Estimation of Approximation with Jacobi Weights by Multivariate Baskakov Operator

Abstract

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.

1. Introduction and Main Results

Let 𝐶_𝐵[0,+∞) be the set of bounded continuous functions on [0,+∞); then the unvariate Baskakov operator is defined by 𝑉_𝑛,1𝑓,𝑥=^∞_𝑘=0𝑣_𝑛,𝑘(𝑥)𝑓𝑘𝑛, (1) where 𝑣_𝑛,𝑘(𝑥)=𝑘𝑥𝑛+𝑘−1_𝑘(1+𝑥)^{−𝑛−𝑘},𝑥∈[0,∞]. (2)

There are many papers to study the univariate Baskakov-type operator (see [1–4]). 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 𝐽𝑎𝑐𝑜𝑏𝑖 weight; that is, for any 𝑓∈𝐶_𝜔,0≤𝜆≤1, 𝜔(𝑥)𝑉_𝑛,1𝑓,𝑥−𝑓(𝑥)≤𝐶𝜔²_𝜑𝜆𝑓;𝑛^−1/2𝜑^1−𝜆(𝑥)_𝜔, (3) where 𝐶_𝜔={𝑓∣𝑓∈𝐶[0,∞],𝜔𝑓∈𝐿_∞[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 𝑇∈𝑅^𝑑(𝑑∈𝑁), let 𝑇=𝑇_𝑑=𝐱={(𝑥₁,𝑥₂,⋅,𝑥_𝑑), and let 0≤𝑥_𝑖<∞,0≤𝑖≤𝑑}, 𝐱=𝑥₁,𝑥₂⋯𝑥_𝑑,,𝐤=𝑘₁,𝑘₂,…,𝑘_𝑑,𝐱^𝐤=𝑥^𝑘1₁𝑥^𝑘2₂⋯𝑥^𝑘𝑑_𝑑,|𝐤|=𝑘₁+𝑘₂+⋯+𝑘_𝑑|𝐱|=𝑥₁+𝑥₂+⋯+𝑥_𝑑,𝑛𝐤=𝑛!(|𝐤,𝐤!𝑛−|)^∞_𝐤=0=^∞_𝑘1^∞_=0𝑘2=0⋯^∞_𝑘𝑑=0,𝐶^𝑟_𝜑(𝑇)=𝑓∈𝐶_𝐵(𝑇)∶𝑓∈𝐶^𝑟_𝐵(𝑇),and𝜑^𝑟_𝑖𝐷^𝑟_𝑖𝑓∈𝐶_𝐵(𝑇),1≤𝑖≤𝑑, (4) where 𝐶_𝐵(𝑇) represents the set of bounded continuous functions in 𝑇. The norm defined on 𝑇 is ‖𝑓‖_∞=max_𝑥∈𝑇|𝑓(𝑥)|.

Using the previous notations, the multivariate Baskakov operator is defined by 𝑉_𝑛,𝑑𝑓;𝐱=^∞_𝐤=0𝑣_𝑛,𝐤(𝐱)𝑓𝐤𝑛, (5) where 𝑣_𝑛,𝐤(𝐱)=(_𝐤^{𝑛+|𝐤|−1})𝐱^𝐤(1+𝐱)^{−𝑛−|𝐤|}. In [7], the authors gave the equivalent relation between 𝐾-functional and modulus of smoothness and obtained the following result: for any 𝑓∈𝐶_𝐵(𝑇), 𝑉_𝑛,𝑑𝑓−𝑓_∞≤𝐶𝜔²_𝜑1𝑓,√𝑛_∞, (6) where 𝜔^𝑟_𝜑𝑓,𝑡_∞=sup^𝑑_{0≤ℎ≤𝑡𝑖=1}Δ^𝑟_{ℎ𝜑𝑖𝐞𝑖}𝑓_∞,Δ^𝑟_ℎ𝐞𝑓(𝐱)=0,else,_𝑟𝑖∑₌₀𝑟𝑖(−1)^𝑖𝑓(𝐱+𝑖ℎ𝐞),𝐱,𝐱+𝑟ℎ𝐞∈𝑇. (7) The 𝑟th K-functional is defined as follows 𝐾^𝑟_𝜑𝑓;𝑡=inf_𝑔𝑓−𝑔_∞+𝑡^𝑟𝑑_𝑖=1𝜑^𝑟_𝑖𝐷^𝑟_𝑖𝑔_∞,𝑔∈𝐶²_𝜑(𝑇). (8) And the 𝑟th modulus of smoothness is 𝜔^𝑟_𝜑𝑓,𝑡_∞=sup^𝑑_{0≤ℎ≤𝑡𝑖=1}Δ^𝑟_{ℎ𝜑𝑖𝐞𝑖}𝑓_∞. (9)

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 (𝜔(𝐱)=𝐱^𝛼(1+|𝐱|)^−𝑏(0<|𝛼|<1,𝑏>0)) is not a simple generalization of normal approximation means. In the norm ‖𝜔𝑓‖_∞, the multivariate Baskakov operator is not bounded. By introducing the weighted norm 𝑓_𝜔=𝜔𝑓_∞+𝑓(𝟎), (10) 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 𝑟th weighted 𝐾-functional is defined as follows: 𝐾^𝑟_𝜑𝑓;𝑡_𝜔=inf_𝑔𝑓−𝑔_𝑤+𝑡^𝑟𝑑_𝑖=1𝜑^𝑟_𝑖𝐷^𝑟_𝑖𝑔_𝜔,𝑔∈𝐶²_𝜑(𝑇),𝑓∈𝐶_𝐵(𝑇), (11) and 𝑟th weighted Ditzian-Totik modulus of smoothness is 𝜔^𝑟_𝜑𝑓,𝑡_𝜔=sup^𝑑_{0≤ℎ≤𝑡𝑖=1}Δ^𝑟_{ℎ𝜑𝑖𝐞𝑖}𝑓_𝜔,𝑔∈𝐶_𝐵(𝑇), (12) where 𝜑²_𝑖(𝐱)=𝑥_𝑖(1+|𝐱|).

Using the previous notations, we will present our results. Firstly, we study the unboundedness of multivariate Baskakov operator in the normal norm.

Theorem 1. Let 𝑓∈𝐶_𝐵(𝑇); then the operator 𝑉_𝑛,𝑑(𝑓;𝑥) is unbounded in the norm ‖𝜔𝑓‖_∞.

In addition, introducing the new norm, we establish the upper bound estimation of weighted approximation by multivariate Baskakov operator.

Theorem 2. Let 𝑓∈𝐶_𝐵(𝑇); then one has 𝑉_𝑛,𝑑𝑓−𝑓_𝜔≤𝐶𝜔²_𝜑1𝑓,√𝑛_𝜔. (13)

Remark 3. Our result reveals two things: (i) for any multivariate bounded continuous function 𝑓∈𝐶_𝐵(𝑇), there is a multivariate polynomial 𝑉_𝑛,𝑑(𝑓) that approximates 𝑓 arbitrarily well (when 𝑛 is sufficiently large) in the continuous space. (ii) Quantitatively, the approximation accuracy of a polynomial 𝑉_𝑛,𝑑(𝑓) can be controlled by the 𝜔²_𝜑(𝑓,1/√𝑛)_𝜔; here 𝜔²_𝜑(𝑓,1/√𝑛)_𝜔 is the weighted smoothness modulus of the function 𝑓.

2. Some Lemmas

In order to prove our results, we will show some lemmas in this section.

Lemma 4 (see [6]). For 𝑓∈𝐶_𝜔,0≤𝜆≤1, one has 𝑉_𝑛,1𝑓−𝑓_𝜔≤𝐶𝜔²_𝜑1𝑓,√𝑛_𝜔. (14)

Lemma 5 (see [5]). For 𝑐,𝑑≥0, ^∞_𝑘=1𝑘𝑛^−𝑐𝑘1+𝑛^𝑑𝑣_𝑛,𝑘(𝑥)≤𝐶𝑥^−𝑐(1+𝑥)^𝑑,𝑥>0. (15)

Lemma 6. For 𝑓∈𝐶_𝐵(𝑇), then one has 𝜔(𝐱)^∞_𝐤=1𝑣_𝑛,𝐤(𝐱)𝑓𝐤𝑛≤𝐶𝜔𝑓_∞. (16)

Proof. For convenience, we only prove the case of 𝑑=2; for a general 𝑑, we can prove similarly. For 𝑑=2, we have 𝜔(𝐱)^∞_𝐤=1𝑣_𝑛,𝐤(𝐱)𝑓𝐤𝑛=𝜔(𝐱)^∞_𝑘1=1𝑣_𝑛,𝑘1𝑥₁^∞_𝑘2=1𝑣_{𝑛+𝑘1,𝑘2}𝑥₂1+𝑥₁𝑓𝑘₁𝑛,𝑘₂𝑛=𝜔(𝐱)^∞_𝑘1=1𝑣_𝑛,𝑘1𝑥₁^∞_𝑘2=1𝑣_{𝑛+𝑘1,𝑘2}𝑥₂1+𝑥₁𝑓𝑘₁𝑛,𝑘₂𝑛𝑘×𝜔¹𝑛,𝑘₂𝑛𝑘₁𝑛^−𝛼1𝑘₂𝑛^−𝛼2×𝑘1+¹𝑛+𝑘₂𝑛^−𝑏≤𝐶𝜔𝑓_∞𝜔(𝐱)^∞_𝑘1=1𝑣_𝑛,𝑘1𝑥₁𝑘₁𝑛^−𝛼1×^∞_𝑘2=1𝑣_{𝑛+𝑘1,𝑘2}𝑥₂1+𝑥₁𝑘₂𝑛^−𝛼2×𝑘1+¹𝑛+𝑘₂𝑛^−𝑏=𝐶𝜔𝑓_∞𝜔(𝐱)^∞_𝑘1=1𝑣_𝑛,𝑘1𝑥₁𝑘₁𝑛^−𝛼1×^∞_𝑘2=1𝑣_{𝑛+𝑘1,𝑘2}𝑥₂1+𝑥₁𝑘1+²𝑛+𝑘₁^𝑏×𝑘₂𝑛+𝑘₁^−𝛼2≤𝐶𝜔𝑓_∞∣𝜔(𝐱)𝑥^−𝛼1₁𝑥^−𝛼2₂1+𝑥₁+𝑥₂^𝑏≤𝐶𝜔𝑓_∞. (17) In the previous derivation, Lemma 5 and the following inequality were used: ^∞_𝑘=0𝑣_𝑛,𝑘(𝑥𝑛)𝑘+1^𝑙≤𝐶𝑥^−𝑙,𝑙∈𝑁. (18)

3. 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 𝑓_𝑚𝑥₁,𝑥₂=1𝑥^𝛼1₁𝑥^𝛼2₂+().1/𝑚 (19) Then we have 𝜔𝑓_𝑚∞=max_𝑥>0𝑥^−𝛼1₁𝑥^−𝛼2₂1+𝑥₁+𝑥₂^−𝑏1𝑥^𝛼1₁𝑥^𝛼2₂+()≤11/𝑚1+𝑥₁+𝑥₂^𝑏≤1. (20) On the other hand, 𝜔𝑉_𝑛,𝑑𝑓_∞=max_𝑥>0𝜔(𝐱)^∞_𝑘=0𝑉_𝑛,𝑘(𝑥)𝑓𝑘𝑛≥max_𝑥>0𝜔(𝐱)𝑓_𝑚(0,0)𝑣_𝑛,0(𝑥)−max_𝑥>0𝜔(𝐱)^∞_𝑘=1𝑣_𝑛,𝑘(𝑥)𝑓𝑘𝑛≥max_𝑥>0𝜔(𝐱)𝑓_𝑚(0,0)1+𝑥₁+𝑥₂^−𝑛3−𝐶≥𝑚2^𝑛+𝑏−𝐶⟶∞(𝑚⟶∞). (21)

As we know, multivariate Baskakov operator 𝑉_𝑛,𝑑𝑓 has the property of preserve linear, for convenience the following discussion, we may suppose 𝑓(𝟎)=0.

Proof of Theorem 2. In order to prove Theorem 2, we first show the following inequality: 𝑉_𝑛,𝑑𝑓−𝑓_𝜔≤𝑓_𝜔,𝑓∈𝐶_𝐵(𝑇),𝑛^𝑑−1∑_𝑖=0𝜑²_𝑖𝐷²_𝑖𝑓_𝜔,𝑓∈𝐶²_𝜑(𝑇). (22)
For the first inequality, it is obviously correct since ‖𝑉_𝑛,𝑑𝑓‖_𝜔≤𝐶‖𝑓‖_𝜔. In the following we will certify the second inequality by the mathematical induction. Our proof is based on an induction argument for the dimension 𝑑. We will also use a method called decomposition technique for the multivariate Baskakov operator.(i) For 𝑑=1, from Lemma 4, we have 𝑉_𝑛,𝑑𝑓−𝑓_𝜔≤𝑛⁻¹𝜑²𝑓^′′_𝜔. (23)(ii)When 𝑑=𝑟, suppose that 𝑉_𝑛,𝑑𝑓−𝑓_𝜔≤𝜑²_𝑖𝐷²_𝑖𝑓_𝜔 (24)is established. So when 𝑑=𝑟+1, using the multivariate decomposition technique in [7], we have 𝑉_𝑛,𝑑𝑓,𝑥=^∞_𝑘1=0𝑣_𝑛,𝑘1(𝑥)^∞_𝑘∗=0𝑣_{𝑛+𝑘1,𝐤}^∗𝐱^∗1+𝑥₁, (25) where 𝐱^∗=𝑥₂,𝑥₃,…,𝑥_𝑛,𝐱=𝑥₁,𝐱^∗,𝐤^∗=𝑘₂,𝑘₃,…,𝑘_𝑑,𝐤=𝑘₁,𝐤^∗. (26) Let 𝑔_𝑘1(𝐮)𝑘=𝑓¹𝑛,𝑘1+¹𝑛𝐮,𝐮∈𝑇_𝑑−1,ℎ(𝑡)()()𝐱=ℎ𝑡,𝐱=𝑓𝑡,1+𝑡^∗1+𝑥₁. (27) We get 𝜔(𝐱)𝑉_𝑛,𝑑(𝐱)(𝐱)𝑓,𝐱−𝑓=𝜔^∞_𝑘1=0𝑣_𝑛,𝑘1𝑥₁𝑉_{𝑛+𝑘1,𝑑−1}𝑔_𝑘1(⋅),𝐳−𝑔_𝑘1(𝐳)+𝑉_𝑛,1ℎ(⋅),𝑥₁−ℎ𝑥₁=𝑃+𝑄, (28) where 𝑧=𝐱^∗/(1+𝑥₁), and we have |𝑃|=𝜔(𝐱)^∞_𝑘1=0𝑣_𝑛,𝑘1𝑥₁𝑉_{𝑛+𝑘1,𝑑−1}𝑔_𝑘1(⋅),𝐳−𝑔_𝑘1(𝐳)=𝑥^𝛼1₁1+𝑥₁^{−𝑏+|𝛼∗|∞}_𝑘1=0𝑣_𝑛,𝑘1𝑥₁𝜔(𝐳)×𝑉_{𝑛+𝑘1,𝑑−1}𝑔_𝑘1(⋅),𝐳−𝑔_𝑘1(𝐳)≤𝑥^𝛼1₁1+𝑥₁^{−𝑏+|𝛼∗|∞}_𝑘1=01𝑛+𝑘₁𝑣_𝑛,𝑘1𝑥₁^𝑟_𝑖=1𝜑²_𝑖𝐷²_𝑖𝑓_𝜔. (29) The previous derivation used the following inequality: 𝜔(𝐱)=𝐱^𝛼(1+|𝐱|)^−𝑏=𝑥^𝛼1₁𝑥^∗1+𝑥₁^𝛼∗1+𝑥₁^{−𝑏+|𝛼∗|}𝑥1+^∗1+𝑥₁^−𝑏=𝑥^𝛼1₁1+𝑥₁^{−𝑏+|𝛼∗|}𝜔(𝐳),𝜑²_𝑖(𝐮)𝐷²_𝑖(𝐮)𝑔_𝑘1(𝐮)𝜔(𝐮)=𝐮_𝑖(|𝐮𝑘1+|)1+¹𝑛𝐷²_𝑖+1𝑓𝑘₁𝑛,𝑘1+¹𝑛𝐮×𝐮^𝛼∗(1+|𝐮|)^−𝑏=𝑛𝑘₁^𝛼1𝑛𝑛+𝑘₁^{−𝑏+|𝛼∗|}𝜔𝜑²_𝑖+1𝐷²_𝑖+1𝑓𝑘₁𝑛,𝑘1+¹𝑛𝐮. (30) So we get 𝜑²_𝑖𝐷²_𝑖𝑔_𝑘1𝜔≤𝑛𝑘₁^𝛼1𝑛𝑛+𝑘₁^{−𝑏+|𝛼∗|}𝜑²_𝑖𝐷²_𝑖𝑓_𝜔. (31) From Lemma 6, we have |𝑃|≤𝑥^𝛼1₁1+𝑥₁^{−𝑏+|𝛼∗|∞}_𝑘1=01𝑛+𝑘₁𝑝_𝑛,𝑘1𝑥₁×𝑛𝑘₁^𝛼1𝑛𝑛+𝑘₁^{−𝑏+|𝛼∗|}𝜑²_𝑖𝐷²_𝑖𝑓_𝜔≤𝐶𝑛𝜑²_𝑖𝐷²_𝑖𝑓_𝜔. (32) In the following, we will establish the estimation of 𝑄: |𝑄|=𝑉_𝑛,1ℎ(⋅),𝑥₁−ℎ𝑥₁𝜔(𝐱=𝑥_𝛼11)𝐱^∗𝛼∗(1+|𝐱|)^−𝑏𝐵_𝑛,1ℎ(⋅),𝑥₁−ℎ𝑥₁=𝐱^∗𝛼∗1+|𝑥|1+𝑥₁^−𝑏1+𝑥₁^−𝑏𝑥^𝛼1₁×𝑉_𝑛,1ℎ(⋅),𝑥₁−ℎ𝑥₁≤𝐶𝑛𝐱^∗𝛼∗1+|𝑥|1+𝑥₁^−𝑏𝜑²ℎ^′′_𝜔, (33) and for the term ‖𝜑²ℎ^′′‖_𝜔, we have the following estimation: 𝜑²ℎ^′′_𝜔=max_0≤𝑡<∞𝑡(1+𝑡)𝑡^𝛼1(1+𝑡)^−𝑏ℎ^′′()𝐱𝑡,1+𝑡^∗1+𝑥₁=max_0≤𝑡<∞𝑡(1+𝑡)𝑡^𝛼1(1+𝑡)^−𝑏𝐷²₁𝑓+^𝑑_𝑖=2𝐱_𝑖1+𝑥₁𝐷²_1𝑖𝑓+^𝑑_𝑖=2𝐱_𝑖1+𝑥₁𝐷²_𝑖1𝑓+^𝑑𝑑_𝑖=2𝑗=2𝑥_𝑖𝑥_𝑗1+𝑥₁²𝐷²_𝑖,𝑗𝑓()𝐱𝑡,1+𝑡^∗1+𝑥₁=max_0≤𝑡<∞𝑡(1+𝑡)𝑡^𝛼1(1+𝑡)^−𝑏𝐿₁, (34) where 𝐿₁=1+𝑥₁|𝐱|𝜑1+²¹𝐷²₁𝑓+^𝑑_𝑖=2𝜑²_1𝑖𝐷²_1𝑖𝑓+^𝑑_𝑖=2𝜑²_𝑖1𝐷²_𝑖1𝑓+^𝑑_𝑖=2𝑡𝐱1+𝑡^𝑖|𝐱|𝜑1+^2𝑖𝐷²_𝑖𝑓+^𝑑_{𝑖,𝑗=2,𝑖≠𝑗}𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑓()𝐱𝑡,1+𝑡^∗1+𝑥₁. (35) Note that 𝜔()𝐱𝑡,1+𝑡^∗1+𝑥₁=𝐱^∗𝛼∗1+𝑡1+𝑥₁^|𝛼∗|𝑡^𝛼1(1+𝑡)^−𝑏1+|𝐱|1+𝑥₁^−𝑏. (36) We have 𝜑²ℎ^′′_𝜔=𝐱^{∗−𝛼∗}1+|𝑥|1+𝑥₁^𝑏1+𝑡1+𝑥₁^−|𝛼∗|𝐿₂. (37) Here 𝐿₂=1+𝑥₁1+|𝐱|𝜔𝜑²₁𝐷²₁𝑓+^𝑑_𝑖=2𝜔𝜑²_1𝑖𝐷²_1𝑖𝑓+^𝑑_𝑖=2𝜔𝜑²_𝑖1𝐷²_𝑖1𝑓+^𝑑_𝑖=2𝑡𝐱1+𝑡1+^𝑖|𝐱|𝜔𝜑²_𝑖𝐷²_𝑖𝑓+^𝑑_{𝑖,𝑗=2,𝑖≠𝑗}𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑓()𝐱𝑡,1+𝑡^∗1+𝑥₁. (38) From [7], we have 𝐷²_𝑖,𝑗𝑓(𝐱)≤sup_{1≤𝑖≤𝑑}𝐷²_𝑖𝑓(𝐱,𝜑_𝑖,𝑗)(𝐱)≤min𝜑_𝑖(𝐱),𝜑_𝑗(𝐱,𝜑₂)ℎ^′′_𝜔=𝐱^{∗−𝛼∗}1+|𝑥|1+𝑥₁^𝑏1+𝑡1+𝑥₁^−|𝛼∗|𝜑²_𝑖𝐷²_𝑖𝑓_𝜔. (39)(40) Hence |𝑄|≤𝐶𝑛𝜑²_𝑖𝐷²_𝑖𝑓_𝜔. (41) Thus we have 𝑉_𝑛,𝑑𝑓−𝑓_𝜔=𝑉_𝑛,𝑑𝑓−𝑉_𝑛,𝑑𝑔+𝑉_𝑛,𝑑𝑔−𝑔+𝑔−𝑓_𝜔≤𝐶𝑉_𝑛,𝑑𝑓−𝑉_𝑛,𝑑𝑔_𝜔+𝑓−𝑔_𝜔≤𝐶𝑛𝜑²_𝑖𝐷²_𝑖𝑓_𝜔+𝑓−𝑔_𝜔≤𝐶𝐾²_𝑟1𝑓,√𝑛_𝜔≤𝐶𝜔²_𝜑1𝑓,√𝑛_𝜔. (42) Here we used the following facts: for 𝑓∈𝐶_𝐵(𝑇), then 1𝐶𝜔^𝑟_𝜑𝑓,𝑡_𝜔≤𝐾^𝑟_𝜑𝑓;𝑡_𝜔≤𝐶𝜔^𝑟_𝜑𝑓,𝑡_𝜔. (43) The result is easily obtained from [7–9]. Theorem 2 is completed.