The purpose of the present paper is to introduce $q-$ analouge of a sequence of linear and positive operators which was introduced by A. Lupas [2]. First, we estimate moments of the operators and then prove a basic convergence theorem. Next, a local direct approximation theorem is established. Further, we study the rate of convergence and point-wise estimate using the Lipschitz type maximal function

Asha Ram Gairola

Deepmala .

Karunesh Kumar Singh

International Journal of Analysis and Applications

Directory of Open Access Journals

International Journal of Analysis and ApplicationsISSN 2291-8639Volume 12, Number 1 (2016), 30-37http://www.etamaths.comAPPROXIMATION THEOREMS FOR q− ANALOUGE OF A LINEAR POSITIVEOPERATOR BY A. LUPASKARUNESH KUMAR SINGH1, ASHA RAM GAIROLA2 AND DEEPMALA3,∗Abstract. The purpose of the present paper is to introduce q− analouge of a sequence of linearand positive operators which was introduced by A. Lupas [1]. First, we estimate moments of theoperators and then prove a basic convergence theorem. Next, a local direct approximation theoremis established. Further, we study the rate of convergence and point-wise estimate using the Lipschitztype maximal function.1. IntroductionAt the International Dortmund Meeting held in Written (Germany, March, 1995), A. Lupas [1]introduced the following Linear positive operators:(1) Ln(f ;x) = (1− a)nx∞∑k=0(nx)kk!akf(kn), x ≥ 0.with f : [0,∞] → R. If we impose that Lne1 = e1 we find that a = 1/2. Therefore operator (1)becomesLn(f ;x) = 2−nx∞∑k=0(nx)k2kk!f(kn), x ≥ 0,where(α)0 = 1, (α)k = α(α+ 1)...(α+ k − 1), k ≥ 1.The q− analouge of the above operators is defined as:Ln,q(f ;x) = 2−[n]qx∞∑k=0([n]qx)k2k[k]q!f([k]q[n]q), x ≥ 0,We denote CB [0,∞) the space of real valued bounded continuous function f on the interval [0,∞),the norm on the space is defined as‖f‖ = sup0≤x<∞|f(x)|.Let W 2 = {g ∈ CB [0,∞) : g′, g′′ ∈ CB [0,∞)}. The Peetre’s K− functional is defined asK2(f, δ) = infg∈W 2{‖f − g‖+ δ‖g′′‖},where δ > 0.For f ∈ CB [0,∞) a usual modulus of continuity is given byω(f, δ) = sup0<h≤δsup0≤x<∞|f(x+ h)− f(x))|.The second order modulus of smoothness is given byω2(f,√δ) = sup0<h≤√δsup0≤x<∞|f(x+ 2h)− 2f(x+ h) + f(x))|.2010 Mathematics Subject Classification. 41A36, 41A99, 41A25.Key words and phrases. linear positive operators; q−operators; rate of convergence.c©2016 Authors retain the copyrights oftheir papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.30LINEAR POSITIVE OPERATORS BY A. LUPAS 31By [[3], p.177, Theorem 2.4] there exists an absolute constant C > 0 such thatK2(f, δ) ≤ Cω2(f,√δ).In recent years, many results about the generalization of linear positive operators have been obtainedby several mathematicians ([6]-[17]).2. Moment estimatesLemma 1. The following relations hold:Ln,q(1;x) = 1, Ln,q(t;x) = x and Ln,q(t2;x) = qx2 +1 + q[n]x.Proof. We haveLn,q(1;x) = 2−[n]qx∞∑k=0([n]qx)k2k[k]q!= 1Now,Ln,q(t;x) = 2−[n]qx∞∑k=0([n]qx)k2k[k]q![k]q[n]q= 2−[n]qx∞∑k=0([n]qx)k2k[k − 1]q![n]q=2−[n]qx−1[n]q∞∑k=1[n]qx([n]qx+ 1)k−12k−1[k − 1]q!= 2−[n]qx−1x∞∑k=1([n]qx+ 1)k−12k−1[k − 1]q!= 2−[n]qx−1x∞∑k=0([n]qx+ 1)k2k[k]q!= x.Next,Ln,q(t2;x) = 2−[n]qx∞∑k=0([n]qx)k2k[k]q![k]2q[n]2q= 2−[n]qx∞∑k=0[n]qx([n]qx+ 1)k−12k[k]q[k − 1]q![k]2q[n]2q= 2−[n]qx−1x∞∑k=1([n]qx+ 1)k−12k−1[k − 1]q![k]q[n]q=2−[n]qx−1x[n]q∞∑k=1([n]qx+ 1)k−1[k]q2k−1[k − 1]q!=2−[n]qx−1x[n]q∞∑k=0([n]qx+ 1)k[k + 1]q2k[k]q!=2−[n]qx−1x[n]q∞∑k=0([n]qx+ 1)k(1 + q[k]q)2k[k]q!=2−[n]qx−1x[n]q∞∑k=0([n]qx+ 1)k2k[k]q!+2−[n]qx−1x[n]q∞∑k=0([n]qx+ 1)kq[k]q2k[k]q!= I1 + I2, say.32 SINGH, GAIROLA AND DEEPMALAWe find that I1 =x[n]q.Now,I2 =2−[n]qx−1x[n]q∞∑k=0([n]qx+ 1)kq[k]q2k[k]q!=2−[n]qx−2qx[n]q∞∑k=1([n]qx+ 1)([n]qx+ 2)k−12k−1[k − 1]q!=2−[n]qx−2qx([n]qx+ 1)[n]q∞∑k=1([n]qx+ 2)k−12k−1[k − 1]q!=2−[n]qx−2qx([n]qx+ 1)[n]q∞∑k=0([n]qx+ 2)k2k[k]q!=qx([n]qx+ 1)[n]q.Hence, on combining I1 and I2, we getLn,q(t2;x) =(1 + q)x[n]q+ qx2.Let us define mth order moment by ψn,m(q;x) = Ln,q((t− x)m;x).Lemma 2. Let 0 < q < 1, then for x ∈ [0,∞) we haveψn,1(q;x) = 0 and ψn,2(q;x) =x([2]− (1− q)[n]qx)[n]q.Proof. We haveψn,1(q;x) = Ln,q(t− x;x) = 0.Now,ψn,2(q;x) = Ln,q((t− x)2;x)= Ln,q(t2 + x2 − 2tx;x)=(1 + q)x[n]q+ (q − 1)x2.3. Basic Pointwise ConvergenceThe operators Ln,q do not satisfy the conditions of the Bohman-Korovkin theorem in case 0 < q < 1.To make this theorem applicable, we can choose a sequence (qn) in place of the number q such thatqn → 1 and qnn → 0 as n→∞. With this modification we obtain the following Korovkin type result:Theorem 1. Let f ∈ CB [0,∞) and qn be a real sequence in (0, 1) such that qn → 1 and qnn → 0 asn→∞. Then, for each x ∈ [0,∞) we havelimn→∞Ln,qn(f ;x) = f(x).Proof. The proof is based on the well known Korovkin theorem regarding the convergence of a sequenceof linear positive operators. So, it is enough to prove the conditionslimn→∞Ln,qn(tm;x) = xm,m = 0, 1, 2.Now, using Lemma 1 we obtainlimn→∞Ln,qn(1;x) = 1,limn→∞Ln,qn(t;x) = xLINEAR POSITIVE OPERATORS BY A. LUPAS 33andlimn→∞Ln,qn(t;x) = limn→∞qnx2 +1 + qn[n]qnx= x2.This completes the proof. 4. Direct resultsTheorem 2. Let f ∈ CB [0,∞) and q ∈ (0, 1). Then, for each x ∈ [0,∞) and n ∈ N there exists anabsolute constant C > 0 such that|Ln,q(f ;x)− f(x)| 6 Cω2(f,√x([2]− (1− q)[n]qx)[n]q).Proof. Let g ∈W 2 and x, t ∈ [0,∞). Using Taylor’s expansion we can writeg(t) = g(x) + g′(x)(t− x) +t∫x(t− v)g′′(v)dv.On application of Lemma 2 we obtainLn,q(g(t);x)− g(x))= Ln,q t∫x(t− v)g′′(v)dv;x .Now, we have∣∣∣∣ t∫x(t− v)g′′(v)dv∣∣∣∣ ≤ (t− x)2‖g′′‖. Therefore|Ln,q(g(t);x)− g(x)| ≤ Ln,q((t− x)2;x)‖g′′‖ = x([2]− (1− q)[n]qx)[n]q‖g′′‖.By Lemma 1, we have|Ln,q(f ;x)| ≤ 2−[n]qx∞∑k=0([n]qx)k2k[k]q!∣∣∣∣f ( [k]q[n]q)∣∣∣∣ ≤ ‖f‖.Thus|Ln,q(f ;x)− f(x)| ≤ |Ln,q(f − g;x)− (f − g)(x)|+ |Ln,q(g;x)− g(x)|≤ 2‖f − g‖+ x([2]−(1−q)[n]qx)[n]q ‖g′′‖.At last, taking the infimum over all g ∈W 2 and on application of the inequalityK2(f, δ) ≤ Cω2(f, δ1/2), δ >0, we get the required result. This completes the proof of the theorem.5. Pointwise EstimatesIn this section, we obtain some pointwise estimates of the rate of convergence of the q− Baskakov-Durrmeyer operators. First, we discuss the relationship between the local smoothness of f and thelocal approximation.Theorem 3. Let 0 < α ≤ 1 and E be any bounded subset of the interval [0,∞). If f ∈ CB [0,∞) ∩LipM (α) then we have|Ln,q(f ;x)− f(x)| ≤M{ψα2n,2(q;x) + 2(d(x,E))α}, x ∈ [0,∞),where M is a constant depending on α and f , d(x,E) is the distance between x and E defined asd(x,E) = inf{|t− x|; t ∈ E} and ψn,2(q;x) = Ln,q((t− x)2;x) .34 SINGH, GAIROLA AND DEEPMALAProof. From the property of infimum, it follows that there exists a point t0 ∈ Ē such that d(x,E) =|t0 − x|.In view of the triangle inequality we have|f(t)− f(x)| ≤ |f(t)− f(t0)|+ |f(t0)− f(x)|.Using the definition of LipM (α), we get|Ln,q(f ;x)− f(x)| ≤ Ln,q(|f(t)− f(t0)|;x) + Ln,q(|f(x)− f(t0)|;x)≤ M{Ln,q(|t− t0|α;x) + |x− t0|α}≤ M{Ln,q(|t− x|α;x) + 2|x− t0|α}.Choosing p1 =2α and p2 =22−α , we get1p1+ 1p2 = 1. Then, Hölder’s inequality yields|Ln,q(f ;x)− f(x)| ≤ M{(Ln,q(|t− x|αp1 ;x))1/p1 [Ln,q(1p2 ;x)]1/p2 + 2(d(x,E))α}≤ M{(Ln,q((t− x)2;x))α/2 + 2(d(x,E))α}= M{ψα/2n,2 (q;x) + 2(d(x,E))α}.This completes the proof of the theorem. Next, we obtain a local direct estimate of operators Ln,q using the Lipschitz-type maximal functionof order α introduced by Lenze [2] as(2) ω̃α(f, x) = supt 6=x,t∈[0,∞)|f(t)− f(x)||t− x|α, x ∈ [0,∞) andα ∈ (0, 1].Theorem 4. Let 0 < α ≤ 1 and f ∈ CB [0,∞), then for all x ∈ [0,∞) we have|Ln,q(f ;x)− f(x)| ≤ ω̃α(f, x)ψα/2n,2 (q;x).Proof. In view of (2), we get|f(t)− f(x)| ≤ ω̃α(f, x)|t− x|αand hence|Ln,q(f ;x)− f(x)| ≤ Ln,q(|f(t)− f(x)|;x) ≤ ω̃α(f, x)Ln,q(|t− x|α;x).Now, using the Hölder’s inequality with p = 2α and1q = 1−1p , we obtain|Ln,q(f ;x)− f(x)| ≤ ω̃α(f, x)(Ln,q(|t− x|2;x))α/2 = ω̃α(f, x)ψα/2n,2 (x).Thus, the proof is completed.6. Weighted ApproximationIn this section, we discuss about the weighted approximation theorem for the operators Ln,q(f).Let C∗x2 [0,∞) be the subspace of all functions f ∈ Cx2 [0,∞) for which limx→∞|f(x)|1+x2 is finite.Theorem 5. Let qn be a sequence in (0, 1) such that qn → 1 and qnn → 0, as n → ∞. For eachC∗x2 [0,∞), we have(3) limn→∞‖Ln,qn(f)− f‖x2 = 0.LINEAR POSITIVE OPERATORS BY A. LUPAS 35Proof. In order to proof (3) it is sufficient to show that ([5])(4) limn→∞‖Ln,qn(tν ;x)− xν‖x2 = 0, ν = 0, 1, 2.Since, Ln,qn(1;x) = 1, (4) holds true for ν = 0.Now, by Lemma 1, we have‖Ln,qn(t;x)− x‖x2 = supx∈[0,∞)|Ln,qn(t;x)− x|1 + x2→ 0, as n→∞.Therefore, (4) is true for ν = 1.Again, by Lemma 1, we may write‖Ln,qn(t2;x)− x2‖x2 = supx∈[0,∞)|Ln,qn(t2;x)− x2|1 + x2= supx∈[0,∞)∣∣∣ (1+qn)x+qn[n]qnx2[n]qn − x2∣∣∣1 + x2≤ 1 + qn[n]qnsupx∈[0,∞)x1 + x2+ (qn − 1) supx∈[0,∞)x21 + x2=1 + qn[n]qn+ (qn − 1).Hence, (4) follows for ν = 2. This completes the proof of the theorem. Theorem 6. Let f ∈ Cx2 [0,∞), q = qn ∈ (0, 1) such that qn → 1 and qnn → 0 as n → ∞ and ωa+1beits modulus of continuity on the finite interval [0, a+ 1] ⊂ [0,∞), a > 0. Then, for every n ≥ 1‖Ln,q(f)− f‖C[0,a] ≤12Mf (1+a2)a[n]q+ 2ωa+1(f,√2a[n]q).Proof. For x ∈ [0, a] and t > a+ 1. Since t− x > 1, we have|f(t)− f(x)| ≤ Mf (2 + x2 + t2)≤ Mf (2 + 3x2 + 2(t− x)2)≤ 3Mf (1 + x2 + (t− x)2)≤ 6Mf (1 + x2)(t− x)2≤ 6Mf (1 + a2)(t− x)2.(5)For x ∈ [0, a] and t ≤ a+ 1, we have|f(t)− f(x)| ≤ ωa+1(f, |t− x|) ≤(1 +|t− x|δ)ωa+1(f, δ),(6)where δ > 0.From (5) and (6), we can write|f(t)− f(x)| ≤ 6Mf (1 + a2)(t− x)2 +(1 +|t− x|δ)ωa+1(f, δ)(7)For x ∈ [0, a] and t ≥ 0 and applying Schwarz inequality, we obtain|Ln,q(f ;x)− f(x)| ≤ Ln,q(|f(t)− f(x)|;x)≤ 6Mf (1 + a2)Ln,q((t− x)2;x)+ ωa+1(f, δ)(1 +1δLn,q((t− x)2;x)12).36 SINGH, GAIROLA AND DEEPMALAHence, using Lemma 2, for every q ∈ (0, 1) and x ∈ [0, a]|Ln,q(f ;x)− f(x)| ≤ 6Mf (1 + a2)x([2]− (1− q)[n]qx)[n]q+ Cωa+1(f, δ)(1 +1δ√x([2]− (1− q)[n]qx)[n]q≤ 12Mf (1 + a2)a[n]q+ ωa+1(f, δ)(1 +1δ√2a[n]q).Taking δ =√2a[n]q, we get the required result.This completes the proof of Theorem. Now, we prove a theorem to approximate all functions in Cx2 [0,∞). Such type of results are givenin [4] for locally integrable functions.Theorem 7. Let q = qn ∈ (0, 1) such that qn → 1 and qnn → 0, as n → ∞. For each f ∈ C∗x2 [0,∞),and α > 1, we havelimn→∞supx∈[0,∞)|Ln,qn(f ;x)− f(x)|(1 + x2)α= 0.Proof. For any fixed x0 > 0,supx∈[0,∞)|Ln,qn(f ;x)− f(x)|(1 + x2)α≤ supx≤x0|Ln,qn(f ;x)− f(x)|(1 + x2)α+ supx>x0|Ln,qn(f ;x)− f(x)|(1 + x2)α≤ ‖Ln,qn(f)− f‖C[0,x0] + ‖f‖x2 supx≥x0|Ln,qn(1 + t2;x)|(1 + x2)α(8)+ supx≥x0|f(x)|(1 + x2)α.Since, |f(x)| ≤Mf (1 + x2), we havesupx≥x0|f(x)|(1 + x2)α≤ supx≥x0Mf(1 + x2)α−1≤ Mf(1 + x20)α−1 .Let ε > 0 be arbitrary. We can choose x0 to be large that(9)Mf(1 + x20)α−1 <ε3and in view of Lemma 1, we obtain‖f‖x2 limn→∞|Ln,qn(1 + t2;x)|(1 + x2)α=1 + x2(1 + x2)α‖f‖x2=‖f‖x2(1 + x2)α−1≤ ‖f‖x2(1 + x20)α−1<ε3.(10)Using Theorem 6 we can see that the first term of the inequality (8) implies that(11) ‖Ln,qn(f ; .)− f‖C[0,x0] <ε3, as n→∞.Combining (8)-(11), we get the desired result.LINEAR POSITIVE OPERATORS BY A. LUPAS 37AcknowledgementThe research work of the third author Deepmala is supported by the Science and Engineering ResearchBoard (SERB), Government of India under SERB NPDF scheme, File Number: PDF/2015/000799.References[1] A. Lupas, The approximation by some positive linear operators. In: proceedings of the International Dortmundmeeting on Approximation Theory(M.W. Müller et al., eds.), akademie Verlag, Berlin, 1995, 201-229.[2] B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces, Nederl. Akad. Wetensch. Indag. Math.50(1) (1988), 531-763.[3] R.A. DeVore and G. G. Lorentz, Constructive Approximation, Volume 303. Springer-Verlag, Berlin, 1993[4] A.D. Gadjiev, R. O. Efendiyev, and E. Ibikli, On Korovkin type theorem in the space of locally integrable functions,Czechoslovak Mathematical Journal, 53 (1) (2003), 45-53.[5] A.D. Gadjiev, Theorems of the type of P. P. Korovkin’s theorems, Math. Zametki, 20(5) (1976), 7811-7786.[6] A.R. Gairola, Deepmala, and L.N. Mishra, Rate of Approximation by Finite Iterates of q-Durrmeyer Operators,Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 86(2) (2016), 229-234.[7] M. A. Özarslan, O. Duman and H.M. Srivastava, Statistical approximation results for Kantorovich-type operatorsinvolving some special polynomials, Math. Comput. Modelling, 48 (2008), 388-401.[8] V.N. Mishra, P. Sharma, and L.N. Mishra; On statistical approximation properties of q−Baskakov-Szász-Stancuoperators, Journal of Egyptian Mathematical Society, 24 (3) (2016), 396-401.[9] H.M. Srivastava, M. Mursaleen, and A. Khan, Generalized equi-statistical convergence of positive linear operatorsand associated approximation theorems, Math. Comput. Model. 55 (2012), 2040-2051.[10] V.N. Mishra, K. Khatri, and L.N. Mishra, Statistical approximation by Kantorovich type Discrete q−Beta operators,Advances in Difference Equations 2013 (2013), Art. ID 345.[11] M. Mursaleen, A. Khan, H.M. Srivastava, and K.S. Nisar, Operators constructed by means of q-Lagrange polyno-mials and A-statistical approximation, Appl. Math. Comput. 219 (2013), 6911-6918.[12] N.L. Braha, H.M. Srivastava and S.A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functionsvia the statistical summability of the generalized de la Vallée Poussin mean, Appl. Math. Comput. 228 (2014), 162-169.[13] V.N. Mishra, K. Khatri, L.N. Mishra, and Deepmala; Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications 2013 (2013), Art. ID 586.[14] V.N. Mishra, H.H. Khan, K. Khatri, and L.N. Mishra; Hypergeometric Representation for Baskakov-Durrmeyer-Stancu Type Operators, Bulletin of Mathematical Analysis and Applications, 5 (3) (2013), 18-26.[15] V.N. Mishra, K. Khatri, and L.N. Mishra; On Simultaneous Approximation for Baskakov-Durrmeyer-Stancu typeoperators, Journal of Ultra Scientist of Physical Sciences, 24 (3) (2012), 567-577.[16] V.N. Mishra, K. Khatri, and L.N. Mishra, Some approximation properties of q-Baskakov-Beta-Stancu type opera-tors, Journal of Calculus of Variations, 2013 (2013), Article ID 814824.[17] A. Wafi, N. Rao, and Deepmala, Approximation properties by generalized-Baskakov-Kantorovich-Stancu type op-erators, Appl. Math. Inf. Sci. Lett., 4 (3) (2016), 1-8.1Government Polytechnic, Rampur, Uttar Pradesh, India2Department of Mathematics, Doon University, Dehradun, Uttarakhand-248 001, India3SQC & OR Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700 108, West Bengal, India∗Corresponding author: dmrai23@gmail.com, deepmaladm23@gmail.com

Approximation Theorems for q- Analouge of a Linear Positive Operator by A. Lupas

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals

Directory of Open Access Journals