Analysis of Approximation by Linear Operators on Variable 𝐿_𝜌^𝑝(⋅) Spaces and Applications in Learning Theory

Abstract

This paper is concerned with approximation on variable 𝐿_𝜌^𝑝(⋅) spaces associated with a general exponent function 𝑝 and a general bounded Borel measure 𝜌 on an open subset Ω of ℝ^𝑑. We mainly consider approximation by Bernstein type linear operators. Under an assumption of log-Hölder continuity of the exponent function 𝑝, we verify a conjecture raised previously about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on the 𝐿_𝜌^𝑝(⋅) space. Quantitative estimates for the approximation are provided for high orders of approximation by linear combinations of such positive linear operators. Motivating connections to classification and quantile regression problems in learning theory are also described.

1. Introduction

Approximation by Bernstein type positive linear operators has a long history and is an important topic in approximation theory. It started with Bernstein operators [1] for proving the Weierstrass theorem about the denseness of the set of polynomials in the space 𝐶[0,1] of continuous functions on the interval [0,1]. These classical operators are defined as 𝐵_𝑛(𝑓,𝑥)=∑^𝑛_𝑘=0𝑓(𝑘/𝑛)𝑝_𝑛,𝑘(𝑥) for 𝑥∈[0,1] and 𝑓∈𝐶[0,1] with the Bernstein basis given by 𝑝_𝑛,𝑘(𝑥)=(^𝑛_𝑘)𝑥^𝑘(1−𝑥)^𝑛−𝑘. The Bernstein operators have been extended in various forms for the purpose of approximating discontinuous functions, by replacing the point evaluation functionals by some integrals. The classical examples for approximation in 𝐿^𝑝[0,1] (with 1≤𝑝<∞), the Banach space of all integrable functions 𝑓 on [0,1] with the norm ‖𝑓‖_𝐿𝑝=(∫¹₀|𝑓(𝑥)|^𝑝𝑑𝑥)^1/𝑝, are Bernstein-Kantorovich operators [2] 𝐾_𝑛=𝑓,𝑥_𝑛𝑘=0(𝑛+1)^{(𝑘+1)/(𝑛+1)}_{𝑘/(𝑛+1)}𝑓(𝑡)𝑑𝑡𝑝_𝑛,𝑘(𝑥),𝑥∈[0,1] (1) and Bernstein-Durrmeyer operators [3] 𝐷_𝑛𝑓,𝑥=^𝑛_𝑘=0(𝑛+1)¹₀𝑝_𝑛,𝑘(𝑡)𝑓(𝑡)𝑑𝑡𝑝_𝑛,𝑘(𝑥),𝑥∈[0,1]. (2) Quantitative estimates for approximation by Bernstein type positive linear operators in 𝐶[0,1] or 𝐿^𝑝[0,1] have been presented in a large literature (e.g., [4, 5]). See the book [6] and references therein for details and extensions to infinite intervals and linear combinations of positive operators for achieving high orders of approximation.

In this paper we provide a general framework for approximation by linear operators on variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces on an open subset Ω of ℝ^𝑑. Here 𝑝∶Ω→[1,∞) is a measurable function called the exponent function and 𝜌 is a positive bounded Borel measure on Ω. The variable space 𝐿_𝜌^𝑝(⋅)(Ω) is a generalization of the weighted 𝐿^𝑝 spaces with a constant exponent 𝑝∈[1,∞]. It consists of all the measurable functions 𝑓 on Ω such that ∫_Ω(|𝑓(𝑥)|/𝜆)^𝑝(𝑥)𝑑𝜌<∞ for some 𝜆>0. The norm is defined by 𝑓_𝐿𝜌^𝑝(⋅)=inf𝜆>0∶_Ω𝑓(𝑥)𝜆^𝑝(𝑥)𝑑𝜌≤1. (3) The space 𝐿_𝜌^𝑝(⋅)(Ω) is a Banach space [7]. The idea of variable 𝐿^𝑝(⋅) spaces was introduced by Orlicz [8]. Motivated by connections to variational integrals with nonstandard growth related to modeling of electrorheological fluids [9], these function spaces have been developed in analysis and research topics include boundedness of maximal operators, continuity of translates, and denseness of smooth functions. We will not go into details which can be found in [7, 10] and references therein. Instead, we only mention the following core condition on the log-Hölder continuity of the exponent function which leads to the boundedness of Hardy-Littlewood maximal operators and the rich theory of the variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces.

Definition 1. We say that the exponent function 𝑝∶Ω→[1,∞) is log-Hölder continuous if there exist positive constants 𝐴_𝑝>0 such that 𝑝(𝑥)𝑦≤𝐴−𝑝^𝑝<1−log𝑥−𝑦,𝑥,𝑦∈Ω,𝑥−𝑦2. (4) We say that 𝑝 is log-Hölder continuous at infinity (when Ω is unbounded) if there holds 𝑝(𝑥)𝑦≤𝐴−𝑝^𝑝(|𝑥𝑦≥|𝑥|.log𝑒+|),𝑥,𝑦∈Ω, (5) Denote 𝑝₋=inf_𝑥∈Ω𝑝(𝑥),𝑝₊=sup_𝑥∈Ω𝑝(𝑥). (6)

The issue of approximation by Bernstein type positive linear operators on variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces was raised by the second author in [11]. It turned out that the variety of the exponent function 𝑝 creates technical difficulty in the study of approximation. In particular, the uniform boundedness of the Bernstein-Kantorovich operators (1) and Bernstein-Durrmeyer operators (2) is already a difficult problem. The key analysis in [11] is to show that the Bernstein-Kantorovich operators and Bernstein-Durrmeyer operators are uniformly bounded when the exponent function 𝑝 is Lipschitz 𝛼 for some 𝛼∈(0,1]. It was conjectured there that the uniform boundedness still holds when 𝑝 is log-Hölder continuous. The first main result of this paper is to confirm this conjecture in Theorem 6 below.

Our second main result is to abandon the positivity and present quantitative estimates for the high order approximation by linear operators including linear combinations of Bernstein type positive linear operators, extending the results in [11] for the first order approximation by positive operators.

2. Motivations from Learning Theory

Our main motivation for considering the approximation of functions by linear operators on variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces is from learning theory. Besides the example of extending the Bernstein-Durrmeyer operators (2) to those associated with a general probability measure 𝜌 on Ω in [12, 13] for the multivariate case, we mention two learning theory settings here. Since error analysis for concrete learning algorithms in terms of the introduced noise conditions involves sample error estimates which are out of the scope of this paper, we leave the detailed error bounds to our further study.

2.1. Noise Conditions for Classification and Approximation

The first learning theory setting related to approximation on variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces is noise conditions for binary classification. Here Ω is an input space consisting of possible events while the output space is denoted as 𝑌={1,−1}. A Borel probability measure 𝑃 on the product space Ω×𝑌 can be decomposed into its marginal distribution 𝑃_Ω on Ω and conditional distributions 𝑃(⋅∣𝑥) for 𝑥∈Ω. A binary classifier 𝑓∶Ω→𝑌 makes predictions 𝑓(𝑥)∈𝑌 for future events 𝑥∈Ω. The best classifier 𝑓_𝑐, called Bayes rule, is given by 𝑓_𝑐(𝑥)=1 if 𝑃(1∣𝑥)>1/2 and −1 otherwise. The probability measure 𝑃 fits the binary classification problem well if the conditional probabilities 𝑃(1∣𝑥) and 𝑃(−1∣𝑥) are well separated from the boundary 1/2 for most events 𝑥. Their separations are equivalent to the separation of the value 𝑓_𝑃(𝑥) of the regression function 𝑓_𝑃(𝑥)=∫_𝑌𝑦𝑑𝑃(𝑦∣𝑥)=𝑃(1∣𝑥)−𝑃(−1∣𝑥) from 0 and can be measured in various quantitative ways. The Tsybakov noise condition [14] with noise exponent 𝑞∈(0,∞] asserts that for some constant 𝑐_𝑞>0, there holds 𝑃_Ω𝑥∈Ω∶0<𝑓_𝑃(𝑥)≤𝑐_𝑞𝑡≤𝑡^𝑞,∀𝑡>0. (7) When 𝑞=∞, Tsybakov noise condition (7) means |𝑓_𝑃(𝑥)|≥𝑐_𝑞 almost surely, and 𝑓_𝑃(𝑥) is well separated from 0. The case 𝑞<∞ means the measure of the set of events 𝑥 with 𝑓_𝑃(𝑥) not well separated from 0 decays polynomially fast as the threshold 𝑐_𝑞𝑡 tends to 0. More details about the Tsybakov noise condition, the so-called Tsybakov function, and its applications to the study of classification problems can also be found in [15]. Here we introduce a noise condition by allowing some noise situations measured by an exponent function 𝑝.

Example 2. We say that the probability measure 𝑃 satisfies the noise condition associated with an exponent function 𝑝∶Ω→[0,∞) if for some 𝜆>0, there holds _Ω𝑓_𝑃(𝑥)𝜆^𝑝(𝑥)𝑑𝑃_Ω<∞. (8)

Remark 3. The above condition can be applied to the regression setting for dealing with unbounded regression functions. When 𝑝 takes values on [1,∞), the above condition is equivalent to the requirement 𝑓_𝑃∈𝐿_𝜌^𝑝(⋅)(Ω) with 𝜌=𝑃_Ω. When 𝑑=1 and Ω=ℝ, we apply the classical identity ∫_ℝ𝑔(𝑥)𝑑𝑃_Ω=∫^∞₀𝑃_Ω({𝑥∈Ω∶𝑔(𝑥)≥𝑡})𝑑𝑡 to the nonnegative function 𝑔(𝑥)=(|𝑓_𝑃(𝑥)|/𝜆)^𝑝(𝑥) and find that the above condition is equivalent to ^∞₀𝑃_Ω𝑥∈Ω∶𝑓_𝑃(𝑥)≥𝜆𝑡^1/𝑝(𝑥)𝑑𝑡<∞. (9) This illustrates some similarity between the noise condition (8) and Tsybakov noise condition (7).

The following is an example to show some differences.

Example 4. Let 𝑑∈ℕ and Ω={𝑥∈ℝ^𝑑∶|𝑥|≤1/2}. If 𝑃_Ω is the normalized Lebesgue measure on Ω and 𝑓_𝑃(𝑥)=𝑒^−1/|𝑥|, then the measure 𝑃 satisfies the noise condition associated with the exponent function 𝑝(𝑥)=|𝑥| but does not satisfy the Tsybakov noise condition (7) with any 𝑞∈(0,∞]. In fact we have ∫_Ω|𝑓_𝑃(𝑥)|^𝑝(𝑥)𝑑𝑃_Ω=1/𝑒<∞ while for any 𝑞∈(0,∞) and 𝑐_𝑞>0, we have 𝑃_Ω({𝑥∈Ω∶0<|𝑓_𝑃(𝑥)|≤𝑐_𝑞𝑡})=(2/−log(𝑐_𝑞𝑡))^𝑑>𝑡^𝑞 for 𝑡∈(0,min{𝑡^∗,1/𝑐_𝑞𝑒²}) with 𝑡^∗ being the positive solution to the equation 𝑡^𝑞/𝑑log1/𝑐_𝑞𝑡=2.

2.2. Noise Conditions for Quantile Regression and Approximation

The second learning theory setting related to approximation on variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces is noise conditions for quantile regression. Here the output space is 𝑌=ℝ. Similar to the least squares regression [16] for learning means of conditional distributions 𝑃(⋅∣𝑥) but providing richer information [17] about response variables such as stretching or compressing tails, the learning problem for quantile regression aims at estimating quantiles of conditional distributions. With a quantile parameter 0<𝜏<1, the value of a quantile regression function 𝑓_𝑃,𝜏 at 𝑥∈Ω is defined by its value 𝑓_𝑃,𝜏(𝑥) as a 𝜏-quantile of 𝑃(⋅∣𝑥), that is, a value 𝑡^∗∈𝑌 satisfying 𝑃𝑦∈𝑌∶𝑦≤𝑡^∗𝑃∣𝑥≥𝜏,𝑦∈𝑌∶𝑦≥𝑡^∗∣𝑥≥1−𝜏. (10) Quantile regression has been studied by kernel-based regularization schemes in a learning theory literature (e.g., [18, 19]). For optimal error analysis of these learning algorithms, asymptotic behaviors of the conditional distributions near the 𝜏-quantiles are needed. In particular, one is interested in how slow the following function decays as 𝑡 decreases: 𝐹_𝑃,𝜏(𝑥)=min𝑃𝑦∈𝑌∶𝑡^∗−𝑡<𝑦<𝑡^∗,𝑃∣𝑥𝑦∈𝑌∶𝑡^∗+𝑡>𝑦>𝑡^∗∣𝑥. (11) A noise condition was introduced in [18] by requiring lower bounds 𝐹_𝑃,𝜏(𝑥)≥𝑏_𝑥𝑡^𝑞−1 for every 𝑡∈[0,𝑎_𝑥] and some 𝑞∈(1,∞), 𝑝∈(0,∞) and constants 𝑏_𝑥, 𝑎_𝑥>0 satisfying (𝑏_𝑥𝑎_𝑥^𝑞−1)⁻¹∈𝐿^𝑝_𝑃Ω. This condition was extended to a logarithmic bound in [19] by replacing 𝑡^𝑞−1 by (log(1/𝑡))^−𝑞 and 𝑎_𝑥^𝑞−1 by (log(1/𝑎_𝑥))^−𝑞. Here we introduce the following noise condition which is more general than the one in [18] by allowing the indices 𝑞, 𝑝 to depend on the events 𝑥∈Ω.

Example 5. We say that the probability measure 𝑃 satisfies the quantile noise condition associated with exponent functions 𝑞∶Ω→(1,∞) and 𝑝∶Ω→(0,∞) if for every 𝑥∈𝑋, there exist a 𝜏-quantile 𝑡^∗∈ℝ and constants 𝑎_𝑥∈(0,2],  𝑏_𝑥>0 such that for each 𝑢∈[0,𝑎_𝑥]𝐹_𝑃,𝜏(𝑥)≥𝑏_𝑥𝑡^{𝑞(𝑥)−1} (12) and that for some 𝜆>0, there holds ∫_Ω(1/𝜆𝑏_𝑥𝑎_𝑥^{𝑞(𝑥)−1})^𝑝(𝑥)𝑑𝑃_Ω<∞.

While the lower bounds (12) imply polynomial decays of the conditional distributions near the 𝜏-quantiles with a power index depending on the event, the finiteness of the integral is equivalent to the requirement that the function 1/𝑏_𝑥𝑎_𝑥^{𝑞(𝑥)−1} lies in the variable 𝐿_𝑃^𝑝(⋅)_Ω(Ω) space (when 𝑝 takes values in [1,∞)).

3. Main Results for Approximation on 𝐿_𝜌^𝑝(⋅)

Our first theorem is about the uniform boundedness of a sequence of linear operators on the variable 𝐿_𝜌^𝑝(⋅) spaces. These operators take the form 𝐿_𝑛𝑓,𝑥=_Ω𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)𝑑𝜌(𝑡),𝑥∈Ω,𝑓∈𝐿_𝜌^𝑝(⋅) (13) in terms of their kernels {𝐾_𝑛(𝑥,𝑡)}^∞_𝑛=1 defined on Ω×Ω. We assume that the kernels satisfy the following three conditions with some positive constants 𝐶₀≥1, 𝑏, 𝐶_𝑏, and 𝐶_𝑟 (depending on 𝑟∈ℕ) sup_𝑡∈ΩΩ𝐾_𝑛(𝑥,𝑡)𝑑𝜌(𝑥)≤𝐶₀,sup_𝑥∈ΩΩ𝐾_𝑛(𝑥,𝑡)𝑑𝜌(𝑡)≤𝐶₀,sup_{𝑥,𝑡∈Ω}𝐾_𝑛(𝑥,𝑡)≤𝐶_𝑏𝑛^𝑏,∀𝑛∈ℕ,_ΩΩ𝐾_𝑛(𝑥,𝑡)|𝑡−𝑥|^2𝑟𝑑𝜌(𝑡)𝑑𝜌(𝑥)≤𝐶_𝑟𝑛^−𝑟,∀𝑛∈ℕ,𝑟∈ℕ. (14)(15)(16)

Then the uniform boundedness follows, which will be proved in Section 5.

Theorem 6. Let Ω⊆ℝ^𝑑 be an open set, and an exponent function 𝑝∶Ω→(1,∞) satisfy 1<𝑝₋<𝑝₊<∞ and the log-Hölder continuity condition (4). If the kernels {𝐾_𝑛(𝑥,𝑡)}^∞_𝑛=1 satisfy conditions (14), (15), and (16), then the operators {𝐿_𝑛}^∞_𝑛=1 on 𝐿_𝜌^𝑝(⋅) defined by (13) are uniformly bounded as 𝐿_𝑛≤𝑀_𝑝,𝑏,∀𝑛∈ℕ (17) by a positive constant 𝑀_𝑝,𝑏 (depending on 𝑝 and the constants in (14), (15), and (16), given explicitly in the proof).

Our second theorem gives orders of approximation when the approximated function has some smoothness stated in terms of a 𝒦-functional. Define a Hölder space with index 𝑟∈ℕ on Ω by 𝑊_𝑝^𝑟,∞=𝑔∈𝐿_𝜌^𝑝(⋅)∶𝑔_{𝑝,𝑟,∞}<∞, (18) where ‖𝑔‖_{𝑝,𝑟,∞} is the norm given by ‖𝑔‖_{𝑝,𝑟,∞}=‖𝑔‖_𝐿𝜌^𝑝(⋅)+∑_{|𝛼|1≤𝑟}‖𝐷^𝛼𝑔‖_∞ with 𝐷^𝛼𝑔=𝜕^|𝛼|1/𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑑_𝑑 and |𝛼|₁=∑^𝑑_𝑗=1𝛼_𝑗 for 𝛼=(𝛼₁,…,𝛼_𝑑)∈ℤ^𝑑₊. The 𝒦-functional 𝒦_𝑟(𝑓,𝑡)_𝑝(⋅) is defined by 𝒦_𝑟𝑓,𝑡_𝑝(⋅)=inf_{𝑔∈𝑊𝑝𝑟,∞}𝑓−𝑔_𝐿𝜌^𝑝(⋅)+𝑡𝑔_{𝑝,𝑟,∞},𝑡>0. (19)

Denote 𝐶^∞₀(Ω) as the space of all compactly supported 𝐶^∞ functions on Ω. From [7], we know that when 𝑝⁺<∞,  𝐶^∞₀(Ω) is dense in 𝐿_𝜌^𝑝(⋅)(Ω). Hence for any 𝑓∈𝐿_𝜌^𝑝(⋅)(Ω), there holds 𝒦_𝑟(𝑓,𝑡)_𝑝(⋅)→0 as 𝑡→0.

The following theorem, to be proved in Section 5 and extending the results for 𝑟=1 in [11], gives orders of approximation by linear operators on 𝐿_𝜌^𝑝(⋅)(Ω) when the 𝒦-functional has explicit decay rates.

Theorem 7. Under the assumption of Theorem 6, if Ω is convex, 𝑟∈ℕ and the kernels satisfy 𝑟𝑝₋≥2, and _Ω𝐾_𝑛(𝑥,𝑡)(𝑡−𝑥)^𝛼𝑑𝜌(𝑡)=𝛿_𝛼,0,∀𝛼∈ℤ^𝑑𝑤𝑖𝑡ℎ|𝛼|₁<𝑟 (20) for almost every 𝑥∈Ω, then there holds for any 𝑓∈𝐿_𝜌^𝑝(⋅), 𝐿_𝑛(𝑓)−𝑓_𝐿𝜌^𝑝(⋅)≤𝐴_{𝑝,𝑏,𝑑}𝒦_𝑟𝑓,𝑛^{−𝑟−/𝑝+}_𝑝(⋅), (21) where 𝑟₋ is the integer part of 𝑟𝑝₋/2 and the constant 𝐴_{𝑝,𝑏,𝑑} is independent of 𝑓∈𝐿_𝜌^𝑝(⋅) (given explicitly in the proof).

The vanishing moment assumption (20) corresponds to Strang-Fix type conditions in the literature of shift-invariant spaces, for example [20, 21]. It has appeared in the literature of Bernstein type operators when linear combinations are considered, as described by (34) in the next section.

4. Approximation by Bernstein Type Operators

In this section we apply our main results to Bernstein type positive linear operators and give high orders of approximation by linear combinations of these operators on variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces. We demonstrate the analysis for the general Bernstein-Durrmeyer operators in detail and describe briefly results for the general Bernstein-Kantorovich operators as an example of other families of operators.

The Bernstein-Durrmeyer operators on an open simplex Ω=𝑆=𝑥∈ℝ^𝑑₊∶|𝑥|₁<1 (22) associated with a general positive Borel measure 𝜌 on 𝑆 are defined as 𝐷_𝑛𝑓,𝑥=_{|𝛼|1≤𝑛}∫_𝑆𝑓(𝑡)𝑝_𝑛,𝛼(𝑡)𝑑𝜌(𝑡)∫_𝑆𝑝_𝑛,𝛼(𝑡)(𝑡)𝑝𝑑𝜌^𝑛,𝛼(𝑥),𝑓∈𝐿¹_𝜌,𝑥∈𝑆, (23) where for 𝑥=(𝑥₁,…,𝑥_𝑑)∈𝑆⊂ℝ^𝑑, 𝛼=(𝛼₁,…,𝛼_𝑑)∈ℤ^𝑑₊, we denote 𝑝_𝑛,𝛼(𝑥)=𝑛!Π^𝑑_𝑗=1𝛼_𝑗!𝑛−|𝛼|₁!^𝑑_𝑗=1𝑥^𝛼𝑗_𝑗1−|𝑥|₁^{𝑛−|𝛼|1}. (24)

The classical Bernstein-Durrmeyer operators (2) on Ω=(0,1) (𝑑=1) with 𝑑𝜌(𝑥)=𝑑𝑥 have been well studied (e.g., [22]) and extended to a multivariate form with respect to Jacobi weights 𝑑𝜌(𝑥)=Π^𝑑_𝑗=1𝑥^𝑞𝑗_𝑗𝑑𝑥 (e.g., [23]). Bernstein-Durrmeyer operators on Ω=(0,1) with respect to an arbitrary Borel probability measure were introduced in [12] and applied to error analysis of learning algorithms for support vector machine classifications. The multidimensional version of such linear operators (23) was introduced in [13]. In [24], the first author showed for a constant exponent function 𝑝(𝑥)≡𝑝∈[1,∞) that lim_𝑛→∞‖𝑓−𝐷_𝑛(𝑓)‖_𝐿^𝑝_𝜌=0 for any 𝑓∈𝐿^𝑝_𝜌. The case 𝑝≡∞ was studied in [25, 26]. Here we consider the case with a general exponent function satisfying 1<𝑝₋<𝑝₊<∞ and the log-Hölder continuity condition (4).

By applying Theorem 6, we can prove the uniform boundedness of the Bernstein-Durrmeyer operators (23).

Proposition 8. Let Ω=𝑆, 𝜌 be a Borel probability measure on 𝑆, and an exponent function 𝑝∶𝑆→(1,∞) satisfy 1<𝑝₋<𝑝₊<∞ and the log-Hölder continuity condition (4). If there exist positive constants 𝑏 and 𝐶^∗_𝑏 such that ∫_𝑆𝑝_𝑛,𝛼(𝑡)𝑑𝜌(𝑡)≥𝐶^∗_𝑏𝑛^−𝑏 for 𝑛∈ℕ and |𝛼|₁≤𝑛, then for the Bernstein-Durrmeyer operators defined on 𝐿_𝜌^𝑝(⋅)(Ω) by (23), there exists a positive constant 𝑀_𝑝,𝑏 depending only on 𝑝,𝑏,𝐶^∗_𝑏 such that 𝐷_𝑛≤𝑀_𝑝,𝑏,∀𝑛∈ℕ. (25)

Proof. Define a sequence of kernels {Ψ_𝑛} on 𝑆×𝑆 by Ψ_𝑛(𝑥,𝑡)=_{|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑡)𝑝_𝑛,𝛼(𝑥)∫_𝑆𝑝_𝑛,𝛼(𝑡)(𝑡).𝑑𝜌 (26) Then the Bernstein-Durrmeyer operators (23) can be written as 𝐷_𝑛𝑓,𝑥=_𝑆Ψ_𝑛(𝑥,𝑡)𝑓(𝑡)𝑑𝜌(𝑡). (27) So we only need to check the three conditions (14), (15), and (16) of Theorem 6.
Since Ψ_𝑛(𝑥,𝑡)≥0, we know that _𝑆Ψ_𝑛(𝑥,𝑡)𝑑𝜌(𝑥)=_𝑆Ψ_𝑛(𝑥,𝑡)𝑑𝜌(𝑥)=_{|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑡)∫_𝑆𝑝_𝑛,𝛼(𝑥)𝑑𝜌(𝑥)∫_𝑆𝑝_𝑛,𝛼(𝑡)𝑑𝜌(𝑡)=_{|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑡)≡1. (28) The same is true for ∫_𝑆|Ψ_𝑛(𝑥,𝑡)|𝑑𝜌(𝑡)≡1. So we know that condition (14) holds with the constant 𝐶₀=1.
Applying the lower bound ∫_𝑆𝑝_𝑛,𝛼(𝑡)𝑑𝜌(𝑡)≥𝐶^∗_𝑏𝑛^−𝑏 and the inequality 𝑝_𝑛,𝛼(𝑡)≤1, we see that for any 𝑛∈ℕ, and 𝑥,𝑡∈𝑆Ψ_𝑛(𝑥,𝑡)≤_{|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑥)𝐶^∗_𝑏𝑛^−𝑏=1𝐶^∗_𝑏𝑛^𝑏. (29) Hence condition (15) holds with 𝐶_𝑏=1/𝐶^∗_𝑏.
As for the last condition, we separate 𝑡−𝑥 into 𝑡−(𝛼/𝑛)+(𝛼/𝑛)−𝑥 and find that for any 𝑟∈ℕ and 𝑛∈ℕ_𝑆𝑆Ψ_𝑛(𝑥,𝑡)|𝑡−𝑥|^2𝑟𝑑𝜌(𝑡)𝑑𝜌(𝑥)≤2^2𝑟_{𝑆𝑆|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑡)𝑝_𝑛,𝛼(𝑥)∫_𝑆𝑝_𝑛,𝛼(𝑡)(𝑡)×𝛼𝑑𝜌𝑡−𝑛^2𝑟(𝑡)(𝑥)+𝑑𝜌𝑑𝜌_{𝑆𝑆|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑡)𝑝_𝑛,𝛼(𝑥)∫_𝑆𝑝_𝑛,𝛼(𝑡)(𝑡)×𝛼𝑑𝜌𝑛−𝑥^2𝑟𝑑𝜌(𝑡)𝑑𝜌(𝑥)=2^2𝑟_{𝑆|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑡𝛼)𝑡−𝑛^2𝑟(𝑡)+𝑑𝜌_{𝑆|𝛼|1≤𝑛}𝑝_𝑛,𝛼(𝑥𝛼)𝑛−𝑥^2𝑟𝑑𝜌(𝑥)=2^2𝑟+1_𝑆𝐵_𝑛(⋅−𝑥)^2𝑟,𝑥𝑑𝜌(𝑥), (30) where 𝐵_𝑛 is the multidimensional Bernstein operators on the closure 𝑆 of 𝑆 defined by 𝐵_𝑛𝑓,𝑥=_{|𝛼|1≤𝑛}𝑓𝛼𝑛𝑝_𝑛,𝛼(𝑥),𝑥∈𝑆,𝑓∈𝐶𝑆. (31) It is well known [6] for the multidimensional Bernstein operators that there exists a constant 𝐶_𝑑,𝑟 depending only on 𝑑 and 𝑟 such that 𝐵_𝑛(⋅−𝑥)^2𝑟,𝑥≤𝐶_𝑑,𝑟𝑛^−𝑟,∀𝑥∈𝑆. (32) It follows that _𝑆𝑆Ψ_𝑛(𝑥,𝑡)|𝑡−𝑥|^2𝑟𝑑𝜌(𝑡)𝑑𝜌(𝑥)≤2^2𝑟+1𝐶_𝑑,𝑟𝑛^−𝑟 (33) and condition (16) holds true with 𝐶_𝑟=2^2𝑟+1𝐶_𝑑,𝑟.
With all the three conditions verified, the desired uniform bound (25) for the Bernstein-Durrmeyer operators follows from Theorem 6. This proves the proposition.

The Bernstein-Durrmeyer operators (23) are positive, which prevent from achieving high order approximation due to a saturation phenomenon. Linear combinations of such operators can be used to get high orders of approximation. The idea and literature review of this method can be found in [6] while further developments will not be mentioned here. The linear combinations are defined as 𝐿_𝑛,𝑟𝑓,𝑥=^{𝑚𝑑,𝑟}_𝑖=0𝐶_𝑖(𝑛)𝐷_𝑛𝑖𝑓,𝑥, (34) where 𝑚_𝑑,𝑟=(𝑑+𝑟−1)!/𝑑!(𝑟−1)! is the dimension of the space of polynomials of degree at most 𝑟−1, and with two positive constants 𝐵₁,𝐵₂ independent of 𝑛, we have 𝑛=𝑛₀<𝑛₁<⋯<𝑛_{𝑚𝑑,𝑟}≤𝐵₁𝑛,^{𝑚𝑑,𝑟}_𝑖=0𝐶_𝑖(𝑛)≤𝐵₂,_{𝑚𝑑,𝑟𝑖=0}𝐶_𝑖(𝑛)𝐷_𝑛𝑖(⋅−𝑥)^𝛼,𝑥=𝛿_𝛼,0,∀0≤|𝛼|₁≤𝑟−1. (35) For the classical Bernstein-Durrmeyer operators with respect to the Lebesgue measure (or even the Jacobi weights), the existence of the above linear combinations can be seen and found in the literature. The existence of such linear combinations with respect to the arbitrary measure 𝜌 is a nontrivial problem and deserves intensive study. This technical question is out of the scope of this paper and will be discussed in our further work. Here we concentrate on the variable 𝐿_𝜌^𝑝(⋅)(Ω) spaces and state the following result for the high orders of approximation under the condition (35) which is an immediate consequence of Theorem 7.

Proposition 9. Under the assumption of Proposition 8, if 2≤𝑟∈𝑁 and the operators {𝐿_𝑛,𝑟}_𝑛∈ℕ defined by (34) satisfy (35), then for any 𝑓∈𝐿_𝜌^𝑝(⋅), we have 𝐿_𝑛,𝑟(𝑓)−𝑓_𝐿𝜌^𝑝(⋅)≤𝐴_{𝑝,𝑏,𝑑}𝒦_𝑟𝑓,𝑛^{−𝑟−/𝑝+}_𝑝(⋅), (36) where 𝑟₋ is the integer part of 𝑟𝑝₋/2 and the constant 𝐴_{𝑝,𝑏,𝑑} is independent of 𝑓∈𝐿_𝜌^𝑝(⋅).

Let us now briefly describe approximation results for the Bernstein-Kantorovich operators on 𝑆 defined [27] as 𝐵𝐾_𝑛𝑓,𝑥=_{|𝛼|1≤𝑛}∫_{𝑆𝑛,𝛼}𝑓(𝑡)𝑑𝜌(𝑡)𝜌𝑆_𝑛,𝛼𝑝_𝑛,𝛼(𝑥),𝑓∈𝐿¹_𝜌,𝑥∈𝑆, (37) where {𝑆_𝑛,𝛼}_𝛼 are subdomains of 𝑆 defined by 𝑆_𝑛,𝛼=𝑥∈𝑆∶𝑥∈^𝑑_𝑖=1𝛼_𝑖,𝛼𝑛+1^𝑖,|𝑥|𝑛+1¹≤|𝛼|₁+1,|𝛼|𝑛+1₁≤𝑛. (38)

In the same way as for the Bernstein-Durrmeyer operators, we have the following results for the Bernstein-Kantorovich operators.

Proposition 10. Under the assumption of Proposition 8 for 𝑝, if there exist positive constants 𝑏 and 𝐶^∗_𝑏 such that 𝜌(𝑆_𝑛,𝛼)≥𝐶^∗_𝑏𝑛^−𝑏 for 𝑛∈ℕ and |𝛼|₁≤𝑛, then for the Bernstein-Kantorovich operators defined on 𝐿_𝜌^𝑝(⋅)(Ω) by (37), there exists a positive constant 𝑀_𝑝,𝑏 depending only on 𝑝,𝑏,𝐶^∗_𝑏 such that 𝐵𝐾_𝑛≤𝑀_𝑝,𝑏,∀𝑛∈ℕ. (39) If 2≤𝑟∈𝑁 and, with 𝐷_𝑛𝑖 replaced by 𝐵𝐾_𝑛𝑖, the operators {𝐿_𝑛,𝑟}_𝑛∈ℕ defined by (34) satisfy (35), then 𝐿_𝑛,𝑟(𝑓)−𝑓_𝐿𝜌^𝑝(⋅)≤𝐴_{𝑝,𝑏,𝑑}𝒦_𝑟𝑓,𝑛^{−𝑟−/𝑝+}_𝑝(⋅),∀𝑓∈𝐿_𝜌^𝑝(⋅). (40)

5. Proof of Main Results

In this section we give detailed proof of our main results. Let us first prove Theorem 6.

Proof of Theorem 6. Let 𝑓∈𝐿^𝑝(⋅) have norm 1, which implies ∫_Ω|𝑓(𝑡)|^𝑝(𝑡)𝑑𝜌(𝑡)≤1. Choose 𝛾=𝑏/(𝑝₋−1)>0.
For 𝑥∈Ω, we define two subsets Ω_𝑛,𝑥 and Ω^′_𝑛,𝑥 of Ω as Ω_𝑛,𝑥=𝑡∈Ω∶𝑓(𝑡)≤𝑛^𝛾,|𝑡−𝑥|≤𝑛^−1/4,Ω^′_𝑛,𝑥=𝑡∈Ω∶𝑓(𝑡)≤𝑛^𝛾,|𝑡−𝑥|>𝑛^−1/4. (41) Set 𝐿_𝑛,1𝑓,𝑥=_{Ω⧵(Ω𝑛,𝑥∪Ω′𝑛,𝑥)}𝐾_𝑛()𝑓(𝑡)(𝑡),𝐿𝑥,𝑡𝑑𝜌_𝑛,2𝑓,𝑥=_Ω𝑛,𝑥𝐾_𝑛()𝑓(𝑡)(𝑡),𝐿𝑥,𝑡𝑑𝜌_𝑛,3𝑓,𝑥=_{Ω′𝑛,𝑥}𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)𝑑𝜌(𝑡). (42) Then the value 𝐿_𝑛(𝑓,𝑥) can be decomposed into three parts as 𝐿_𝑛𝑓,𝑥=𝐿_𝑛,1𝑓,𝑥+𝐿_𝑛,2𝑓,𝑥+𝐿_𝑛,3𝑓,𝑥,𝑥∈Ω, (43) and we have _Ω𝐿_𝑛𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥)≤3^𝑝+3_𝑗=1Ω𝐿_𝑛,𝑗𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥). (44) In the following we estimate the three terms in (44) separately.
Step  1. Estimating the First Term of (44). By the definition of 𝑝₋, we have 𝑝(𝑡)≥𝑝₋>1. For 𝑡∈Ω⧵(Ω_𝑛,𝑥∪Ω^′_𝑛,𝑥), we have |𝑓(𝑡)|>𝑛^𝛾 and thereby 𝑓(𝑡)≤𝑓(𝑡)𝑓(𝑡)𝑛^{−𝛾𝑝(𝑡)−1}=𝑓(𝑡)^𝑝(𝑡)𝑛^{𝛾(1−𝑝(𝑡))}≤𝑓(𝑡)^𝑝(𝑡)𝑛^{𝛾(1−𝑝−)}=𝑛^−𝑏𝑓(𝑡)^𝑝(𝑡). (45) It follows from condition (15) that 𝐿_𝑛,1𝑓,𝑥≤𝑛^−𝑏_{Ω⧵(Ω𝑛,𝑥∪Ω′𝑛,𝑥)}𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)^𝑝(𝑡)𝑑𝜌(𝑡)≤𝑛^−𝑏𝐶_𝑏𝑛^𝑏_{Ω⧵(Ω𝑛,𝑥∪Ω′𝑛,𝑥)}𝑓(𝑡)^𝑝(𝑡)𝑑𝜌(𝑡). (46) So by the assumption ∫_Ω|𝑓(𝑡)|^𝑝(𝑡)𝑑𝜌(𝑡)≤1, 𝐿_𝑛,1𝑓,𝑥≤𝐶_𝑏Ω𝑓(𝑡)^𝑝(𝑡)𝑑𝜌(𝑡)≤𝐶_𝑏. (47) Consequently, _Ω𝐿_𝑛,1𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥)≤𝜌(Ω)𝐶_𝑏^𝑝−+𝐶_𝑏^𝑝+. (48)
Step  2. Estimating the Second Term of (44). By the condition ∫_Ω|𝐾_𝑛(𝑥,𝑡)|𝑑𝜌(𝑡)≤𝐶₀ in (14) with 𝐶₀≥1, we know by the Hölder inequality _Ω𝐿_𝑛,2𝑓,𝑥^𝑝(𝑥)(𝑥)≤𝑑𝜌_{ΩΩ𝑛,𝑥}𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)^𝑝(𝑥)(𝑡)×𝑑𝜌_Ω𝑛,𝑥𝐾_𝑛(𝑥,𝑡)𝑑𝜌(𝑡)^{𝑝(𝑥)−1}𝑑𝜌(𝑥)≤𝐶^𝑝+₀⁻¹_{ΩΩ𝑛,𝑥}𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)^𝑝(𝑡)𝑓(𝑡)^{𝑝(𝑥)−𝑝(𝑡)}×𝑑𝜌(𝑡)𝑑𝜌(𝑥)=𝐶^𝑝+₀⁻¹_{ΩΩ𝑛,𝑥}𝐽_𝑛𝑓,𝑥,𝑡𝑓(𝑡)^𝑝(𝑡)𝑑𝜌(𝑡)𝑑𝜌(𝑥), (49) where 𝐽_𝑛𝑓,𝑥,𝑡∶=𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)^{𝑝(𝑥)−𝑝(𝑡)},𝑡∈Ω_𝑛,𝑥,𝑥∈Ω. (50)
For 𝑡∈Ω_𝑛,𝑥, we have a bound |𝑓(𝑡)|≤𝑛^𝛾 and the restriction |𝑥−𝑡|≤𝑛^−1/4. From the log-Hölder continuity of the exponent function 𝑝, there exists a constant 𝐴_𝑝 only dependent on 𝑝 such that 𝑝(𝑥)(𝑡)≤𝐴−𝑝^𝑝−log𝑛^−1/4=4𝐴_𝑝.log𝑛 (51)
When |𝑓(𝑡)|≥1, we find 𝑓(𝑡)^{𝑝(𝑥)−𝑝(𝑡)}≤𝑛^{𝛾4𝐴𝑝/log𝑛}≤𝑛^{4𝛾𝐴𝑝/log𝑛}≤𝑀_𝑝, (52) where the constant number 𝑀_𝑝 defined by 𝑀_𝑝=sup_𝑛∈ℕ𝑛^{4𝛾𝐴𝑝/log𝑛}, (53) is finite because log𝑛^{4𝛾𝐴𝑝/log𝑛}=4𝛾𝐴_𝑝log𝑛log𝑛⟶4𝛾𝐴_𝑝,as𝑛⟶∞. (54)
When |𝑓(𝑡)|<1, we simply use |𝑓(𝑡)|^{𝑝(𝑥)−𝑝(𝑡)}≤|𝑓(𝑡)|^{−𝑝(𝑡)}. Applying these bounds, we can estimate the core part of Step  2 as _{ΩΩ𝑛,𝑥}𝐽_𝑛𝑓,𝑥,𝑡𝑓(𝑡)^𝑝(𝑡)(𝑡)(𝑥)≤𝑑𝜌𝑑𝜌_{ΩΩ𝑛,𝑥}𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)^𝑝(𝑡)𝑀_𝑝+𝑓(𝑡)^{−𝑝(𝑡)}(𝑡)(𝑥)=𝑀×𝑑𝜌𝑑𝜌_{𝑝ΩΩ𝑛,𝑥}𝐾_𝑛(𝑥,𝑡)𝑓(𝑡)^𝑝(𝑡)(𝑡)(𝑥)+𝑑𝜌𝑑𝜌_{ΩΩ𝑛,𝑥}𝐾_𝑛((𝑡)(𝑥)≤𝑀𝑥,𝑡)𝑑𝜌𝑑𝜌_𝑝𝐶_0Ω𝑓(𝑡)^𝑝(𝑡)𝑑𝜌(𝑡)+𝐶₀𝜌(Ω)≤𝐶₀𝑀_𝑝+𝐶₀𝜌(Ω). (55) Here we have used the assumption ∫_Ω|𝑓(𝑡)|^𝑝(𝑡)𝑑𝜌(𝑡)≤1 and (14). So the second term of (44) can be estimated as _Ω𝐿_𝑛,2𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥)≤𝐶^𝑝+₀𝑀_𝑝+𝜌(Ω). (56)
Step  3. Estimating the Third Term of (44). For 𝑡∈Ω^′_𝑛,𝑥, we have |𝑓(𝑡)|≤𝑛^𝛾 and |𝑥−𝑡|>𝑛^−1/4 yielding 𝑛^1/4|𝑥−𝑡|>1. It follows that 𝐿_𝑛,3𝑓,𝑥≤_{Ω′𝑛,𝑥}𝐾_𝑛(𝑥,𝑡)𝑛^𝛾𝑛^1/4|𝑥−𝑡|^{2𝑟/𝑝(𝑥)}𝑑𝜌(𝑡), (57) where 𝑟 is the integer part of 2𝛾𝑝₊+1. Applying the Hölder inequality and (14), we see that 𝐿_𝑛,3(𝑓,𝑥)^𝑝(𝑥)≤_{Ω′𝑛,𝑥}𝑛^{𝛾𝑝(𝑥)+(𝑟/2)}𝐾_𝑛(𝑥,𝑡)|𝑥−𝑡|^2𝑟𝑑𝜌(𝑡)𝐶₀^{𝑝(𝑥)−1}. (58) So by the bound 𝑝(𝑥)≤𝑝₊ and condition (16), the third term of (44) can be bounded as _Ω𝐿_𝑛,3𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥)≤𝐶^𝑝+₀⁻¹𝑛^{𝛾𝑝++(𝑟/2)}𝐶_𝑟𝑛^−𝑟=𝐶^𝑝+₀⁻¹𝐶_𝑟𝑛^{𝛾𝑝+−(𝑟/2)}≤𝐶^𝑝+₀⁻¹𝐶_𝑟. (59)
Finally, we put the estimates (48), (56), and (59) into (44) to conclude _Ω𝐿_𝑛𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥)≤3^𝑝+𝜌(Ω)𝐶^𝑝−_𝑏+𝐶^𝑝+_𝑏+𝐶^𝑝+₀𝑀_𝑝+𝜌(Ω)+𝐶^𝑝+₀⁻¹𝐶_𝑟. (60) Take 𝜆=𝑀_𝑝,𝑏∶=1+3^𝑝+𝜌(Ω)𝐶^𝑝−_𝑏+𝐶^𝑝+_𝑏+𝐶^𝑝+₀𝑀_𝑝+𝜌(Ω)+𝐶^𝑝+₀⁻¹𝐶_𝑟; (61) we find _Ω𝐿_𝑛𝑓,𝑥𝜆^𝑝(𝑥)(𝑥)≤1𝑑𝜌𝜆^𝑝−_Ω𝐿_𝑛𝑓,𝑥^𝑝(𝑥)𝑑𝜌(𝑥)≤1. (62) This implies ‖𝐿_𝑛(𝑓)‖_𝐿𝜌^𝑝(⋅)≤𝑀_𝑝,𝑏. The bound 𝑀_𝑝,𝑏 is independent of 𝑓. So we have ‖𝐿_𝑛‖≤𝑀_𝑝,𝑏. The proof Theorem 6 is complete.

We are now in a position to prove Theorem 7.

Proof of Theorem 7. We follow the standard procedure in approximation theory and consider the error 𝐿_𝑛(𝑔,𝑥)−𝑔(𝑥) for 𝑔∈𝑊_𝑝^𝑟,∞. Apply the Taylor expansion 𝑔(𝑡)=𝑔(𝑥)+_{|𝛼|1≤1≤𝑟−1}𝐷^𝛼𝑔(𝑥)()𝛼!𝑡−𝑥^𝛼+𝑅_𝑔,𝑟(𝑥,𝑡),𝑥,𝑡∈Ω, (63) where the remainder term 𝑅_𝑔,𝑟(𝑥,𝑡) is given by 𝑅_𝑔,𝑟()=𝑥,𝑡₁₀(1−𝑢)^𝑟−1_|𝛼|1=𝑟𝐷^𝛼𝑔(𝑥+𝑢(𝑡−𝑥))()𝛼!𝑡−𝑥^𝛼𝑑𝑢. (64) We see from the vanishing moment condition (20) that 𝐿_𝑛(𝑥)=𝑔,𝑥−𝑔_Ω𝐾_𝑛(𝑥,𝑡)𝑔(𝑥)+_{|𝛼|1≤1≤𝑟−1}𝐷^𝛼𝑔(𝑥)()𝛼!𝑡−𝑥^𝛼+𝑅_𝑔,𝑟()(𝑡)(𝑥)=𝑥,𝑡𝑑𝜌−𝑔_Ω𝐾_𝑛(𝑥,𝑡)¹₀(1−𝑢)^𝑟−1_|𝛼|1=𝑟𝐷^𝛼𝑔(𝑥+𝑢(𝑡−𝑥))×()𝛼!𝑡−𝑥^𝛼𝑑𝑢𝑑𝜌(𝑡). (65) Since Ω is convex, 𝑥+𝑢(𝑡−𝑥)∈Ω for any 𝑢∈[0,1], 𝑥, 𝑡∈Ω. So |𝐷^𝛼𝑔(𝑥+𝑢(𝑡−𝑥))|≤‖𝑔‖_{𝑝,𝑟,∞} and we have 𝐿_𝑛(𝑥≤𝑔,𝑥−𝑔)₁₀(1−𝑢)^𝑟−1×_|𝛼|1=𝑟𝑔_{𝑝,𝑟,∞}𝛼!_Ω𝐾_𝑛(𝑥,𝑡)|𝑡−𝑥|^|𝛼|𝑑𝜌(𝑡)𝑑𝑢≤𝑑^𝑟𝑔_{𝑝,𝑟,∞Ω}𝐾_𝑛(𝑥,𝑡)|𝑡−𝑥|^𝑟𝑑𝜌(𝑡). (66) By (14) and the Hölder inequality, _Ω𝐾_𝑛(𝑥,𝑡)|𝑡−𝑥|^𝑟𝑑𝜌(𝑡)^𝑝(𝑥)≤_Ω𝐾_𝑛(𝑥,𝑡)|𝑡−𝑥|^{𝑟𝑝(𝑥)}𝑑𝜌(𝑡)𝐶₀^{𝑝(𝑥)−1}. (67) If we denote the largest integer 𝑟₋ satisfying 2𝑟₋≤𝑟𝑝₋ as 𝑟₋, and the smallest integer 𝑟₊ satisfying 2𝑟₊≥𝑟𝑝₊, we find |𝑡−𝑥|^{𝑟𝑝(𝑥)}≤|𝑡−𝑥|^𝑟𝑝+≤|𝑡−𝑥|^2𝑟+,if||||𝑡−𝑥≥1,𝑡−𝑥^𝑟𝑝−≤|𝑡−𝑥|^2𝑟−,if|𝑡−𝑥|<1. (68) We combine this with (16) and see that for 𝜆=𝑑^𝑟‖𝑔‖_{𝑝,𝑟,∞}, _Ω𝐿_𝑛𝑔,𝑥−𝑔(𝑥)𝜆^𝑝(𝑥)𝑑𝜌(𝑥)≤𝐶^𝑝+₀⁻¹_ΩΩ𝐾_𝑛(𝑥,𝑡)|𝑡−𝑥|^2𝑟++|𝑡−𝑥|^2𝑟−𝑑𝜌(𝑡)𝑑𝜌(𝑥)≤𝐶^𝑝+₀⁻¹𝐶_𝑟+𝑛^−𝑟++𝐶_𝑟−𝑛^−𝑟−≤𝐶^𝑝+₀⁻¹𝐶_𝑟++𝐶_𝑟−𝑛^−𝑟−. (69)
When 𝑛≥𝐶^{(𝑝+−1)/𝑟−}₀(𝐶_𝑟++𝐶_𝑟−)^1/𝑟−, we take 𝜆=𝑑^𝑟𝑔_{𝑝,𝑟,∞}𝐶^𝑝+₀⁻¹𝐶_𝑟++𝐶_𝑟−𝑛^{−𝑟−1/𝑝+} (70) and find _Ω𝐿_𝑛𝑔,𝑥−𝑔(𝑥)𝜆^𝑝(𝑥)𝑑𝜌(𝑥)≤1, (71) which implies 𝐿_𝑛(𝑔)−𝑔_𝐿𝜌^𝑝(⋅)≤𝜆≤𝑑^𝑟𝐶^{1−(1/𝑝+)}₀𝐶_𝑟++𝐶_𝑟−^1/𝑝+𝑛^{−𝑟−/𝑝+}𝑔_{𝑝,𝑟,∞}. (72) Thus by Theorem 6 and taking infimum over 𝑔∈𝑊_𝑝^𝑟,∞, we have 𝐿_𝑛(𝑓)−𝑓_𝐿𝜌^𝑝(⋅)≤inf_{𝑔∈𝑊𝑝𝑟,∞}𝐿_𝑛(𝑓−𝑔)_𝐿𝜌^𝑝(⋅)+𝐿_𝑛(𝑔)−𝑔_𝐿𝜌^𝑝(⋅)+𝑔−𝑓_𝐿𝜌^𝑝(⋅)≤inf_{𝑔∈𝑊𝑝𝑟,∞}𝑀_𝑝,𝑏+1𝑓−𝑔_𝐿𝜌^𝑝(⋅)+𝑑^𝑟𝐶^{1−(1/𝑝+)}₀𝐶_𝑟++𝐶_𝑟−^1/𝑝+𝑛^{−𝑟−/𝑝+}𝑔_{𝑝,𝑟,∞}≤𝐴_{𝑝,𝑏,𝑑}𝒦_𝑟𝑓,𝑛^{−𝑟−/𝑝+}_𝑝(⋅), (73) where the constant 𝐴_{𝑝,𝑏,𝑑} is given by 𝐴_{𝑝,𝑏,𝑑}=𝑀_𝑝,𝑏+1+𝑑^𝑟𝐶^{1−(1/𝑝+)}₀𝐶_𝑟++𝐶_𝑟−^1/𝑝+. (74)
When 𝑛<𝐶^{(𝑝+−1)/𝑟−}₀(𝐶_𝑟++𝐶_𝑟−)^1/𝑟−, then from the inequality ‖𝑓−𝑔‖_𝐿𝜌^𝑝(⋅)+𝑡‖𝑔‖_{𝑝,𝑟,∞}≥𝑡‖𝑓‖_𝐿𝜌^𝑝(⋅) valid for 𝑡≤1 we observe 𝒦_𝑟𝑓,𝑛^{−𝑟−/𝑝+}_𝑝(⋅)≥𝑛^{−𝑟−/𝑝+}𝑓_𝐿𝜌^𝑝(⋅), (75) and applying Theorem 6 directly yields 𝐿_𝑛(𝑓)−𝑓_𝐿𝜌^𝑝(⋅)≤𝑀_𝑝,𝑏+1𝑓_𝐿𝜌^𝑝(⋅)≤𝑀_𝑝,𝑏+1𝑛^{𝑟−/𝑝+}𝒦_𝑟𝑓,𝑛^{−𝑟−/𝑝+}_𝑝(⋅) (76) and thereby (21) by setting 𝐴_{𝑝,𝑏,𝑑}=𝑀_𝑝,𝑏+1𝐶^{1−(1/𝑝+)}₀𝐶_𝑟++𝐶_𝑟−^1/𝑝+. (77) The proof of Theorem 7 is complete.

6. Further Topics and Discussion

Approximation by linear operators is an important topic in approximation theory. It mainly consists of two families of approximation schemes: Bernstein type positive linear operators and quasi-interpolation type linear operators in multivariate approximation.

In this paper, we mainly consider Bernstein type positive linear operators. We verified a conjecture in [11] about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators with respect to an arbitrary Borel measure on (0,1) on the variable 𝐿_𝜌^𝑝(⋅) space under the assumption of log-Hölder continuity of the exponent function 𝑝. We also provide quantitative estimates for high orders of approximation on the variable 𝐿_𝜌^𝑝(⋅) by linear combinations of Bernstein type positive linear operators.

The study of quasi-interpolation type linear operators started with the classical work of Schoenberg on cardinal interpolation by B-splines. It has been developed significantly due to important applications in the areas of finite element methods, cardinal interpolation for multivariate approximation, and wavelet analysis. A large class of linear operators for approximating functions on ℝ^𝑑 take the form 𝑇𝑓,𝑥=_ℝ𝑑Φ(𝑥,𝑡)𝑓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑑, (78) where Φ∶ℝ^𝑑×ℝ^𝑑→ℝ is a window function satisfying ∫_ℝ𝑑Φ(𝑥,𝑡)𝑑𝑡=1 and some conditions for decays of |Φ(𝑥,𝑡) as |𝑥−𝑡| increases. Quantitative estimates for the approximation of functions in 𝐶(ℝ^𝑑)∩𝐿^∞(ℝ^𝑑) or 𝐿^𝑝(ℝ^𝑑) with 1≤𝑝<∞ can be found in a large literature of multivariate approximation (see, e.g., [20, 21, 28]). Establishing analysis for approximation by quasi-interpolation type linear operators on the variable 𝐿_𝜌^𝑝(⋅) spaces would be an interesting topic. An immediate barrier we meet with such analysis is the boundedness assumption of the measure 𝜌 (𝜌(Ω)<∞). This assumption is not satisfied for most quasi-interpolation type linear operators or the classical Weierstrass (or Gaussian convolution) operators 𝒲_𝑛(𝑓)=(𝑛/2𝜋)^𝑑∫_ℝ𝑑exp{−𝑛‖𝑡−𝑥‖²₂/2}𝑓(𝑡)𝑑𝑡, for which 𝜌 is often the Lebesgue measure on ℝ^𝑑. It is desirable to overcome the technical difficulty and establish error analysis for linear operators with respect to unbounded measures.

We described motivations of our study in learning theory. It would be interesting to implement detailed error analysis for some related learning algorithms in classification and quantile regression by means of our results on orders of approximation by linear operators.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their constructive suggestions and comments. The work described in this paper is supported partially by the Research Grants Council of Hong Kong (Project no. CityU 105011). The corresponding author is Ding-Xuan Zhou.