On Rate of Approximation by Modified Beta Operators

Abstract

We establish the rate of convergence for the modified Beta operators 𝐵_𝑛(𝑓,𝑥), for functions having derivatives of bounded variation.

1. Introduction

In the year 1995 Gupta and Ahmad [1] proposed modified Beta operators so as to approximate Lebesgue integrable functions on 𝑅⁺ as 𝐵_𝑛(𝑓,𝑥)=𝑛−1𝑛^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)^∞₀𝑝_𝑛,𝑘(𝑡)𝑓(𝑡)𝑑𝑡,𝑥∈𝑅⁺, (1.1) where 𝑏_𝑛,𝑘1(𝑥)=𝑥𝐵(𝑘+1,𝑛)(1+𝑥^𝑘)^𝑛+𝑘+1,𝑝_𝑛,𝑘𝑘(𝑡)=𝑡𝑛+𝑘−1_𝑘(1+𝑡)^{−𝑛−𝑘}. (1.2) These operators are linear positive operators and reproduce only the constant functions. We set 𝜆_𝑛(𝑥,𝑡)=𝑛−1𝑛^𝑡₀^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑠)𝑑𝑠. (1.3) In particular, it can be observed by the definition of Beta function that 𝜆_𝑛(𝑥,∞)=1. (1.4)

Gupta and Ahmad [1] estimated asymptotic formula and error estimate for these operators in simultaneous approximation. Later Gupta et al. [2] studied the approximation properties of these operators in 𝐿_𝑝 norm and they obtained some direct results for the linear combinations. Here we continue our studies on these operators. First we consider the following class of functions.

By 𝐷𝐵_𝑟(0,∞),𝑟≥0 we mean the class of absolutely continuous functions 𝑓 defined on (0,∞), which satisfy the following:

(i)𝑓(𝑡)=𝑂(𝑡^𝑟),𝑡→∞,(ii)this class has a derivative 𝑓^′ on the interval (0,∞) coinciding a.e. with a function which is of bounded variation on every finite subinterval of (0,∞). It can be observed that all functions 𝑓∈𝐵𝐷_𝑟(0,∞) possess for each 𝑐>0 the representation 𝑓(𝑥)=𝑓(𝑐)+^𝑥_𝑐𝜓(𝑡)𝑑𝑡,𝑥≥𝑐. (1.5)

Very recently Gupta and Agrawal [3] and Gupta et al. [4] have obtained an interesting result on the rate of convergence of certain Durrmeyer type operators in ordinary and simultaneous approximation. İspir et al. [5] estimated similar results for Kantorovich operators for functions with derivatives of bounded variation. We now extend the study for the modified Beta operators and estimate the rate of convergence of the operators (1) for functions having derivatives of bounded variation.

2. Auxiliary Results

We shall use the following lemmas to prove our main theorem.

Lemma 2.1 ([1]). Let the function 𝑇_𝑛,𝑚(𝑥), 𝑚∈ℵ⁰, be defined as 𝑇_𝑛,𝑚(𝑥)=𝑛−1𝑛^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)^∞₀𝑝_𝑛,𝑘(𝑡)(𝑡−𝑥)^𝑚𝑑𝑡. (2.1) Then 𝑇_𝑛,0(𝑥)=1,𝑇_𝑛,1(𝑥)=3𝑥+1,𝑇𝑛−2_𝑛,2(𝑥)=2(𝑛+7)𝑥²+2(𝑛+5)𝑥+2.(𝑛−2)(𝑛−3) (2.2) Also for each 𝑥∈[0,∞), one has 𝑇_𝑛,𝑚(𝑥)=𝑂(𝑛^{−[(𝑚+1)/2]}). (2.3)

Remark 2.2. From Lemma 2.1 and using Cauchy-Schwarz inequality, for 𝑛 sufficiently large, it follows that 𝐵_𝑛(|𝑡−𝑥|,𝑥)≤[𝑇_𝑛,2(𝑥)]^1/2≤𝐶𝑥(𝑥+1)𝑛, (2.4) where 𝐶>2.

Lemma 2.3. Let 𝑥∈(0,∞), then for 𝐶>2 and 𝑛 sufficiently large, one has 𝜆_𝑛(𝑥,𝑦)≤𝐶𝑥(𝑥+1)𝑛(𝑥−𝑦)²,0≤𝑦<𝑥,1−𝜆_𝑛(𝑥,𝑧)≤𝐶𝑥(𝑥+1)𝑛(𝑧−𝑥)²,𝑥<𝑧<∞. (2.5)

Proof. By using Lemma 2.1, for 𝑛 sufficiently large, we have 𝜆_𝑛(𝑥,𝑦)=𝑛−1𝑛^𝑦₀^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑑𝑡≤𝑛−1𝑛^𝑦₀(𝑥−𝑡)²(𝑥−𝑦)^2∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑑𝑡≤(𝑥−𝑦)⁻²𝑇_𝑛,2(𝑥)≤𝐶𝑥(𝑥+1)𝑛(𝑥−𝑦)². (2.6) The proof of the second inequality is similar, and we skip the details.

3. Rate of Approximation

Our main result is stated as follows.

Theorem 3.1. Let 𝑓∈𝐷𝐵_𝑟(0,∞),??𝑟∈𝑁, and 𝑥∈(0,∞). Then for 𝑛 sufficiently large, one has |𝐵_𝑛(𝑓,𝑥)−𝑓(𝑥)|≤𝐶(𝑥+1)𝑛(^[√𝑛]_𝑘=1^{𝑥+𝑥/𝑘}_{𝑥−𝑥/𝑘}((𝑓^′)_𝑥𝑥)+√𝑛^{√𝑥+𝑥/𝑛}_{√𝑥−𝑥/𝑛}((𝑓^′)_𝑥+))𝐶(𝑥+1)𝑛𝑥(|𝑓(2𝑥)−𝑓(𝑥)−𝑥𝑓^′(𝑥⁺+)|+|𝑓(𝑥)|)𝐶𝑥(𝑥+1)𝑛(𝑂(𝑛^−𝑟/2)+|𝑓^′(𝑥⁺+)|)𝐶𝑥(𝑥+1)4𝑛|𝑓^′(𝑥⁺)−𝑓^′(𝑥⁻1)|+2|𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)|3𝑥+1,𝑛−2 (3.1) where the auxiliary function 𝑓_𝑥 is given by 𝑓_𝑥(𝑡)=𝑓(𝑡)−𝑓(𝑥⁻),0≤𝑡<𝑥,0,𝑡=𝑥,𝑓(𝑡)−𝑓(𝑥⁺),𝑥<𝑡<∞. (3.2)⋁^𝑏_𝑎𝑓(𝑥) denotes the total variation of 𝑓_𝑥 on [𝑎,𝑏].

Proof. By mean value theorem, we have 𝐵_𝑛(𝑓,𝑥)−𝑓(𝑥)=𝑛−1𝑛^∞₀^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘=(𝑡)(𝑓(𝑡)−𝑓(𝑥))𝑑𝑡_∞0𝑡𝑥𝑛−1𝑛^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)(𝑓^′(𝑢)𝑑𝑢)𝑑𝑡. (3.3) Applying the identity 𝑓^′1(𝑢)=2[𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)]+(𝑓^′)_𝑥1(𝑢)+2[𝑓^′(𝑥⁺)−𝑓^′(𝑥⁻)]sgn(𝑢−𝑥)+[𝑓^′1(𝑥)−2[𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)]]𝜒_𝑥(𝑢), (3.4) Equation (3.3) becomes (keeping in mind that the last term of the above identity vanishes) |𝐵_𝑛(𝑓,𝑥)−𝑓(𝑥)|≤|^∞_𝑥(^𝑡_𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢)𝑛−1𝑛^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘−(𝑡)𝑑𝑡_𝑥0(^𝑡_𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢)𝑛−1𝑛^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘+1(𝑡)𝑑𝑡|2|𝑓^′(𝑥⁺)−𝑓^′(𝑥⁻)|𝐵_𝑛+1(|𝑡−𝑥|,𝑥)2|𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)|𝐵_𝑛((𝑡−𝑥),𝑥). (3.5) As 𝑛−1𝑛^∞₀(^𝑡_𝑥12[𝑓^′(𝑥⁺)−𝑓^′(𝑥⁻)]sgn(𝑢−𝑥)𝑑𝑢)^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘=1(𝑡)𝑑𝑡2[𝑓^′(𝑥⁺)−𝑓^′(𝑥⁻)]𝐵_𝑛(|𝑡−𝑥|,𝑥),𝑛−1𝑛^∞₀(^𝑡_𝑥12[𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)]𝑑𝑢)^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘=1(𝑡)𝑑𝑡2[𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)]𝐵_𝑛((𝑡−𝑥),𝑥). (3.6) Thus in view of the above values, Lemma 2.1, and Remark 2.2, (3.5) reduces to |𝐵_𝑛(𝑓,𝑥)−𝑓(𝑥)|≤|𝐸_𝑛(𝑓,𝑥)+𝐹_𝑛1(𝑓,𝑥)|+2|𝑓^′(𝑥⁺)−𝑓^′(𝑥⁻)|𝐶𝑥(𝑥+1)𝑛+12|𝑓^′(𝑥⁺)+𝑓^′(𝑥⁻)|3𝑥+1.𝑛−2 (3.7) In order to complete the proof of the theorem, it is sufficient to estimate the terms 𝐸_𝑛(𝑓,𝑥) and 𝐹_𝑛(𝑓,𝑥). Applying integration by parts, using Lemma 2.3 and taking 𝑦=𝑥−𝑥/√𝑛, we have |𝐹_𝑛(𝑓,𝑥)|=|^𝑥₀(^𝑡_𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢)𝑑_𝑡𝜆_𝑛(𝑥,𝑡)|_𝑥0𝜆_𝑛(𝑥,𝑡)(𝑓^′)_𝑥(𝑡)𝑑𝑡≤(^𝑦₀+^𝑥_𝑦)|(𝑓^′)_𝑥(𝑡)||𝜆_𝑛≤(𝑥,𝑡)|𝑑𝑡𝐶𝑥(𝑥+1)𝑛^𝑦₀^𝑥_𝑡((𝑓^′)_𝑥)1(𝑥−𝑡)²𝑑𝑡+^𝑥_𝑦^𝑥_𝑡((𝑓^′)_𝑥≤)𝑑𝑡𝐶𝑥(𝑥+1)𝑛^𝑦₀^𝑥_𝑡((𝑓^′)_𝑥)1(𝑥−𝑡)²𝑥𝑑𝑡+√𝑛^𝑥_{√𝑥−𝑥/𝑛}((𝑓^′)_𝑥). (3.8) Let 𝑢=𝑥/(𝑥−𝑡). Then we have 𝐶𝑥(𝑥+1)𝑛^𝑦₀^𝑥_𝑡((𝑓^′)_𝑥)1(𝑥−𝑡)²𝑑𝑡=𝐶𝑥(𝑥+1)𝑛^√𝑛₁^𝑥_{𝑥−𝑥/𝑢}((𝑓^′)_𝑥≤)𝑑𝑢𝐶𝑥(𝑥+1)𝑛^{[√𝑛]𝑥}_{𝑘=1𝑥−𝑥/𝑘}((𝑓^′)_𝑥). (3.9) Thus, |𝐹_𝑛(𝑓,𝑥)|≤𝐶𝑥(𝑥+1)𝑛^{[√𝑛]𝑥}_{𝑘=1𝑥−𝑥/𝑢}((𝑓^′)_𝑥𝑥)+√𝑛^𝑥_{√𝑥−𝑥/𝑛}((𝑓^′)_𝑥). (3.10) On the other hand, we have |𝐸_𝑛(𝑓,𝑥)|=|𝑛−1𝑛^∞_𝑥(^𝑡_𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢)^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑑𝑡|=|𝑛−1𝑛^∞_2𝑥(^𝑡_𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢)^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘+(𝑡)𝑑𝑡_𝑥2𝑥(^𝑡_𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢)𝑑_𝑡(1−𝜆_𝑛(𝑥,𝑡))|≤|𝑛−1𝑛^∞_2𝑥(𝑓(𝑡)−𝑓(𝑥))^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑑𝑡|+|𝑓^′(𝑥⁺)||𝑛−1𝑛^∞_2𝑥(𝑡−𝑥)^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑑𝑡|+|_𝑥^2𝑥(𝑓^′)_𝑥(𝑢)𝑑𝑢||1−𝜆_𝑛(𝑥,2𝑥)|+_𝑥^2𝑥|(𝑓^′)_𝑥(𝑡)||1−𝜆_𝑛≤𝐶(𝑥,𝑡)|𝑑𝑡₁𝑥𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑡^𝑟+|𝑡−𝑥|𝑑𝑡|𝑓(𝑥)|𝑥²𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)(𝑡−𝑥)²𝑑𝑡+|𝑓^′(𝑥⁺)|𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)|𝑡−𝑥|𝑑𝑡+𝐶(𝑥+1)𝑛𝑥|𝑓(2𝑥)−𝑓(𝑥)−𝑥𝑓^′(𝑥⁺+)|𝐶(𝑥+1)𝑛^[√𝑛]_𝑘=1^{𝑥+𝑥/𝑘}_𝑥((𝑓^′)_𝑥𝑥)+√𝑛^{√𝑥+𝑥/𝑛}_𝑥((𝑓^′)_𝑥). (3.11)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10) For the estimation of the first two terms in the right-hand side of (3.11).we proceed as follows.
Applying Holder's inequality, Remark 2.2, and Lemma 2.1, we have 𝐶₁𝑥𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑡^𝑟|𝑡−𝑥|𝑑𝑡+|𝑓(𝑥)|𝑥²𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)(𝑡−𝑥)²≤𝐶𝑑𝑡₁𝑥(𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)𝑡^2𝑟𝑑𝑡)^1/2(^∞₀^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)(𝑡−𝑥)²𝑑𝑡)^1/2+|𝑓(𝑥)|𝑥²(𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)(𝑡−𝑥)²𝑑𝑡)≤𝐶₁𝑂(𝑛^−𝑟/2)𝐶𝑥(𝑥+1)√𝑛+|𝑓(𝑥)|𝐶(𝑥+1).𝑛𝑥 (3.12) Also, by Remark 2.2, the third term of the right side of (3.11) is given by |𝑓^′(𝑥⁺)|𝑛−1𝑛^∞∞_2𝑥𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)|𝑡−𝑥|𝑑𝑡≤|𝑓^′(𝑥⁺)|𝑛−1𝑛^∞₀^∞_𝑘=0𝑏_𝑛,𝑘(𝑥)𝑝_𝑛,𝑘(𝑡)|𝑡−𝑥|𝑑𝑡≤|𝑓^′(𝑥⁺)|𝐶𝑥(𝑥+1)√𝑛. (3.13) Thus 𝐸_𝑛(𝑓,𝑥)≤𝐶₁𝑂(𝑛^−𝑟/2)𝐶𝑥(𝑥+1)√𝑛+|𝑓(𝑥)|𝐶(𝑥+1)𝑛𝑥+|𝑓^′(𝑥⁺)|𝐶𝑥(𝑥+1)√𝑛+𝐶(𝑥+1)𝑛𝑥|𝑓(2𝑥)−𝑓(𝑥)−𝑥𝑓^′(𝑥⁺+)|𝐶(𝑥+1)𝑛^[√𝑛]_𝑘=1^{𝑥+𝑥/𝑘}_𝑥((𝑓^′)_𝑥𝑥)+√𝑛^{√𝑥+𝑥/𝑛}_𝑥((𝑓^′)_𝑥). (3.14) Combining the estimates (3.7), (3.10), and (3.14) we get the desired result. This completes the proof of the theorem.