Estimation of Approximation with Jacobi Weights by Multivariate Baskakov Operator

We first give the unboundedness of multivariate Baskakov operators with the normal weighted norm. By introducing new norms, using the multivariate decomposition technique and the modulus of smoothness with Jacobi weight, the upper bound estimation of multivariate Baskakov operators is obtained. The obtained results not only generalize the corresponding ones for multivariate Baskakov operators without weights, but also give the approximation accuracy with the Jacobi weights approximation.


Introduction and Main Results
Let   [0, +∞) be the set of bounded continuous functions on [0, +∞); then the unvariate Baskakov operator is defined by where There are many papers to study the univariate Baskakovtype operator (see [1][2][3][4]).In [5], the authors discussed the convergence property of Baskakov operator's approximation with weights by introducing weighted smoothness modulus and obtained the characteristics of approximation.And in [6], using the weighted smoothness modulus, the close connection between the derivative of Baskakov operator and the smoothness of function approximated which has been investigated, the upper bound estimation has been established with the  weight; that is, for any  ∈   , 0 ≤  ≤ 1,      () ( ,1 (, ) −  ()) ≤  2   (;  −1/2  1− ())  , (3) where   = { |  ∈ [0, ∞],  ∈  ∞ [0, +∞]}.In this paper, the letter C, appearing in various formulas, denotes a positive constant independent of n, x, and f.Its value may be different at different occurrences, even within the same formula.
Let  ∈   ( ∈ ), let  =   = x = {( 1 ,  2 , ⋅,   ), and let 0 ≤   < ∞, 0 ≤  ≤ }, and where   () represents the set of bounded continuous functions in .The norm defined on  is Using the previous notations, the multivariate Baskakov operator is defined by where In [7], the authors gave the equivalent relation between -functional and modulus of smoothness and obtained the following result: for any  ∈   (), where The th K-functional is defined as follows And the th modulus of smoothness is Naturally, we will consider the following problem: "are there similar results in the approximation with weights by multivariate Baskakov operator?"However, approximation with Jacobi weights ((x) = x  (1 + |x|) − (0 < || < 1,  > 0)) is not a simple generalization of normal approximation means.In the norm ‖‖ ∞ , the multivariate Baskakov operator is not bounded.By introducing the weighted norm          =         ∞ +  (0) , we find that the multivariate Baskakov operator is bounded, and thus we can investigate approximation capability of the operators.In the previous weighted norm, the th weighted -functional is defined as follows: and th weighted Ditzian-Totik modulus of smoothness is where Using the previous notations, we will present our results.Firstly, we study the unboundedness of multivariate Baskakov operator in the normal norm.
In addition, introducing the new norm, we establish the upper bound estimation of weighted approximation by multivariate Baskakov operator.
Theorem 2. Let  ∈   (); then one has Remark 3. Our result reveals two things: (i) for any multivariate bounded continuous function  ∈   (), there is a multivariate polynomial  , () that approximates  arbitrarily well (when  is sufficiently large) in the continuous space.(ii) Quantitatively, the approximation accuracy of a polynomial  , () can be controlled by the  2  (, 1/√)  ; here  2  (, 1/√)  is the weighted smoothness modulus of the function .

Some Lemmas
In order to prove our results, we will show some lemmas in this section.Lemma 4 (see [6]).For  ∈   , 0 ≤  ≤ 1, one has Lemma 5 (see [5]).For ,  ≥ 0, (15) Lemma 6.For  ∈   (), then one has Proof.For convenience, we only prove the case of  = 2; for a general , we can prove similarly.For  = 2, we have In the previous derivation, Lemma 5 and the following inequality were used:

The Proof of Theorems
In this section, we will show the unboundedness of multivariate Baskakov operator in the general weighted norm and then give the proof of Theorem 2.
Proof of Theorem 1.Let Then we have + −  → ∞ ( → ∞) . ( As we know, multivariate Baskakov operator  ,  has the property of preserve linear, for convenience the following discussion, we may suppose (0) = 0.
Proof of Theorem 2. In order to prove Theorem 2, we first show the following inequality: For the first inequality, it is obviously correct since ‖ , ‖  ≤ ‖‖  .In the following we will certify the second inequality by the mathematical induction.Our proof is based on an induction argument for the dimension .We will also use a method called decomposition technique for the multivariate Baskakov operator.is established.So when  =  + 1, using the multivariate decomposition technique in [7], we have where Let ) . ( We get where  = x * /(1 +  1 ), and we have The previous derivation used the following inequality: So we get where Note that  (, (1 + ) We have The result is easily obtained from [7][8][9].Theorem 2 is completed.