Approximation by Multivariate Baskakov Durrmeyer Operator

Volume 2011, Article ID 158219,7pages doi:10.1155/2011/158219

Research Article

L

p

Approximation by Multivariate

Baskakov-Durrmeyer Operator

Feilong Cao and Yongfeng An

Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, China

Correspondence should be addressed to Feilong Cao,[email protected]

Received 14 November 2010; Accepted 17 January 2011

Academic Editor: Jewgeni Dshalalow

Copyrightq2011 F. Cao and Y. An. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The main aim of this paper is to introduce and study multivariate Baskakov-Durrmeyer operator, which is nontensor product generalization of the one variable. As a main result, the strong direct inequality ofLp_{approximation by the operator is established by using a decomposition technique.}

1. Introduction

LetPn,kx nk−1 k

xk1x−n−k,x∈0,∞,n∈N. The Baskakov operator defined by

Bn,1

f, x

∞

k0

Pn,kxf

k n

1.1

was introduced by Baskakov1and can be used to approximate a function f defined on

0,∞. It is the prototype of the Baskakov-Kantorovich operatorsee2and the Baskakov-Durrmeyer operator defined bysee3,4

Mn,1f, x

∞

k0

Pn,kxn−1

_∞

0

Pn,ktftdt, x∈0,∞, 1.2

wheref∈Lp_{0,∞1≤p <∞.}

By now, the approximation behavior of the Baskakov-Durrmeyer operator is well understood. It is characterized by the second-order Ditzian-Totik modulussee3

ω_ϕ2f, t_p sup

0<h≤t _f

More precisely, for any function defined onLp_{0,∞1≤p <∞, there is a constant such that}

_Mn,1f−f

p≤const.

ω_ϕ2

f,√1 n

p 1

nfp

, 1.4

ω2 ϕ

f, t_pOt2α_⇐⇒M

n,1f−f_pO

n−α_{, 1.5}

where 0< α <1.

LetT ⊂Rd _{d∈N, which is defined by}

T :Td:{x: x1, x2, . . . , xd: 0≤xi<∞,1≤i≤d}. 1.6

Here and in the following, we will use the standard notations

x: x1, x2, . . . , xd, k: k1, k2, . . . , kd∈Nd₀,

xk _:xk1

1 x k2

2 · · ·x kd

d , k!k1!k2!· · ·kd!, |x|: d

i1

xi, |k|: d

i1 ki,

n

k

: n! k!n− |k|!,

∞

k0

: ∞

k10

∞

k20 · · ·∞

kd0 .

1.7

By means of the notations, for a functionf defined onTthe multivariate Baskakov operator is defined assee5

Bn,df,x: ∞

k0 f

k

n

Pn,kx, 1.8

where

Pn,kx

n|k| −1 k

xk1|x|−n−|k|. 1.9

Naturally, we can modify the multivariate Baskakov operator as multivariate Baskakov-Durrmeyer operator

Mn,df :Mn,d

f,x: ∞

k0

Pn,kxφn,k,d

f, f ∈LpT, 1.10

where

φn,k,d

f:

TPn, kufudu TPn,kudu

n−1n−2· · ·n−d

T

It is a multivariate generalization of the univariate Baskakov-Durrmeyer operators given in

1.2and can be considered as a tool to approximate the function inLp_T.

2. Main Result

We will show a direct inequality ofLp_{approximation by the Baskakov-Durrmeyer operator}

given in1.10. By means of K-functional and modulus of smoothness defined in5, we will extend1.4to the case of higher dimension by using a decomposition technique.

Foxx∈T, we define the weight functions

ϕix xi1|x|, 1≤i≤d. 2.1

Let

D_ir ∂ r

∂xr i

, r ∈N, DkDk1

1 D k2

2 · · ·D kd

d , k∈N d

0 2.2

denote the differential operators. For 1 ≤ p < ∞, we define the weighted Sobolev space as follows:

Wϕr,pT

f ∈LpT:Dkf∈Lloc _˙

T, ϕr_iDr_if∈LpT, 2.3

where|k| ≤ r,k ∈ Nd₀, and ˙T denotes the interior ofT. The Peetre K-functional on Lp_T 1≤p <∞, are defined by

K_ϕrf, tr_pinf

f−g_ptr d

i1

ϕr_iDr_ig_p

, t >0, 2.4

where the infimum is taken over allg ∈Wϕr,pT.

For any vector e in Rd_{, we write therth forward difference of a function f in the}

direction ofeas

Δr hefx

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r

i0 ⎛ ⎝r

i ⎞

⎠₋1ifxihe, x,xrhe∈T,

0, otherwise.

2.5

We then can define the modulus of smoothness off∈Lp_{T1≤p <∞, as}

ω_ϕrf, t_p sup

0<h≤t d

i1 _Δr

hϕieif_p, 2.6

whereeidenotes the unit vector inRd, that is, itsith component is 1 and the others are 0.

Lemma 2.1. There exists a positive constant, dependent only onpandr, such that for anyf ∈Lp_T,

1≤p <∞

1 const.ω

r ϕ

f, t_p≤Kr_ϕf, tr_p≤const. ωr_ϕf, t_p. 2.7

Now we state the main result of this paper.

Theorem 2.2. Iff ∈Lp_{T,1≤p <∞, then there is a positive constant independent ofnandfsuch}

that

Mn,df−f_p≤const.

ω2 ϕ

f,√1

n

p 1

nfp

. 2.8

Proof. Our proof is based on an induction argument for the dimensiond. We will also use a decomposition method of the operatorMn,df. We report the detailed proof only for two

dimensions. The higher dimensional cases are similar.

Our proof depends onLemma 2.1and the following estimates:

_Mn,2f−f

p≤const. ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

f_p, f∈Lp_T,

1

n ₂

i1 _ϕ2

iDi2fpfp

, f ∈Wϕ2,pT.

2.9

The first estimate is evident as the Mn,df are positive and linear contractions on

Lp_{T1 ≤ p < ∞. We can demonstrate the second estimate by reducing it to the one}

dimensional inequality

Mn,1f−f_p≤

const. n

ϕ2f

pfp

, 2.10

which has been proved in3

Now we give the following decomposition formula:

Mn,2

f,x ∞

k10

∞

k20

Pn,k1x1Pnk1,k2

x2

1x1

n−1n−2

×

_∞

0

Pn,k1u1Pnk1,k2

u2

1u1

fu1, u2du1du2

∞

k10

Pn,k1x1n−2

_∞

0

Pn−1,k1u1

∞

k20

Pnk1,k2

x2

1x1

×nk1−1

_∞

0

Pnk1,k2tfu1,1u1tdt du1

∞

k10

Pn,k1x1n−2

_∞

0

Pn−1,k1u1Mnk1,1

gu1, z

du1,

where

gu1t fu1,1u1t, 0≤t <∞, z

x2

1x1

, 2.12

which can be checked directly and will play an important role in the following proof. From the decomposition formula, it follows that

Mn,2

f,x−fx ∞

k10

Pn,k1x1n−2

×

∞

0

Pn−1,k1u1

Mnk1,1

gu1, z

−gu1z

du1

M_n,1∗ h·, x1−hx1

:JL,

2.13

where

hu1:hu1,x:f

u1,1u1 x2

1x1

, 0≤u1<∞,

M∗_n,1g, y

∞

l0 Pn,l

yn−2

_∞

0

Pn−1,ltgtdt.

2.14

Then by the Jensen’s inequality, we have

Jpp≤

T

∞

k10

Pn,k1x1

n−2

_∞

0

Pn−1,k1u1Mnk1,1gu1, z−gu1zdu1

pdx ≤ T ∞

k10

Pn,k1x1n−2

_∞

0

Pn−1,k1u1Mnk1,1

gu1, z

−gu1z

p du1dx

_∞

0

∞

k10

Pn,k1x11x1dx1n−2

_∞

0

Pn−1,k1u1

×Mnk1,1

gu1, z

−gu1z

p dzdu1

≤ ∞

k10

nk1−1

n−1

_∞

0

Pn−1,k1u1

_∞

0

_Mnk

1,1

gu1, z

−gu1z

p dzdu1

≤const.

∞

k10

nk1−1 n−1

_∞

0

Pn−1,k1u1

1

nk1 p

ϕ2g_{u 1}p

pgu1 p p

du1.

2.15

However, by definition, one also has

ϕ2_tg

u1t t1t1u1

2_D2

2fu1,1u1t

ϕ2 2D22f

Therefore,

Jpp≤const.

∞

k10

nk1−1

n−1nk1p _∞

0

Pn−1,k1u1

×ϕ2₂D2₂fu1,1u1t p

fu1,1u1tp

dt du1

const.

∞

k10

nk1−1 n−1nk1p

_∞

0

1 1u1

Pn−1,k1u1

×

_∞

0 _ϕ2

2u1, u2D22fu1, u2 p

fu1, u2p

du1du2

≤ const.

np

∞

k10

_∞

0

Pn,k1u1

_∞

0

ϕ2₂u1, u2D₂2fu1, u2pfu1, u2pdu1du2

const. np

ϕ2₂D₂2fp pf

p p

.

2.17

To estimate the second termL, we use a similar method as to estimate2.10 see3

and can get

L_p≤ const. n

ϕ2_h

php

. 2.18

Denotingϕ12x ϕ21x : √x1x2,D2

12 : ∂2/∂x1∂x2, andD212 : ∂2/∂x2∂x1, we

have

ϕ2sh s

s1s

D2₁f x2

1x1

D2₁₂f x2

1x1

D2₂₁f x 2 2 1x12D

2 22f

×

s,1s x2

1x1

1x1

1x1x2ϕ 2

1D21fϕ212D122 fϕ221D221f s

1s x2

1x1x2ϕ 2 2D22f

s,1s x2

1x1

. 2.19

Recalling thatϕ12xis no bigger thanϕ1xorϕ2x, and the fact

D2

12fx≤supD21fx,D22fx

2.20

proved in6 see6, Lemma 2.1, we obtain

ϕ2_h

p≤const. 2

i1 ϕ2

and hence

L_p≤ const. n

₂

i1

ϕ2_iD2_if

pfp

. 2.22

The second inequality of 2.9has thus been established, and the proof of Theorem 2.2is finished.

Acknowledgment

The research was supported by the National Natural Science Foundation of China no. 90818020.

References

1 V. A. Baskakov, “An instance of a sequence of linear positive operators in the space of continuous functions,”Doklady Akademii Nauk SSSR, vol. 113, pp. 249–251, 1957.

2 Z. Ditzian and V. Totik,Moduli of Smoothness, vol. 9 of Springer Series in Computational Mathematics, Springer, New York, NY, USA, 1987.

3 M. Heilmann, “Direct and converse results for operators of Baskakov-Durrmeyer type,”Approximation Theory and its Applications, vol. 5, no. 1, pp. 105–127, 1989.

4 A. Sahai and G. Prasad, “On simultaneous approximation by modified Lupas operators,”Journal of Approximation Theory, vol. 45, no. 2, pp. 122–128, 1985.

5 F. Cao, C. Ding, and Z. Xu, “On multivariate Baskakov operator,”Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 274–291, 2005.