f (x)| p are integrable, there is growing interest in problems of approximation in Orlicz spaces.
For smoothly proceeding, we recall from [17] some definitions and related results.
For a given Young function Φ(t), its complementary Young function is denoted by Ψ(t).
Let Φ(t) be a Young function.We define the Orlicz class L Φ [0, ∞) as the collection of all Lebesgue measurable functions u(x) on [0, ∞).Since the integral The Orlicz norm, which is equivalent to the Luxemburg norm on L * Φ [0, ∞), is given by For f ∈ L * Φ [0, ∞), the weighted K-functional K r,ϕ ( f , t r ), the modified weighted K-functional K r,ϕ ( f , t r ), and the weighted modulus of smoothness ω r,ϕ ( f , t) Φ are given, respectively, by and x, and g (r−1) ∈ AC loc means that g is r − 1 times differentiable and g (r−1) is absolutely continuous in every closed finite interval Between the weighted modulus of smoothness and the modified weighted K-functional, there exists the following equivalent theorem.
Then there exist some constants C and t 0 such that Between the weighted modulus of smoothness and the weighted K-functional, there exists the following equivalent theorem.
Then there are some constants C and t 0 such that Currently, there are few results about linear combinations of modified summation operators of integral type B n ( f , x).In this article, we investigate the approximation of linear combinations of modified summation operators of integral type The linear combinations of modified summation operators of integral type B n ( f , x) are defined as where (5) Our main results in this paper can be stated as the following three theorems.
These main results improve some conclusions in [19] and increase the approximating speed of corresponding operators.

Proof of the Direct Theorem
In order to prove the direct theorem, we need several lemmas below.
Lemma 1.The modified summation operator of integral type B n ( f , x) defined in Equation (1) satisfies Proof.This follows from simple calculation.

Lemma 2 ([19]
).If u locates between x and t, then Proof.By Lemma 3.2 in [16], we have Using Equation ( 5), we obtain The proof of Lemma 3 is complete.
We are now in a position to prove Theorem 6.

Proof of Theorem 6. Let
Taylor's formula with integral remainder of g ∈ U reads where From Equation ( 5), it follows that L n,r (g, x) − g(x) = L n,r (R 2r (g, t, x), x) and Now we estimate |R 2r (g, t, x)|.

Proofs of the Inverse and Equivalent Theorems
For proving Theorems 7 and 8, we need the following lemmas.
we have Therefore, by Jensen's inequality [20] and the inequality (2), we obtain where b n,k+r−i (t) = 0 for n + r − i ≤ 0. Combining this with Equation ( 5) leads to Lemma 5 is thus proved.