Approximation by multivariate Baskakov–Durrmeyer operators in Orlicz spaces

Employing some properties of multivariate Baskakov–Durrmeyer operators and utilizing modified K-functional and a decomposition technique, the authors obtain the direct theorem and weak type inverse theorem in the Orlicz spaces.


Preliminaries
For proceeding smoothly, we recall from [27]  For a Young function (t), its complementary Young function is denoted by (t).
A Young function (t) is said to satisfy the 2 -condition, denoted by ∈ 2 , if there exist t 0 ≥ 0 and C ≥ 1 such that (2t) ≤ C (t) for t ≥ t 0 .
Throughout the paper we shall use the following standard notations: for m ∈ N and r ∈ N. Let (t) be a Young function.We define the Orlicz class L (R m 0 ) as the collection of all Lebesgue measurable functions f (x) on R m 0 for which We also define the Orlicz space L * (R m 0 ) as the set of all Lebesgue measurable functions f (x) on R m 0 , such that R m 0 (|αf (x)|) dx < ∞ for some α > 0. The Orlicz space is a Banach space under the Luxemburg norm The Orlicz norm, an equivalence of the Luxemburg norm on L * (R m 0 ), is given by and satisfies If (u) = u p p for 1 < p < ∞, then the complementary function becomes (u) = |u| q q with 1 p + 1 q = 1, and then Throughout this paper we use C to denote a constant independent of n and x, which may be not necessarily the same in different cases.
For x ∈ R m 0 , we introduce weight functions for m = 1 and for m > 1 and 1 ≤ i ≤ m.We also define the weighted Sobolev space where |k| ≤ r and • R m 0 is the interior of R m 0 .The modified Peetre K -functionals are defined by For any vector e ∈ R m , we write for the rth forward difference of a function f in the direction of e.We define the modulus of smoothness of f ∈ L * (R m 0 ) as

Motivations and main results
Between the modulus of smoothness and the K -functional there exists the following equivalent theorems.
Then there exist some constants C and t 0 such that Theorem B ( [31]) Let f ∈ L * (R m 0 ) and r ∈ N. Then there exist some constants C and t 0 such that Let The well-known Baskakov operators were defined in [2] as These operators can be used to approximate any function f defined on [0, ∞).For f ∈ L p [0, ∞) and 1 ≤ p < ∞, the Baskakov-Durrmeyer operators were defined in [17] as For a function f defined on R m 0 , the multivariate Baskakov operators were defined in [5] as where The multivariate Baskakov-Durrmeyer operators were defined in [4] as where There are many approximation results about one variable operator of the Baskakov type in C[0, ∞) or L p [0, ∞), see [1, 2, 7-9, 15, 17-19, 29, 30].But there are few approximation results about multivariate Baskakov type operators (see [4,5,13,22]) or multivariate Durrmeyer type operators (see [3,20]).
In this paper, we will discover not only the direct theorem, but also the weak type inverse theorem for the multivariate Baskakov-Durrmeyer operators V n,m (f , x).
Our main results can be stated in the following two theorems.

Proof of direct theorem
In order to prove the direct theorem, we need several lemmas.
Proof Employing the decomposition formula and Jensen's inequality, we obtain By the double Inequality (1), we complete the proof of Lemma 1. where the notation means "define", and Now we start out to estimate From [17], we obtain From the Inequality (6), Jensen's inequality, and the convexity of (t), it follows On the other hand, by definition, we can deduce To estimate the second term J 2 = V * n,1 (h(•), x 1 )h(x 1 ), we use a similar method as estimating (6) and acquire By the Inequality (8) and the convexity of (t), we arrive at When denoting By virtue of the facts that ϕ 12 (x) is not bigger than ϕ 1 (x) or ϕ 2 (x) and that Combining the above inequality with ( 5) and (7) and paying attention to computation formulas of norm and the double Inequality (1) yield The proof of Lemma 2 is complete.
Proof of Theorem 1 Our proof is based on induction on the dimension m and on a decomposition for the Baskakov-Durrmeyer operator.
For m ≥ 1, the proof of Theorem 1 follows from combining Lemmas 1 and 2 with the estimates The first estimate in ( 9) can be derived from Lemma 1.By Lemma 2, the second estimate in ( 9) is valid for m = 1, 2. If the second estimate in ( 9) is valid for m = r ≥ 2, that is then we have to further verify its validity for m = r + 1. Let We claim that the decomposition formula is valid, where g u 1 (t) = f (u 1 , (1 + u 1 )t) for 0 ≤ t < ∞.From the above formula, it follows that where By the inequality which can be obtained from (10) and Jensen's inequality, we arrive at On the other hand, by definition, we can deduce As a result, we obtain By the Inequality (8) and the convexity of (t), we acquire Recalling that ϕ ij (x) is not bigger than ϕ i (x) or ϕ j (x), and using the fact Combining the Inequalities ( 11), (12), and ( 13) and paying attention to computation of norm and the Inequality (1), we obtain the second estimate of (9) for any m ≥ 2.

Proof of inverse theorem
In order to prove the inverse theorem, we need several lemmas.
and n > m for n, m ∈ N. Then Proof By straight computation, we have where (n+k+1)(k+1) (k+2)(n+k) ≤ 1 for k ≥ 0 and n ∈ N. Using Jensen's inequality, we derive Let Then, by the Inequality (4) and for m > 1, we have Using the Inequalities ( 14), (15), and Jensen's inequality, we see that (n -2 Hence, from the computation formula of the form, it follows that Similarly, we can prove the same results for i = 1, 2, . . ., m -1.The proof of Lemma 3 is thus complete.
Proof By straight calculation, for m = 1, we have where Moreover, we can verify that for n ≥ 2. By the Inequalities (4) and ( 16), for m > 1, we have Hence, from the computation formula of the form, it follows that Similarly, we can prove the same results for i = 1, 2, . . ., m -1.The proof of Lemma 4 is thus complete.It is obvious that v 1 = 0. From Lemmas 3 and 4, it follows that

Proof of Theorem 2 Let
By [28, Lemma 2.1], we acquire v n ≤ C n n k=1 τ k .Therefore, Using the double Inequality (1), we obtain For n ≥ 2, there exists s ∈ N such that n 2 ≤ s ≤ n and Accordingly, we have Hence, by the definition of the K -functional, we deduce Finally, using (3), we finish the proof of Theorem 2.

Conclusions
In this paper, using the equivalent theorem between the modified K -functional and modulus of smoothness, employing a decomposition technique, and considering some properties of multivariate Baskakov-Durrmeyer operators in the form of Lemmas 1, 2, 3, and 4, we obtained a direct theorem and weak type inverse theorem in the Orlicz spaces f ∈ L * (R m 0 ).
some definitions and related results.
A continuous convex function (t) on [0, ∞) is called a Young function if it satisfies t→∞ (t) t = ∞.