Pointwise approximation for a type of Bernstein Durrmeyer operators

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R E S E A R C H

Open Access

Pointwise approximation for a type of

Bernstein-Durrmeyer operators

Guofen Liu

*

*_{Correspondence:}

[email protected] College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang, 050024, People’s Republic of China

Abstract

We give the direct and inverse approximation theorems for a new type of Bernstein-Durrmeyer operators with the modulus of smoothness. MSC: 41A25; 41A27; 41A36

Keywords: Bernstein-Durrmeyer type operator; modulus of smoothness;

K-functional; direct and inverse approximation theorems

1 Introduction

Durrmeyer [] introduced the integral modiﬁcation of the well-known Bernstein polyno-mials given by

Dn(f,x) = (n+ )

n

k=

pn,k(x)

 

pn,k(t)f(t)dt,

wherepn,k(x) =

_n

k

xk( –x)n–k. Derriennic [] established some direct results in ordinary and simultaneous approximation for Durrmeyer operators. Then Durrmeyer type opera-tors were studied widely [–]. Recently Guptaet al.[] considered a family of Durrmeyer type operators:

Pn,m(f,x) =

nn_k_=pn,k(x)



pn–,k–(t)f(t)dt+pn,(x)f(), m= ; nm

(n+m–)m–

n–m

k=pn–m,k(x)



pn+m–,k+m–(t)f(t)dt, m> ,

wherem,n∈Nwithm≤nand for anya,b∈N,ab=a(a– )· · ·(a–b+ ),a=  is the falling factorial; and we get the rate of convergence for these operators for a function having derivatives of bounded variation and the result in the simultaneous approximation. In the present note our main aim is to get the direct and inverse approximation theorem for this type of operators. Here we shall utilize modulus of smoothness andK-functional as the tools, which are deﬁned by []

ω_ϕrλ(f,t) = sup

<h≤t sup x±hrϕλ₍_x₎_∈_[,]

r

hϕλ₍_x₎f(x) , Kϕλ

f,tr= inf g(r–)_∈_A_._C_.

loc

f–g+trϕrλg(r),

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whereϕ(x) =√x( –x), ≤λ≤,r∈N. It is well known thatωr

ϕλ(f,t)∼Kϕλ(f,tr), where a∼bmeans that there exists some constantC>  such thatC–b≤a≤Cb. We denote

Mn,m(f,x) =(m+n) m

nm Pn,m(f,x) and state our main results as follows.

Theorem  For f ∈C[, ],  <λ< ,ϕ(x) =√x( –x),one has

Mn,m(f,x) –f(x) ≤C

ω_ϕλ

f,√

n

+

nf

.

Theorem  Let f ∈C[, ], ≤λ≤,ϕ(x) =√x( –x),δn(x) =max{ϕ(x),√_n},r∈N,  < α< r,for m> ,and from

Mn,m(f,x) –f(x) =O

n_δ–λ

n (x)

α

,

we getω_ϕrλ(f,t) =O(tα).

Throughout this paper · = · ∞andCdenotes a positive constant independent of

nandxnot necessarily the same at each occurrence.

2 Lemmas

To prove the above theorems we need the following lemmas. First we deﬁne the moments, for anys∈N,Tn,m,s(x) =Mn,m((t–x)s,x).

Lemma ([]) The following claims hold.

() For anys,m∈N,x∈[, ],the following recurrence relation is satisﬁed:

(n+m+s+ )Tn,m,s+(x) =x( –x)

T_n_,_m_,_s(x) + sTn,m,s–(x)

+(s+m) –x( + s+ m)Tn,m,s(x),

where fors= ,we denoteTn,m,–(x) = .

() For anym∈Nandx∈[, ],

Tn,m,(x) = , Tn,m,(x) =

m–x( + m)

n+m+  ,

Tn,m,(x) =

nx( –x) +m( +m) – mx(m+ ) + x_(_m_{+ }_m_{+ )} (n+m+ )(n+m+ ) .

() For anys,m∈N,x∈[, ],Tn,m,s(x) =O(n–[(s+)/]).

Remark Fornsuﬃciently large andx∈(, ), it can be seen from Lemma  that

x( –x)

n ≤Tn,m,(x)≤

Cx( –x)

n , (.)

for anyC> .

Lemma  For f(x)∈C[, ],ϕ(x) =√x( –x),δn(x) =max{ϕ(x),√_n}, ≤λ≤,r∈N, m> ,we have

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Proof To complete the proof we consider two cases ofx∈En= [_n,  –_n] andx∈Ec n=

[,_n]∪[ –_n, ]. Forx∈Ec

n,ϕ(x)≤Cn,δn(x)∼n. Using

M(r)

n,m(f,x) =

(n–m)! (n–m– r)!

n–m–r k=

pn–m–r,k(x)rak(n),

where ak(n) = (n + m)_pn+m–,k+m–(t)f(t)dt, ak(n) = ak+(n) – ak(n), rak(n) = (r–_ak₍_n_{)); and}_|r_ak₍_n₎_{| ≤}_Cf_, (n–m)!

(n–m–r)!≤(n–m)r<nr, one has

ϕrλ(x)M(_n_,_mr)(f,x) ≤Cn–rλnrf ≤Cnrδ_nr(λ–)f. (.)

Forx∈En,δn(x)∼ϕ(x). From [] we have

M(_n_,_mr)(f,x) =ϕ–r(x) r

i=

QB_i(x,n)ni n–m

k=

pn–m,k(x)

k n–m–x

i ak(n),

whereQB_i(x,n) a polynomial innϕ(x) of degree [(r–i)/] with non-constant bounded coeﬃcients. Therefore,

ϕ–r(x)QB_i(x,n)ni ≤C

n ϕ₍_x₎

r+_i .

By Holder’s inequality we get

n–m

k=

pn–m,k(x)

k n–m–x

i ≤

_n_–_m

k=

pn–m,k(x)

k n–m–x

i

≤Cn–i_ϕi₍_x_).

Consequently|ϕr(x)Mn(,mr)(f,x)| ≤Cnrf, hence

ϕrλ(x)M(_n_,_mr)(f,x) =ϕr(λ–)(x) ϕr(x)M(_n_,_mr)(f,x) ≤Cnrδ_nr(λ–)f. (.)

From (.) and (.), (.) holds.

Lemma  For f(r–)₍_x₎_∈_A_._C_.loc,_ϕrλ_f(r)_<_∞_,_m_{> ,}_{we have}

ϕrλ(x)M(_n_,_mr)(f,x) ≤Cϕrλf(r). (.)

Proof Fromp(_n_–r_m)_,_k(x) =₍_n(_–n_m–m_–)!_r_)!_i_=r (–)ir i

pn–m–r,k–(r–i)(x), we have

M(_n_,_mr)(f,x)

= (n–m)

n–m

k=

p(_n_–r_m)_,_k(x)

 

pn+m–,k+m–(t)f(t)dt

=(n–m)(n–m)! (n–m– r)!

n–m–r k=

r

i= (–)i

r i

pn–m–r,k–(r–i)(x)

 

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=(n–m)(n–m)! (n–m– r)!

n–m–r k=

pn–m–r,k(x)

 

r

i= (–)i

r i

pn+m–,k+m+r–i–(t)f(t)dt

=(n–m)(n–m)! (n–m– r)!

(n+m– )! (n+m–  + r)!

× n–m–r

k=

pn–m–r,k(x)

 

pn+m+r–,k+m+r–(t)f(r)(t)dt.

LetI= (n–m)_kn–_=m–rpn–m–r,k(x)|



pn+m+r–,k+m+r–(t)f

(r)₍_t₎_dt|_{. For }_≤_λ_≤_{ one}

has

ϕrλ(x)I

≤ϕrλf(r)ϕrλ(x)

n–m–r k=

pn–m–r,k(x)(n–m)

 

pn+m+r–,k+m+r–(t)ϕ–rλ(t)dt

≤ϕrλf(r)

_n_–_m_–_r

k=

pn–m–r,k(x)ϕr(x)(n–m)

 

pn+m+r–,k+m+r–(t)ϕ–r(t)dt

λ

.

Noting that

pn–m–r,k(x)ϕr(x) =

(n–m– r)!(k+r)!(n–m–k–r)!

k!(n–m– r–k)!(n–m)! pn–m,k+r(x) =:αn,kpn–m,k+r(x),

pn+m+r–,k+m+r–(t)ϕ–r(t)

=(n+m+ r– )!(k+m+r– )!(n–k–r)!

(k+m+ r– )!(n–k)!(n+m– )! pn+m–,k+m+r–(t)

=:βn,kpn+m–,k+m+r–(t)

andαn,kβn,k≤C, we getϕrλ(x)I≤ ϕrλf(r). This completes the proof of Lemma .

Lemma ([]) For <t<__r, rt_ <x<  –rt_, ≤β≤r,we have

· · ·

t



–t_

ϕ–β(x+u+· · ·+ur)du· · ·dur≤Ctrϕ–β(x).

3 Proof of the theorems

In this section we will give the proof of Theorem  and Theorem .

Proof of Theorem By the deﬁnition ofKϕλ(f,tr) and the equivalence betweenωr ϕλ(f,t)

andKϕλ(f,tr), for the ﬁxednandx, we can chooseg=gn_,_xsuch that

f–g+

nϕ

λ_g _≤_Cω

ϕλ

f,√

n

. (.)

We know that

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and we have to estimate the second term on the right side of (.). By Taylor’s formula,

g(t) =g(x) +g(x)(t–x) +_xt(t–u)g (u)du, and Lemma  we have

Mn,m(g,x) –g(x) ≤ g(x) Mn,m

(t–x),x + Mn,m

t x

(t–u)g (u)du,x

≤C

n g(x) +

Mn,m

t x

(t–u)g (u)du,x

=:I+I. (.)

We considerIﬁrst. For ≤x≤_,

g(x) –g

  =   x

g (u)du ≤xλ_g

 

x



uλdu≤Cϕ

λ_g _,

together with|g(_)| ≤C(g _L_∞_[

,]+gL∞[,])≤C(ϕ

λ_g ₊_g_{), one has}

g(x) ≤Cg+ϕλg . (.)

It is similar for 

<x≤.

Now we addressI. By the process of (..) in []

Rr(f,u,x) ≤

|u–x|r–

ϕr₍_x₎

_uxϕr(v) f(r)(v) dv ,

we get _ϕ|t–λu₍_u|₎≤

|t–x|

ϕλ₍_x₎, and combining with (.) we deduce

I≤

ϕλ_g ϕλ₍_x₎ Mn,m

(t–x),x≤Cϕ

λ_g

ϕλ₍_x₎ ϕ(x)

n ≤C



nϕ

λ

g . (.)

By (.)-(.), we complete the proof of Theorem .

Proof of Theorem  For convenience let γn,λ(x) = n–

 _δ–λ

n (x). If Mn,m(f,x) – f(x) = O(γα

n,λ(x)), for everyn:n> r, we have

r

tϕλ₍_x₎f(x) ≤ _tϕrλ₍_x₎

Mn,m(f,x) –f(x) + _tϕrλ₍_x₎Mn,m(f,x)

≤Cγα

n,λ(x) +

· · ·

tϕλ(x) 

–tϕλ(x)

M(r)

n,m

f,x+ r

j=

uj

du· · ·dur

≤Cγα

n,λ(x) +

· · ·

tϕλ(x) 

–tϕλ(x)

M(_n_,_mr)

f –g,x+ r

j=

uj

du· · ·dur

+

· · ·

tϕλ(x) 

–tϕλ_(x)

M(_n_,_mr)

g,x+ r

j=

uj

du· · ·dur

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Combining Lemma , Lemma , and Lemma , we have

J≤Ctrγn–,λr(x)f –g, (.)

J≤Ctrϕrλg(r). (.)

Utilizing (.), (.), and (.), choosing the appropriateg, we obtain

r

tϕλ₍_x₎f(x) ≤C

γ_nα_,λ(x) +trγn–,λr(x)ωϕrλ

f,γn,λ(x)

.

For every ﬁxedh:  <h< 

rand everyx:x≥rt, we can choosensuch thatγn,λ(x)≤h<

γn,λ(x). Then

r

tϕλ₍_x₎f(x) ≤C

hα+

t h

r

ω_ϕrλ(f,h)

.

So,

ω_ϕrλ(f,t)≤C

hα₊

t h

r

ω_ϕrλ(f,h)

,

which yields the assertion of Theorem  by the Berens-Lorentz lemma.

Competing interests

The author declares that they have no competing interests.

Received: 13 June 2013 Accepted: 20 February 2014 Published:04 Mar 2014 References

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10.1186/1029-242X-2014-106