Approximation degree for generalized integral operators

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Revista de la Unión Matemática Argentina

versão ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

Approximation degree for generalized integral operators

S. Jain and R. K. Gangwar

Abstract. Very recently Jain et al. [4] proposed generalized integrated Baskakov operators Vn,α (f, x),α > 0 and estimated some approximation properties in simultaneous approximation. In the present paper we establish the rate of convergence of these operators and its Bezier variant, for functions which have derivatives of bounded variation.

1. Introduction

For x ∈ [0,∞ ) and α > 0, the general family of Baskakov type operators considered in [3] is defined as

∞ ∫ ∑ ∞ - n∕α Vn,α(f,x ) = pn,v,α(x) bn,v- 1,α(t)f (t)dt + (1 + αx ) f(0) v=1 0

(1)

where

Γ (n∕α + v) (αx)v α Γ (n∕α + v + 1) (αt)v pn,v,α(x) = --------------.--------n+v,bn,v,α(t) =----------------.-------n+v+1-. Γ (v + 1)Γ (n ∕α) (1+ αx )α Γ (v + 1)Γ (n∕α) (1 + αt)α

In the alternative form the above operators (1), can be defined as

∫ ∞ Vn,α(f (t),x) = 0 Wn,α(x, t)f (t)dt,

where the kernel W (x,t) n,α in terms of Dirac delta function δ(t), is given by

∑∞ Wn,α(x,t) = pn,v,α(x)bn,v-1,α(t) + (1 + αx)- n∕αδ (t). v=1

We define ∫ t βn,α (x, t) = 0 Wn, α(x,s)ds, then as a special case we have ∫ ∞ βn,α(x,∞ ) = 0 Wn, α(x,s)ds = 1. Let DB γ(0, ∞ ),γ ≥ 0 be the class of absolutely continuous functions defined on (0,∞ ) satisfying the growth condition f(t) = O(tγ),t → ∞ and having a derivative on the interval 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 γ(0,∞ ) posses for each c > 0 a representation

∫ x f (x) = f(c) + ψ (t)dt, x ≥ c. c

In [4] the authors studied some direct results in simultaneous approximation for the operators (1). Very recently the rate of convergence for bounded function for the operators Vn,α (f, x) has been obtained by Gangwar and Jain [3]. For certain operators Bojanic and Khan [2] and Taberska [10] estimated the rate of convergence for functions having derivative of B.V. The analogous problem on the convergence rate for the Bernstein polynomials and certain other integral operators were studied in [1],[5], [6] and [8]. Very recently Ispir et al. [9] considered the Kantorovich process of a generalized sequence of linear positive operators and estimated the rate of convergence for absolutely continuous functions having a derivative coinciding a.e., with a function of bounded variation.

As the operators defined by (1) are the generalized operators, this motivated us to extend the studies and here we study the approximation properties of the operators Vn,α(f,x) and its Bezier variant. We estimate the convergence rate for functions whose first derivative is of bounded variation.

2. Auxiliary Results

We shall use the following lemmas to prove our main theorem.
Lemma 1 [4]. Let the m - th order central moment be of the operators (1) be defined by

∑∞ ∫ ∞ Tn,m,α(x) = pn,v,α(x ) bn,v- 1,α(t)(t - x)mdt + (1 + αx )- n∕α (- x)m. v=1 0

Then, we have

αx Tn,0,α(x) = 1,Tn,1,α = ------, n - α

and

x (1 + αx )(2n - α) + αx (1 + 3αx) Tn,2,α(x) = --------------------------------- (n - α)(n - 2α )

and for n > α(m + 1), we have

pict

It is easily checked that for all x ∈ [0,∞ ), one has

Tn,m,α(x) = O (n-[(m+1 )∕2]).

Remark 1. In particular given any number λ > 2 and x > 0 , by Lemma 1, we have for n sufficiently large one has

Vn,α((t - x)2,x) ≡ Tn,2,α(x) ≤ λx-(1 +-αx). n

(2)

Remark 2. In view of Remark 1, it can be observed by Holder's inequality that

1∕2 ∘ -------------- Vn,α(| t - x |,x) ≤ [Tn,2,α(x )] ≤ λx (1 + αx )∕n.

(3)

Lemma 2. Let x > 0,λ > 2 and the kernel Wn,α(x, t) is defined by (1), then for sufficiently large, we have

∫ y (i) ηn,α(x,y) = Wn, α(x,t)dt ≤ λx(1-+-αx-), 0 ≤ y < x 0 n(x - y)2

∫ ∞ (ii) 1 - η (x, z) = W (x,t)dt ≤ λx(1-+-αx-), x < z < ∞ n,α z n,α n(z - x)2

Proof. First we prove (i), by (2), we have

∫ y ∫ y 2 W (x,t)dt ≤ (x---t)-W (x, t)dt 0 n,α 0 (x - y)2 n,α

-2 λx (1 + αx ) ≤ (x - y) Tn,2,α(x) ≤ ---------2-. x(x - y)

The proof of (ii) is similar, we omit the details.

3. Main Result

In this section, we prove the following main theorem.

Theorem 1. Let f ∈ DB γ(0,∞ ) , γ > 0 and x ∈ (0,∞ ) . Then for λ > 2 and for sufficiently large, we have

pict

where ∨ baf (x) denotes the total variation of on [a, b], and is defined by

( ||| f (t) - f(x - ), 0 ≤ t < x; |{ fx(t) = 0, t = x; ||| |( + f (t) - f(x ), x < t < ∞.

Proof. Using the fact that ∫ 0∞ Wn,α(x,t)dt = 1, we can write

∫ ∞ Vn,α(f, x) - f (x) = Wn, α(x,t)(f(t) - f (x))dt 0

∫ ∞ ∫ t = ( Wn,α(x, t)(f ′(u)du )dt). 0 x

Also, we can write

′ + ′ - ′ + ′ - f ′(u) = [f-(x-)-+-f-(x-)] + (f′)x(u ) + [f-(x--) --f-(x-)]sgn (u - x) 2 2

′ + ′ - +[f′(x) - [f(x--) +-f(x--)]]χx (u). 2

Next, we have ∫ ∫ ′ + ′ - ∞ ( tf ′(x) - [f(x-)+f-(x-)]χx (u )du))Wn,α(x, t)dt = 0, 0 x 2 thus

∫ ∫ ∞ t [f′(x+ ) + f′(x - )] ′ Vn,α(f,x )- f (x) = ( Wn (x,t)(--------2--------+ (f)x(u))du )dt 0 x

∫ ∞ ∫ t [f ′(x+ ) - f ′(x- )] ( Wn (x,t)-----------------sgn(u - x)du )dt. 0 x 2

Also

∫ ∞ ∫ t ′ + ′ - ′ + ′ - ( [f-(x--) --f-(x-)]sgn (u - x)du )Wn, α(x,t)dt = [f-(x--) --f-(x-)]Vn,α(| t- x |,x) 0 x 2 2

and

∫ ∞ ∫ t 1- ′ + ′ - 1- ′ + ′ - 0 ( x 2 [f (x ) + f (x )]du )Wn, α(x,t)dt = 2 [f (x )+f (x )]Vn,α((t- x),x ).

Thus we can write
| Vn,α(f,x) - f(x) |

∫ ∞ ∫ t ∫ x ∫ t ≤| ( (f ′)x(u)du )Wn,α(x,t)dt - ( (f′)x(u )du)Wn,α (x, t)dt | x x 0 x

1- ′ + ′ - + 2 | f (x ) - f (x ) | Vn,α(| t - x |,x )

+ 1-| f ′(x+ ) + f ′(x- ) | V ((t - x),x) 2 n,α

= | A (f, x) + B (f,x) | + 1-| f ′(x+ ) - f ′(x- ) | V (| t - x |,x) n,α n,α 2 n,α

1 ′ + ′ - + --| f (x ) + f (x ) | Vn,α((t - x),x). 2

(4)

To complete the proof of the theorem it is sufficient to estimate the terms An (f,x) and Bn (f, x) . Applying integration by parts, using Lemma 2 and taking √ -- y = x - x∕ n ,we have

∫ x∫ t | Bn, α(f,x) |= | ( (f ′)x(u)du)dtηn,α(x,t)dt | 0 x

∫ x ∫ y ∫ x ′ ′ 0 ηn,α(x,t)(f )x(t)dt ≤ ( 0 + y )| (f )x(t) || ηn,α(x,t) | dt

∫ ∫ λx (1 + αx) y∨x ′ 1 x∨x ′ ≤ ----------- ((f )x)-------2dt + ((f )x)dt n 0 t (x - t) y t

λx (1 + αx) ∫ y∨x 1 x ∨x ≤ ----------- ((f ′)x)--------dt + √--- ((f ′)x). n 0 t (x - t)2 n x-√x- n

Let u = -x- x-t . Then we have

∫ x ∫ √- x λx(1-+-αx-) y∨ ′ ----1--- λx-(1-+-αx) n ∨ ′ n ((f )x)(x - t2)dt = n ((f )x)du 0 t 1 x- xu

√- [∑n] x∨ ≤ λ-(1-+-αx-) ((f ′)x). n v=1x- x u

Thus

[√n-] x x λ(1-+-αx-)∑ ∨ ′ -x-- ∨ ′ | Bn,α(f,x ) | ≤ n ((f )x) + √n- ((f )x). v=1 x- xu x-√xn-

(5)

On the other hand, we have

pict

Next applying Holder's inequality, and Lemma 1, we proceed as follows for the estimation of the first two terms in the right hand side of (6):

C ∫ ∞ | f(x) |∫ ∞ -- Wn,α(x,t)tγ| t - x |dt +----2-- Wn, α(x,t)(t - x)2dt x 2x x 2x

∫ ∞ ∫ ∞ 12 ≤ C-( Wn,α(x, t)t2γdt)12( Wn, α(x,t)(t - x )2dt) x 2x 0

| f (x) | ∫ ∞ + ----2--( Wn, α(x,t)(t - x )2dt) x 2x

∘ ----------- λx (1 + αx) λ (1 + αx ) ≤ C2 γO (n-γ∕2)-----√--------+ | f (x) |--------. n nx

(7)

Also the third term of the right side of (6) is estimated as

∫ ∞ ∫ ∞ | f′(x+) | W (x, t)| t - x |dt ≤ | f′(x+) | W (x, t)| t - x |dt 2x n,α 0 n,α

∫ 1 ∫ 1 ′ + ∞ 2 2 ∞ 2 ≤ | f (x ) |( Wn,α(x, t)(t - x) dt) ( Wn, α(x,t)dt) 0 0

∘ ----------- = | f′(x+) |-λx-(√1-+-αx-). n

Collecting the estimates (4)-(7), we get the required result.
This completes the proof of Theorem 1.

4. Bezier Variant

For β ≥ 1 or 0 < β < 1 , the Bezier variant of the operators (1) can be defined as

∑∞ ∫ ∞ V (f,x) = Q (β) (x ) b (t)f(t)dt + Q (β) (x)f(0) (8 ) n,α,β n,v,α 0 n,v- 1,α n,0,α v=1

∫ ∞ = Wn, α,β(x,t)f(t)dt 0

where Q(nβ,)v,α(x) = (Jn,v,α(x))β - (Jn,v+1,α(x))β,Jn,v,α(x) = ∑ ∞ pn,v,α(x) j=v and the kernel is given by

∑∞ Wn, α,β(x, t) = Q (nβ,v),α(x)bn,v-1,α(t) + Q (βn),0,α (x )δ(t), v=1

δ(t) being Dirac delta function. In case β = 1 the operators defined by (8) reduce to the operators (1).
We define

∫ y ηn,α,β(x,y ) = Wn, α,β(x,t)dt 0

Remark 3. By Lemma 2 for β ≥ 1, we can write

∫ y β-λx(1-+-αx-) (i) ηn,α,β(x, y) = Wn, α,β(x,t)dt ≤ n(x - y)2 , 0 ≤ y < x 0

∫ ∞ β λx(1 + αx ) (ii) 1 - ηn,α,β (x, z) = Wn, α,β(x,t)dt ≤ ----------2--, x < z < ∞. z n(z - x)

Remark 4. In view of Lemma 1 and Remark 1, for β ≥ 1, it can be observed that

αβx Vn,α,β((t - x ),x ) ≤------ n - α

∘ ------------- βλx (1 + αx) Vn,α,β (| t - x |,x) ≤ [βTn,2,α(x)]1∕2 ≤ ------------. n

Our main result is stated as follows:

Theorem 2.Let f ∈ DB γ(0,∞ ),γ > 0 , also suppose α > 0,β ≥ 1, and x ∈ (0,∞ ) . Then for λ > 2 and sufficiently large, we have

pict

where ∨ baf (x) denotes the total variation of on [a,b] and the auxiliary function is as given in Theorem 1.

Proof. Following the methods presented in [7], we can write

| Vn,α,β(f,x) - f(x) |

∫ ∞ ∫ t ∫ x ∫ t ≤ | ( (f′)(u )du )W (x,t)dt - ( (f ′) (u)du )W (x, t)dt | x x x n,α,β 0 x x n,α,β

+ --β---| f ′(x+ ) - f ′(x- ) | Vn,α,β(| t - x |,x) β + 1

+---1--[f′(x+ ) + βf′(x- )]V ((t - x),x) β + 1 n,α,β

=| An,α,β(f,x ) + Bn, α,β(f,x) |

+ --β--|f′(x+) - f′(x- )|Vn,α,β(| t - x |,x ) β + 1

+ --1---[f ′(x+ ) + βf ′(x - )]Vn,α,β((t - x),x ). β + 1

Using Remark 4, we have

| Vn,α,β(f,x ) - f (x ) |≤ | An,α,β (f, x) + Bn,α,β(f,x ) |

∘ ------------- --β--- ′ + ′ - β-λx(1-+-αx-) --1--- ′ + ′ - -α-βx--- + β + 1|f (x ) - f (x )| n + β + 1[f (x ) + βf (x )](n - α). (9)

In order to complete the proof of the theorem, it is sufficient to estimate the terms An, α,β(f,x ) and Bn,α,β(f, x) given in (9) above. Using Remark 3 and proceeding along the lines of proof of Theorem 1, we get the desired estimate. Here we omit the details.

Acknowledgment

We are thankful to the referee for his critical review leading to overall improvement of the paper.

References

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[3] R. K. Gangwar and V. K. Jain, Rate of approximation for certain generalized operators, General Math., to appear. [ Links ]

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S. Jain
Guru Nanak Institute of Management
Road No. 75, Punjabi Bagh, New Delhi 110026, India
jainshipra11@rediffmail.com

Ravindra Kumar Gangwar
Department of Mathematics, Bareilly College
Bareilly 243001, India
ravindra1402@yahoo.co.in

Recibido: 6 de marzo de 2008
Aceptado: 7 de octubre de 2008