2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.We establish direct and converse theorems for generalized parameter dependent Bernstein-type operators. The direct estimate is given
using a K-functional and the inverse result is a strong converse inequality
of type A, in the terminology of [2]

Finta, Zoltán

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 30 (2004), 33–42DIRECT AND CONVERSE THEOREMS FORGENERALIZED BERNSTEIN-TYPE OPERATORSZolta´n FintaCommunicated by S. L. TroyanskiAbstract. We establish direct and converse theorems for generalized pa-rameter dependent Bernstein-type operators. The direct estimate is givenusing a K-functional and the inverse result is a strong converse inequalityof type A, in the terminology of [2].1. Introduction. The following operator was introduced by T. N. T.Goodman and A. Sharma in [4]:(Unf)(x) ≡ Un(f, x) == f(0) pn,0(x)+f(1) pn,n(x)+n−1∑k=1pn,k(x) ·∫ 10(n−1) pn−2,k−1(t) f(t) dt,where pn,k(x) =( nk)xk(1− x)n−k, k = 0, n.2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.Key words: Goodman-Sharma operator, direct and converse approximation theorems, K-functional.34 Zolta´n FintaIt was studied for this operator by P. E. Parvanov and B. D. Popov [5]the relation between the rate of approximation of Unf and the K-functionalK(f,1n)≡ K(f,1n; C[0, 1], W 2∞(ϕ))= infg∈W 2∞(ϕ){‖f − g‖+ 1n‖ϕ2g′′‖},where W 2∞(ϕ) consists of all functions g : [0, 1] → R such that g′ is absolutelycontinuous on [0, 1] and ‖ϕ2g′′‖ is finite. Here ϕ(x) =√x(1− x), x ∈ [0, 1] and‖ · ‖ is the uniform norm on C[0, 1]. Originally, in the expression of K(f, 1/n)was considered L∞[0, 1] instead of C[0, 1]. They have proved a direct inequalityand a strong converse inequality of type A, in the terminology of [2], that is forevery f ∈ C[0, 1] were established12· ‖Unf − f‖ ≤ K(f,1n)≤ (6 +√8) · ‖Unf − f‖(1)The Bernstein-type operator discussed in this paper will be given byUαn : C[0, 1]→ C[0, 1], (Uαn f)(x) ≡ Uαn (f, x) == f(0)wn,0(x, α) + f(1)wn,n(x, α) +n−1∑k=1wn,k(x, α) ·1∫0(n− 1)pn−2,k−1(t)f(t) dt(see also [3]), wherewn,k(x, α) =( nk)· Πk−1i=0 (x+ iα) Πn−k−1j=0 (1− x+ jα)(1 + α)(1 + 2α) . . . (1 + (n− 1)α) ,k = 0, n and α ≥ 0 is a parameter which may depend only on the natural numbern. In the case α = 0, U0n is the Goodman-Sharma operator defined above. Thepurpose of this paper is to establish for Uαn direct inequality and strong converseinequality of type A.2. Direct theorem. The theorem in question can be stated as follows:Theorem 1. Let α ≥ 0 andK(f,11 + α·(2n+ 1+ α))= infg∈W 2∞(ϕ){‖f−g‖+ 11+α·(2n+ 1+ α)·‖ϕ2g′′‖}Then for every f ∈ C[0, 1] we have‖Uαn f − f‖ ≤ 2 K(f,11 + α·(2n+ 1+ α))(2)Direct and converse theorems. . . 35P r o o f. By [6, p. 1180, Lemma 3.1] we have for α > 0 and x ∈ (0, 1) thefollowing identitywn,k(x, α) =( nk)· B(xα−1 + k, (1− x)α−1 + n− k)B (xα−1, (1− x)α−1) ,where B( · , · ) denotes the Beta function. Consequently, Uαn f can be representedby means of the Goodman-Sharma operator, as follows(Uαn f)(x) =1B(xα, 1−xα) ·1∫0txα−1 (1− t) 1−xα −1 · (Unf)(t) dt.Hence, by simple computations and [5, p. 166, (2.1)–(2.4)] we obtain that Uαn islinear and positive,Uαn (u− x, x) = 0,(3)Uαn((u− x)2, x) =(2n+ 1+ α)· x(1− x)1 + α(4)and‖Uαn f‖ ≤ ‖f‖(5)for every f ∈ C[0, 1]. Now, the proof is standard (cf. [1, Chapter 9]): usingTaylor’s formulag(u) = g(x) + (u− x) g′(x) +u∫x(u− v) g′′(v) dv,for g ∈W 2∞(ϕ), and (3), [1, p. 141, (9.6.1)] and (4), we obtain|(Uαn g)(x)− g(x)| ≤ Uαn(∣∣∣∫ ux|u− v|ϕ(v)2· ϕ(v)2 |g′′(v)| dv∣∣∣ , x)≤(2n+ 1+ α)· 11 + α· ‖ϕ2g′′‖36 Zolta´n FintaHence, by (5), we have|(Uαn f)(x)− f(x)| ≤ |(Uαn (f − g))(x) − (f − g)(x)| + |(Uαn g)(x) − g(x)|≤ 2 ‖f − g‖+(2n+ 1+ α)· 11 + α· ‖ϕ2g′′‖Taking infimum over all g ∈W 2∞(ϕ), we obtain (2). 3. Direct and converse theorems. In order to prove our results,we need two lemmas:Lemma 1. We havesupf∈C[0,1]f 6=linear‖ϕ2(Unf)′′‖‖Unf − f‖ ={0, if n = 1c0n, if n ≥ 2(6)where 1/2 ≤ c0 ≤ 4 + 3√2.P r o o f. Using [5, p. 175, Lemma 5.2] and the estimate given in the proofof [5, p. 177, Theorem 5] we obtain‖ϕ2(Unf)′′‖ ≤ ‖ϕ2(Un(f − Unf))′′‖+ ‖ϕ2(U2nf)′′‖≤√2 n · ‖f − Unf‖+ (4 + 2√2) n · ‖Unf − f‖= (4 + 3√2) n ‖Unf − f‖Sosupf∈C[0,1]f 6=linear‖ϕ2(Unf)′′‖‖Unf − f‖ ≤ (4 + 3√2)n.On the other hand, forf1(x) =c12· x2 + x+ 1, x ∈ [0, 1],where c1 > 0 is a given constant, we have f1 ∈W 2∞(ϕ){0, 1} (see [5, p. 171]) and,in view of [5, p. 171, Lemma 4.2], we getϕ(x)2(Unf1)′′(x) = (Un(ϕ2f ′′1 ))(x) = c1 Un(u(1 − u) , x)= c1(1− 2n+ 1)ϕ(x)2Direct and converse theorems. . . 37Thus‖ϕ2(Unf1)′′‖ = c14·(1− 2n+ 1)Moreover, by [5, p. 173, (4.6)] and [5, p. 171, Lemma 4.2] we obtain(Unf1)(x)− f1(x) =∞∑k=n+11k(k − 1) · ϕ(x)2 (Ukf1)′′(x)=∞∑k=n+11k(k − 1) · Uk(ϕ2f ′′1 , x)=∞∑k=n+11k(k − 1) · c1(1− 2k + 1)· x(1− x)Therefore‖Unf1 − f1‖ = c14·∞∑k=n+11k(k − 1) ·(1− 2k + 1)=c14· 1n(1− 1n+ 1)Hence‖ϕ2(Unf1)′′‖‖Unf1 − f1‖ =1− 2n+11n(1− 1n+1) = n− 1 ≥ n2for n ≥ 2. Thussupf∈C[0,1]f 6=linear‖ϕ2(Unf)′′‖‖Unf − f‖ ≥n2, n ≥ 2In conclusion, for n ≥ 2 we get (6) with c0 ∈[1/2, 4 + 3√2].For n = 1 we have (Unf)(x) = f(0)(1−x)+f(1)x. Therefore ‖ϕ2(Unf)′′‖ =0, which implies the conclusion. Lemma 2. We havesupf∈C[0,1]f 6=linear‖Uαn f − Unf‖‖ϕ2(Unf)′′‖ = α0 ·α1 + α,(7)38 Zolta´n Fintawhere 1/2 ≤ α0 ≤ 1 and n ≥ 2.P r o o f. By Taylor’s formula(Unf)(t) = (Unf)(x) + (t− x)(Unf)′(x) +t∫x(t− u) · (Unf)′′(u) du(8)and1∫0txα−1 (1− t) 1−xα −1 (t− x) dt = 0,(9)we obtain|(Uαn f)(x)− (Unf)(x)| ==1B(xα, 1−xα) · ∣∣∣1∫0txα−1 (1− t) 1−xα −1 [(Unf)(t)− (Unf)(x)] dt∣∣∣=1B(xα, 1−xα) · ∣∣∣1∫0txα−1 (1− t) 1−xα −1 ··(t− x) · (Unf)′(x) +t∫x(t− u)(Unf)′′(u) du dt ∣∣∣≤ 1B(xα, 1−xα) ·1∫0txα−1 (1− t) 1−xα −1·∣∣∣t∫x(t− u) · (Unf)′′(u) du∣∣∣ dt≤ ‖ϕ2(Unf)′′‖B(xα, 1−xα) ·1∫0txα−1 (1− t) 1−xα −1·∣∣∣t∫x|t− u|u(1− u) du∣∣∣ dtUsing again [1, p. 141, (9.6.1)] and1∫0txα−1 (1− t) 1−xα −1 (t− x)2 dt = αx(1− x)1 + α·B(xα,1− xα),(10)Direct and converse theorems. . . 39we have|(Uαn f)(x)− (Unf)(x)| ≤≤ ‖ϕ2(Unf)′′‖ϕ(x)2 ·B ( xα, 1−xα) ·1∫0txα−1 (1− t) 1−xα −1 · (t− x)2 dt=α1 + α· ‖ϕ2(Unf)′′‖Hence‖Uαn f − Unf‖ ≤α1 + α· ‖ϕ2(Unf)′′‖(11)and thereforesupf∈C[0,1]f 6=linear‖Uαn f − Unf‖‖ϕ2(Unf)′′‖ ≤α1 + αOn the other hand, for f1(x) = (c1/2) · x2 + x + 1 (see Lemma 1) we have, asabove‖ϕ2(Unf1)′′‖ = c14·(1− 2n+ 1)and(Unf1)′′(x) = c1(1− 2n+ 1)Hence, by (8), (9) and (10), replacing f with f1, we obtain(Uαn f1)(x)− (Unf1)(x) ==1B(xα, 1−xα) ·∫ 10txα−1 (1− t) 1−xα −1 ·{∫ tx(t− u) (Unf1)′′(u) du}dt=1B(xα, 1−xα) ·∫ 10txα−1 (1− t) 1−xα −1 ·{∫ tx(t− u) · c1(1− 2n+ 1)du}dt=1B(xα, 1−xα) ·∫ 10txα−1 (1− t) 1−xα −1 · c1(1− 2n+ 1)· 12(t− x)2 dt40 Zolta´n Finta=α1 + α· c12·(1− 2n+ 1)· x(1− x)So‖Uαn f1 − Unf1‖ =α1 + α· c18·(1− 2n+ 1)and‖Uαn f1 − Unf1‖‖ϕ2(Unf1)′′‖ =12· α1 + αHencesupf∈C[0,1]f 6=linear‖Uαn f − Unf‖‖ϕ2(Unf)′′‖ ≥12· α1 + αIn conclusion we get (7) with α0 ∈ [1/2, 1]. Now we can prove the following result:Theorem 2. If α = α(n) and c0 α0 · (n α)/(1 + α) ≤ α1 < 1 forn = 1, 2, . . . then for every f ∈ C[0, 1] we have(1− α1) ‖Unf − f‖ ≤ ‖Uαn f − f‖ ≤ (1 + α1) ‖Unf − f‖(12)and1− α16 +√8·K(f,1n)≤ ‖Uαn f − f‖ ≤ 2 (1 + α1) ·K(f,1n)(13)P r o o f. By (7) and (6) we have‖Uαn f − Unf‖ ≤ α0 ·α1 + α· ‖ϕ2(Unf)′′‖ ≤ c0 α0 · nα1 + α· ‖Unf − f‖≤ α1 · ‖Unf − f‖Hence‖Unf − f‖ ≤ ‖Uαn f − Unf‖+ ‖Uαn f − f‖ ≤ α1 ‖Unf − f‖+ ‖Uαn f − f‖Direct and converse theorems. . . 41or(1− α1) ‖Unf − f‖ ≤ ‖Uαn f − f‖and‖Uαn f − f‖ ≤ ‖Uαn f − Unf‖+ ‖Unf − f‖ ≤ (1 + α1) ‖Unf − f‖,respectively. So we obtain (12). For the second statement we use (1) and (12)obtaining (13), which completes the proof. Furthermore, we have the following property:Theorem 3. If α = α(n) and (n2α)/(1 + α) ≤ α2 < 1 for n = 1, 2, . . .then for every f ∈ C[0, 1] we have(1− α2) ‖Unf − Un−1f‖ ≤ ‖Uαn f − Un−1f‖ ≤ (1 + α2) ‖Unf − Un−1f‖(14)P r o o f. By (11) and [5, p. 169, Lemma 4.1 ] we obtain‖Uαn f − Unf‖ ≤α1 + α· n(n− 1) · ‖Unf − Un−1f‖ ≤ α2 · ‖Unf − Un−1f‖Then we have‖Unf − Un−1f‖ ≤ ‖Uαn f − Un−1f‖+ ‖Uαn f − Unf‖≤ ‖Uαn f − Un−1f‖+ α2 · ‖Unf − Un−1f‖or(1− α2) ‖Unf − Un−1f‖ ≤ ‖Uαn f − Un−1f‖and‖Uαn f − Un−1f‖ ≤ ‖Uαn f − Unf‖+ ‖Unf − Un−1f‖≤ (1 + α2) · ‖Unf − Un−1f‖,respectively. Thus we have proved (14). 42 Zolta´n FintaREFERENCES[1] Z. Ditzian, V. Totik. Moduli of Smoothness. Springer-Verlag, New YorkBerlin Heidelberg London, 1987.[2] Z. Ditzian, K. G. Ivanov. Strong converse inequalities. J. Anal. Math.61 (1993), 61–111.[3] Z. Finta. Quantitative estimates for some linear and positive operators.Stud. Univ. Babes¸-Bolyai, Ser. Mathematica 47, 3 (2002), 71–84.[4] T. N. T. Goodman, A. Sharma. A Bernstein-type Operator on the Sim-plex. Math. Balkanica 5, 2 (1991), 129–145.[5] P. E. Parvanov, B. D. Popov. The limit case of Bernstein’s operatorswith Jacobi-weights. Math. Balkanica 8, 2–3 (1994), 165-177.[6] D. D. Stancu. Approximation of functions by a new class of linear polyno-mial operators. Rev. Roumaine Math. Pures Appl. 13, 8 (1968), 1173–1194.Babes¸-Bolyai UniversityDepartment of Mathematics and Computer Science1, M. Koga˘lniceanu st.3400 Cluj, Romaniae-mail: fzoltan@math.ubbcluj.ro Received July 21, 2003

Direct and Converse Theorems for Generalized Bernstein-Type Operators

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Direct and Converse Theorems for Generalized Bernstein-Type Operators

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