Certain family of Durrmeyer type operators

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXIII, 2009 SECTIO A 109–115

VIJAY GUPTA

Certain family of Durrmeyer type operators

Abstract. The present paper is a continuation of the earlier work of the author. Here we study the rate of convergence of certain Durrmeyer type operators for function having derivatives of bounded variation.

1. Introduction. The integral modification of the well-known Bernstein polynomials was introduced by Durrmeyer [2]. Derriennic [1] established some direct results in ordinary and simultaneous approximation for the Durrmeyer operators. The rate of convergence of the Durrmeyer operators was studied by Zeng and Chen [6]. Very recently Gupta et al. [3] considered a family of Durrmeyer type operators P_n,r(f, x) defined in the following way.

Let p_n,k(x) = ⁿ_k
x^k(1 − x)^n−k. Then

Pn,r(f, x) =













 n

n

X

k=1

p_n,k(x) Z ₁

0

p_n−1,k−1(t)f (t)dt + p_n,0(x)f (0), r = 0, n^r

(n + r − 1)^r−1

n−r

X

k=0

p_n−r,k(x) Z ₁

0

pn+r−1,k+r−1(t)f (t)dt, r > 0, where r, n ∈ N0with r ≤ n and for any a, b ∈ N0, a^b = a(a−1) · · · (a−b+1), a⁰ = 1 is the falling factorial. One can notice that the family P_n,r(f, x), as a particular case (r = 0), contains the sequence P_n,0(f, x) introduced by Srivastava and Gupta in [5]. The rate of convergence of functions of

2000 Mathematics Subject Classification. 41A25, 41A35.

Key words and phrases. Bernstein polynomials, Durrmeyer operators, bounded varia- tion, total variation.

bounded variation of the B´ezier variant of the special case of these operators was obtained by Gupta and Maheshwari in [4].

The operators P_n,r(f, x) also admit the integral representation P_n,r(f, x) =

Z 1 0

K_r,n(x, t)f (t)dt, where the kernel K_n,r(x, t) is given by

Kn,r(x, t) =













 n

n

X

k=1

p_n,k(x)p_n−1,k−1(t) + p_n,0(x)δ(t), r = 0, n^r

(n + r − 1)^r−1

n−r

X

k=0

p_n−r,k(x)pn+r−1,k+r−1(t), r > 0.

Here, δ(t) denotes the Dirac delta function.

The aim of the present paper is to extend the study on the operators P_n,r(f, x). Here we study the rate of convergence for these operators for functions having derivatives of bounded variation. We also present a corol- lary in the end, which gives the result in simultaneous approximation.

2. Auxiliary results. In the sequel we shall need several lemmas. First we will study the moments of the operators P_n,r(f, x). For this purpose we define for any m ∈ N0

Tn,r,m(x) = (n + r)^r

n^r Pn,r((t − x)^m, x)

=













 n

n

X

k=1

pn,k(x) Z 1

0

pn−1,k−1(t)(t − x)^mdt + (−x)^mpn,0(x), r = 0,

(n + r)

n−r

X

k=0

pn−r,k(x) Z 1

0

pn+r−1,k+r−1(t)(t − x)^mdt, r > 0.

Lemma 1 ([3]). The following claims hold:

(i) For any r, m ∈ N0 and x ∈ [0, 1], the recurrence relation is satisfied (n + m + r + 1)T_n,r,m+1(x) = x(1 − x)[DT_n,r,m(x) + 2mT_r,n,m−1(x)]

+ [(m + r) − x(1 + 2m + 2r)]Tn,r,m(x), where, for m = 0, we denote T_n,r,−1(x) = 0.

(ii) For all r ∈ N0 and x ∈ [0, 1],

Tn,r,0(x) = 1, Tn,r,1(x) = r − x(1 + 2r) n + r + 1 ,

T_n,r,2(x) = 2nx(1 − x) + r(1 + r) − 2rx(2r + 3) + 2x²(2r²+ 4r + 1)

(n + r + 1)(n + r + 2) .

(iii) For all r, m ∈ N0 and x ∈ [0, 1],

T_n,r,m(x) = O n^−[(m+1)/2]
.

Lemma 2 ([3]). For any r, s ∈ N0 and x ∈ [0, 1], D^sPn,r(f, x) = Pn,r+s(D^sf, x).

Remark 1. For n sufficiently large and x ∈ (0, 1), it can be seen from Lemma 1, that

x(1 − x)

n ≤ T_n,r,2(x) ≤ Cx(1 − x)

n ,

for any C > 2.

We denote

Kˆ_n,r(x, t) =













 n

n

X

k=1

p_n,k(x)p_n−1,k−1(t) + p_n,0(x)δ(t), r = 0,

(n + r)

n−r

X

k=0

p_n−r,k(x)pn+r−1,k+r−1(t), r > 0.

δ(t) denotes the Dirac delta function.

Remark 2. For n sufficiently large and x ∈ (0, 1), it can be seen from Schwarz inequality and Remark 1, that

Z 1 0

Kˆn,r(x, t)|t − x|dt ≤ [Tn,r,2(x)]^1/2 ≤

rCx(1 − x)

n ,

for any C > 2.

Lemma 3 ([3]). Let x ∈ (0, 1) and C > 2, then for n sufficiently large, we have

λ_n,r(x, y) = Z y

0

Kˆ_n,r(x, t)dt ≤ Cx(1 − x)

n(x − y)², 0 ≤ y < x, 1 − λ_n,r(x, z) =

Z 1 z

Kˆ_n,r(x, t)dt ≤ Cx(1 − x)

n(z − x)², x < z < 1.

3. Rate of convergence. By DB(0, 1), we denote the class of absolutely continuous functions f defined on (0, 1) which have a derivative f⁰ on the interval (0, 1), coinciding a.e. with a function which is of bounded variation on every finite subinterval of (0, 1).

It can be observed that all functions f ∈ DB(0, 1) possess for each c > 0 the representation

f (x) = f (c) + Z x

c

ψ(t)dt, x ≥ c.

In this section, we prove the following main theorem.

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

(n + r)^r

n^r Pn,r(f, x) − f (x) − 1

2(f⁰(x⁺) − f⁰(x⁻))

rCx(1 − x) n

−1

2(f⁰(x⁺) + f⁰(x⁻))r − x(1 + 2r) n + r + 1

≤ C

nx(1 − x)

[√ n]

X

k=1

x+(1−x)/k

_

x−x/k

((f⁰)x) + 1

√n

x+(1−x)/√ n

_

x−x/√ n

((f⁰)x)

!

whereW_b

af (x) denotes the total variation of f_xon [a, b], and f_x is defined by

fx(t) =







f (t) − f (x⁻), 0 ≤ t < x;

0, t = x;

f (t) − f (x⁺), x < t < 1.

Proof. Using the mean value theorem, we can write 

(n + r)^r

n^r P_n,r(f, x) − f (x)    

≤ Z 1

0

Kˆ_n,r(x, t)|f (t) − f (x)|dt

= Z 1

0

Z t x

Kˆ_n,r(x, t)f⁰(u)du    

dt.

Also, we use the identity

f⁰(u) = f⁰(x⁺) + f⁰(x⁻)

2 + (f⁰)x(u) + f⁰(x⁺) − f⁰(x⁻)

2 sgn(u − x) +



f⁰(x) − f⁰(x⁺) + f⁰(x⁻) 2

 χx(u),

where

χx(u) =

(1, u = x;

0, u 6= x.

Obviously, we have Z 1

0

Z t x



f⁰(x) − f⁰(x⁺) + f⁰(x⁻) 2



χ_x(u)du



Kˆ_n,r(x, t)dt = 0.

Thus, using above identities, we can write

(1)    

(n + r)^r

n^r Pn,r(f, x) − f (x)    

≤ Z 1

0

Z t x

Kˆn,r(x, t) [f⁰(x⁺) + f⁰(x⁻)]

2 + (f⁰)x(u)

 du

 dt +

Z 1 0

Z t x

Kˆn,r(x, t)[f⁰(x⁺) − f⁰(x⁻)]

2 sgn(u − x)du

 dt.

Also, it can be verified that

(2)

Z 1 0

Z t x

[f⁰(x⁺) − f⁰(x⁻)]

2 sgn(u − x)du



Kˆ_n,r(x, t)dt

≤ |f⁰(x⁺) − f⁰(x⁻)|

2 [Tn,r,2(x)]^1/2 and

(3)

Z 1 0

Z t x

1

2[f⁰(x⁺) + f⁰(x⁻)]du



Kˆn,r(x, t)dt

= 1

2[f⁰(x⁺) + f⁰(x⁻)]T_n,r,1(x).

Combining (1)–(3), we have

(4)    

(n + r)^r

n^r Pn,r(f, x) − f (x) − 1

2(f⁰(x⁺) − f⁰(x⁻))[Tn,r,2(x)]^1/2

− 1

2(f⁰(x⁺) + f⁰(x⁻))T_n,r,1(x)    

≤    

Z 1 x

Z t x

(f⁰)_x(u)du



Kˆ_n,r(x, t)dt+

Z x 0

Z t x

(f⁰)_x(u)du



Kˆ_n,r(x, t)dt    

=: |An,r(f, x) + Bn,r(f, x)|.

Making use of Remark 1 and Remark 2 in (4), we have

(5)    

(n + r)^r

n^r Pn,r(f, x) − f (x) − 1

2(f⁰(x⁺) − f⁰(x⁻))

rCx(1 − x) n

−1

2(f⁰(x⁺) + f⁰(x⁻))r − x(1 + 2r) n + r + 1

≤ |A_n,r(f, x)| + |B_n,r(f, x)|.

Since R_b

adt(λn,r(x, t)) ≤ 1 for all [a, b] ⊆ [0, 1], applying integration by parts and Lemma 3 with y = x − x/√

n, we have

|B_n,r(f, x)| =    

Z x 0

Z t x

(f⁰)_x(u)du



d_t(λ_n,r(x, t))    

=    

Z x 0

λn,r(x, t)(f⁰)x(t)dt    

≤

Z y 0

+ Z x

y



|(f⁰)x(t)||λn,r(x, t)|dt

≤ Cx(1 − x) n

Z y 0

x

_

t

((f⁰)x) 1

(x − t)²dt + Z x

y x

_

t

((f⁰)x)dt

≤ Cx(1 − x) n

Z y 0

x

_

t

((f⁰)x) 1

(x − t)²dt + x

√n

x

_

x−^√x

n

((f⁰)x).

Let u = _x−t^x . Then we have Cx(1 − x)

n

Z y 0

x

_

t

((f⁰)x) 1

(x − t)²dt = Cx(1 − x) nx²

Z

√n

1 x

_

x−^x_u

((f⁰)x)du

≤ C(1 − x) nx

[√ n]

X

k=1 x

_

x−^x_k

((f⁰)_x).

Thus

(6) | B_n,r(f, x) | ≤ C(1 − x) nx

[√ n]

X

k=1 x

_

x−^x_k

((f⁰)_x) + x

√n

x

_

x−^√x_n

((f⁰)_x).

On the other hand, we have

(7)

|A_n,r(f, x)| =    

Z 1 x

Z t x

(f⁰)_x(u)du



Kˆ_n,r(x, t)dt    

=    

Z 1 z

Z t x

(f⁰)_x(u)du



Kˆ_n,r(x, t)dt

+ Z z

x

Z t x

(f⁰)_x(u)du



d_t(1 − λ_n,r(x, t))dt    

≤ C

n(1 − x) Z

√n

1

x+(1−x)/√ n

_

x

((f⁰)_x)du +(1 − x)

√n

x+(1−x)/√ n

_

x

((f⁰)_x)

≤ C

n(1 − x)

[√ n]

X

k=1

x+(1−x)/k

_

x

((f⁰)_x) +(1 − x)

√n

x+(1−x)/√ n

_

x

((f⁰)_x).

Collecting the estimates (5)–(7), and using the fact that (1−x)²+x² ≤ 1, we get the required result. This completes the proof of Theorem 1. 

Furthermore, as a corollary of Theorem 1, if we take into account the differential formula given by Lemma 2 we can easily prove the following result for the derivatives of the operators P_n,r.

Corollary 1. Let f^(s) ∈ DB(0, 1), and x ∈ (0, 1). Then for C > 2 and n sufficiently large, we have

(n + r + s)^r+s

n^r+s D^sP_n,r(f, x) − f^(s)(x)

−1

2(D^s+1f (x⁺) − D^s+1f (x⁻))

rCx(1 − x) n

−1

2(D^s+1f (x⁺) + D^s+1f (x⁻))r + s − x(1 + 2(r + s)) n + r + s + 1

≤ C

nx(1 − x)





[√ n]

X

k=1

x+(1−x)/k

_

x−x/k

((f⁰)_x) + 1

√n

x+(1−x)/√ n

_

x−x/√ n

((f⁰)_x)



 whereWb

af (x) denotes the total variation of fxon [a, b], and f_x is defined by D^s+1fx(t) =







D^s+1f (t) − D^s+1f (x⁻), 0 ≤ t < x;

0, t = x;

D^s+1f (t) − D^s+1f (x⁺), x < t < 1.

References

[1] Derriennic, M.-M., Sur l’approximation de functions integrable sur [0, 1] par des poly- nomes de Bernstein modifies, J. Approx. Theory 31 (1981), 323–343.

[2] Durrmeyer, J. L., Une formule d’ inversion de la Transformee de Laplace, Applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de l’ Universite de Paris, 1967.

[3] Gupta, V., López-Moreno, A. J. and Latorre-Palacios, J.-M., On simultaneous approx- imation of the Bernstein Durrmeyer operators, Appl. Math. Comput. 213 (1) (2009), 112–120.

[4] Gupta, V., Maheshwari, P., B´ezier variant of a new Durrmeyer type operators, Riv.

Mat. Univ. Parma (6) 7 (2) (2003), 9–21.

[5] Srivastava, H. M., Gupta, V., A certain family of summation-integral type operators, Math. Comput. Modelling 37 (2003), 1307–1315.

[6] Zeng, X. M., Chen, W., On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation, J. Approx. Theory 102 (2000), 1–12.

Vijay Gupta

School of Applied Sciences

Netaji Subhas Institute of Technology Sector 3 Dwarka New Delhi-110078 India

e-mail: vijaygupta2001@hotmail.com Received April 9, 2009