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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005
Simultaneous approximation with linear combination of integral Baskakov type operators
Kareem J. Thamer, May A. Al-Shibeeb, A.I. Ibrahem
Abstract:
The aim of the present paper is to study some direct results in simultaneous approximation for the linear combination of integral Baskakov type operators.
1991 Mathematics Subject Classification. 41A28, 41A36.
Key words and phrases. Linear positive operators; Linear combinations; Simultaneous approximation.
1 INTRODUCTION
Agrawal and Thamer [1] introduced a new sequence of linear positive operators called integral Baskakov – type operators to approximate unbounded continuous functions on
and it is defined as follow
Let ,
.
Then,
(1.1) ,
where is the kernel of Lupas operators
. We may also write (1.1) as :
![∞∫ Mn (f (t) ;x) = Wn (t,x) f (t) dt , 0](/img/revistas/ruma/v46n1/1a019x.png)
where being the Dirac delta function.
The space is normed by
.
The operator (1.1) was used to study the degree of approximation in simultaneous approximation by Agrawal and Thamer [1]. It turned out that the order of approximation by the operator (1.1) is, at best, , howsoever smooth the function may be. Thus, if we want to have a better order of approximation, we have to slacken the positivity condition. This is achieved by considering some carefully chosen linear combination introduced by May [6] and Rathore [7] of the operator (1.1). The linear combination is defined as follows:
Let be
arbitrary but fixed distinct positive integers. Then, following Agrawal and Sinha [3] , the linear combination
of
,
is given by
![∣ ∣ ∣∣ Md0n (f;x) d-0 1 d -02 ... d-0 k ∣∣ 1 ∣∣ Md1n (f;x) d-1 1 d -12 ... d-1 k ∣∣ (1.2) Mn (f,k,x) = -- ∣ . . . . ∣, Δ ∣∣ .. .. .. ... .. ∣∣ ∣ Mdkn (f;x) d-k 1 d -k2 ... d-k k ∣](/img/revistas/ruma/v46n1/1a0119x.png)
where is the Vandermonde determinant obtained by replacing the operator column of the above determinant by the entries 1. We have
![∑k (1.3) Mn (f,k,x) = C (j,k) Mdjn (f ;x) , j=0](/img/revistas/ruma/v46n1/1a0121x.png)
where
![∏k dj (1.4) C (j,k) = ------- , k ⁄= 0 and C (0,0) = 1. i = 0 dj - di i ⁄= j](/img/revistas/ruma/v46n1/1a0122x.png)
The object of the present paper is to investigate the degree of approximation of the operator . First we establish a Voranovskaja type asymptotic formula and then obtain an error estimate in terms of the local modulus of continuity for the operator
.
2 AUXILIARY RESULTS
Throughout our work, denotes the set of natural numbers,
integers, and
an open interval containing
.
LEMMA 2.1 [4]. If for (the set of nonnegative integers), the
order moment of Lupas operators is defined by
![∑∞ ( υ )m μn,m (x) = pn,υ (x) --- x . υ=0 n](/img/revistas/ruma/v46n1/1a0131x.png)
Hence, , and there holds the recurrence relation
![n μn,m+1 (x) = x(1 + x) [μ ′ (x) + m μn,m-1 (x)], m ∈ N. n,m](/img/revistas/ruma/v46n1/1a0133x.png)
Consequently
(i) is a polynomial in
of degree at most
.
(ii) For every , where
denotes the integral part of
.
LEMMA 2.2 [1]. Let the function be defined as
![∑∞ ∫∞ T (x) = (n - 1) p (x) p (t) (t - x)m dt + (- x)m (1 + x)- n. n,m n,υ n,υ-1 υ=1 0](/img/revistas/ruma/v46n1/1a0141x.png)
Then,
![-2x--- Tn,0(x) = 1, Tn,1 (x) = n - 2](/img/revistas/ruma/v46n1/1a0142x.png)
and
![′ (n - m - 2) Tn,m+1 (x) = x (1 + x)T n,m (x) + [(2x + 1) m + 2x] Tn,m (x)+ + 2mx (1 + x) T (x) , m ∈ N. n,m -1](/img/revistas/ruma/v46n1/1a0143x.png)
Hence,
(i) is a polynomial in
of degree
(ii) For every .
(iii) The coefficients of in
and
are given by
and
.
LEMMA 2.3 [5]. There exist polynomials independent of
and
such that
![dr ∑ tr (1 + t)r --r pn,υ (t) = ni (υ - nt)j qi,j,r (t) pn,υ (t) . dt 2i + j ≤ r i,j ≥ 0](/img/revistas/ruma/v46n1/1a0156x.png)
LEMMA 2.4 [6].If are defined as in (1.4), then
![∑k { C (j,k) d-jm = 1, m = 0 . j=0 0, m = 1,...,k](/img/revistas/ruma/v46n1/1a0158x.png)
LEMMA 2.5 [8].Let be
times differentiable on
such that
for some
as
. Then for
, we have
![(r) (n-+-r---1)!-(n---r --1)! M n (f (t),x) = (n - 1)! (n - 2)! × ∞ ∑∞ ∫ × pn+r,υ (x) pn-r,υ+r- 1(t) f(r)(t) dt. υ=1 0](/img/revistas/ruma/v46n1/1a0166x.png)
LEMMA 2.6 [2]. For and
sufficiently large, there holds
![Mn ((t - x)r , k,x) = n-(k+1) {Q (r,k,x) + o(1)},](/img/revistas/ruma/v46n1/1a0169x.png)
where is a certain polynomial in
of degree
.
3 MAIN RESULTS
In this section we shall state and prove the main results.
Theorem 3.1.Let and be bounded on every finite subinterval of
admitting a derivative of order
at a fixed point
. Let
as
for some
, then we have
![2k+r+2 k+1 [ (r) (r) ] ∑ (i) (3.1) nL-i→m∞ n M n (f,k,x) - f (x) = f (x) Q (i,k,r,x) i=r](/img/revistas/ruma/v46n1/1a0180x.png)
and
![[ ](3.2) Lim nk+1 M (nr)(f,k + 1,x) - f(r)(x) = 0, n-→ ∞](/img/revistas/ruma/v46n1/1a0181x.png)
where are certain polynomials in
.
Further, the Limits (3.1) and (3.2) hold uniformly in , if
exists and is continuous on
.
Proof. By the Taylor expansion, we have
![2k+r+2 ∑ f(i)(x) i 2k+r+2 f (t) = i! (t - x) + ɛ(t,x) (t - x) , i=0](/img/revistas/ruma/v46n1/1a0187x.png)
where .
Thus, using Lemma 2.5, we have for sufficiently large
![[ ] k+1[ (r) (r) ] k+1 ∑k (r) (r) n M n (f,k, x) - f (x) = n C (j,k) M djn(f;x) - f (x) j=0](/img/revistas/ruma/v46n1/1a0190x.png)
![= I1 + I2,](/img/revistas/ruma/v46n1/1a0191x.png)
where
![[ k+1 2k+∑r+2 f(i)(x) ∑k (djn - r - 2)! (djn + r - 1)! I1 = n --i-!-- C (j,k) ----(d-n---1)!-(d-n---2)!---- i=0 j=0 j j ∞ ∫∞ r ∑ d-- i (r) × pdjn+r,υ (x) pdjn-r,υ+r -1(t) dtr (t - x) dt - f (x) υ=1 0 ] 2k+r+2 (n + 2k + r + 1) ! - n-2k-r-2 + (- 1) -----------------(1 + x) f (0) . (n - 1) !](/img/revistas/ruma/v46n1/1a0192x.png)
![[k ∞ k+1 ∑ ∑ (r) I2 = n C (j,k) (djn - 1) pdjn,υ (x) j=0 υ=1](/img/revistas/ruma/v46n1/1a0193x.png)
![∫∞ × pdjn,υ- 1(t) ɛ (t,x) (t - x)2k+r+2 dt 0](/img/revistas/ruma/v46n1/1a0194x.png)
![(n + 2k + r + 1) ! ]+ (- 1)2k+r+2------------------(1 + x)-n-2k-r-2 f (0) . (n - 1) !](/img/revistas/ruma/v46n1/1a0195x.png)
It's clear that
![2k+r+2 (n-+-2k-+-r-+-1)-! -n- 2k-r- 2 (- 1) (n - 1) ! (1 + x) f (0) -→ 0 as n -→ ∞.](/img/revistas/ruma/v46n1/1a0196x.png)
Let , where
![[ k+1 2k∑+r+2f (i)(x) ∑ k (djn - r - 2) ! (djn + r - 1) ! I3 = n ---i!-- C (j,k) ----(d-n----1)-! (d-n---2)-!--- i=r+1 j=0 j j](/img/revistas/ruma/v46n1/1a0198x.png)
![∞ ⌋ ∑∞ ∫ dr i × pdjn+r,υ (x) pdjn-r,υ+r-1(t)--r (t - x) dt⌉ . υ=1 0 dt](/img/revistas/ruma/v46n1/1a0199x.png)
![[ ] ∑ k (djn - r - 2) ! (djn + r - 1) ! I4 = nk+1 f (r)(x) C (j,k) ------------------------------ - f(r)(x) . j=0 (djn - 1) ! (djn - 2) !](/img/revistas/ruma/v46n1/1a01100x.png)
Thus, by (1.4),
![[k∑ ]I = nk+1f (r)(x) C (j,k) (djn---r --2)-! (djn-+-r---1)-!- 1 . 4 (djn - 1) ! (djn - 2)! j=0](/img/revistas/ruma/v46n1/1a01101x.png)
Now, in view of Lemma 2.4, we have
![(r) I4 = f (x) K (r,k) + o(1) , n -→ ∞ ,](/img/revistas/ruma/v46n1/1a01102x.png)
where is a constant depending only on
and
.
Next, by Lemma 2.4 and Lemma 2.6,we get
![2k∑+r+2 I3 = f (i)(x) Q (i,k, r,x) + o (1), n -→ ∞. i=r+1](/img/revistas/ruma/v46n1/1a01106x.png)
Thus
![2k+∑r+2 I1 -→ f(r)(x) K (r,k) + f(i)(x) Q (i,k,r,x) i=r+1](/img/revistas/ruma/v46n1/1a01107x.png)
![2k+∑r+2 = f(i)(x) Q (i,k,r,x) as n -→ ∞. i=r](/img/revistas/ruma/v46n1/1a01108x.png)
Now we must prove that . For this, it is sufficient to prove that
![( 2k+r+2 ) I ≡ xrnk+1M (nr) ɛ(t,x) (t - x) ;x -→ 0 as n -→ ∞.](/img/revistas/ruma/v46n1/1a01110x.png)
Using Lemma 2.3, we get
![∞ k+1 ∑ i∑ j ∣I∣ ≤ n (n - 1) M (x) n pn,υ (x) ∣υ - nx∣ 2i + j ≤ r υ=1 i,j ≥ 0](/img/revistas/ruma/v46n1/1a01111x.png)
![∞ ∫ ∣∣ ∣∣ × pn,υ-1(t) ∣ɛ (t,x)∣∣(t - x)2k+r+2 ∣ dt, 0](/img/revistas/ruma/v46n1/1a01112x.png)
where , and then applying the Schwarz inequality we get:
![{ } ∑ ∞∑ 1∕2 ∣I∣ ≤ nk+1(n - 1) M (x) ni pn,υ (x) (υ - nx)2j υ=1 2i + j ≤ r i,j ≥ 0](/img/revistas/ruma/v46n1/1a01114x.png)
![( ∞ ( ∫∞ ) 2) 1∕2 { ∑ ∣∣ 2k+r+2∣∣ } × ( pn,υ (x) ( pn,υ-1 (t) ∣ɛ(t,x)∣∣(t - x) ∣ dt) ) . υ=1 0](/img/revistas/ruma/v46n1/1a01115x.png)
Since , for a given
such that
, whenever
, and for
there exists a constant
such that
.
Hence, as , we have
![( ) 2 ∫∞ ∣ ∣ ( pn,υ-1 (t) ∣ɛ(t,x)∣∣∣(t - x)2k+r+2 ∣∣ dt) ≤ 0](/img/revistas/ruma/v46n1/1a01124x.png)
![( ) ( ) ∫∞ ∫∞ ≤ ( p (t) dt) ( p (t) (ɛ (t,x))2 (t - x)4k+2r+4dt) . n,υ-1 n,υ-1 0 0](/img/revistas/ruma/v46n1/1a01125x.png)
![⌊ 1 ∫ ≤ ------|⌈ pn,υ-1(t) ɛ2(t - x)4k+2r+4 dt n - 1 0< ∣t-x∣< δ](/img/revistas/ruma/v46n1/1a01126x.png)
![⌋ ∫ 2 4k+2r+2β+4 | + pn,υ-1(t) C (t - x) dt⌉ . ∣t-x∣≥δ](/img/revistas/ruma/v46n1/1a01127x.png)
Now, by Lemma 2.2, we get
![( ) ∞ ∫∞ ∣ ∣ 2 ∑ p (x) ( p (t) ∣ɛ (t,x)∣∣(t - x)2k+r+2∣ dt) ≤ n,υ n,υ-1 ∣ ∣ υ=1 0](/img/revistas/ruma/v46n1/1a01128x.png)
![∑∞ ∫∞ ≤ --1--- pn,υ (x) pn,υ-1 (t) ɛ2 (t - x)4k+2r+4 dt n - 1 υ=1 0](/img/revistas/ruma/v46n1/1a01129x.png)
![2 ∑∞ ∞∫ + -C---- pn,υ (x) pn,υ-1(t) (t - x)4k+2r+2β+4dt. n - 1 υ=1 0](/img/revistas/ruma/v46n1/1a01130x.png)
![[ ]≤ ɛ2--1--- Tn,4k+2r+4 (x) - (- x)4k+2r+4 (1 + x)- n n - 1](/img/revistas/ruma/v46n1/1a01131x.png)
![C2 [ ]+ ------ Tn,4k+2r+2β+4(x) - (- x)4k+2r+2 β+4 (1 + x)-n . n - 1](/img/revistas/ruma/v46n1/1a01132x.png)
![( ) ( ) = ɛ2 O n-(2k+r+2) + O n -(2k+r+ β+2) .](/img/revistas/ruma/v46n1/1a01133x.png)
By Lemma 2.1, we have
![k+1 ∑ i+j ( -j∕2) ( -(2k+r+2)∕2) ∣I ∣ ≤ n M (x) n O n O n 2i + j ≤ r i,j ≥ 0](/img/revistas/ruma/v46n1/1a01134x.png)
![{ ( )}1∕2 × ɛ2 + O n- β .](/img/revistas/ruma/v46n1/1a01135x.png)
![= O (1) { ɛ2 + O (n-β)} 1∕2](/img/revistas/ruma/v46n1/1a01136x.png)
![≤ ɛ O (1).](/img/revistas/ruma/v46n1/1a01137x.png)
Since is arbitrary, it follows that
. The assertion (3.2) follows along similar lines by using Lemma 2.4 for
in place of
.
The last assertion follows, due to the uniform continuity of (enabling
to became independent of
) and the uniform of
term in the estimate of
(because, in fact, it is a polynomial in
.
The next result provides an estimate of degree approximation in .
Theorem 3.2. Let and
be bounded on every finite subinterval of
. Let
If
exists and is continuous on
, then for
sufficiently large
![∥∥ (r) (r)∥∥ ( - p∕2 ( - 1∕2) -(k+1)) M n (f, k,x) - f ≤ max C1n ωf(p+r) n , C2n ,](/img/revistas/ruma/v46n1/1a01156x.png)
where denotes the modulus continuity of
on
,
denotes the sup-norm on
.
Proof: For by the hypothesis we have
![p+r ( (p+r) (p+r) ) ∑ f-(i)(x) i -f-----(ξ) --f----(x)-- (p+r) (3.3)f (t) = i! (t - x) + (p + r)! (t - x) (1 - χ (t)) i=0](/img/revistas/ruma/v46n1/1a01163x.png)
![+h (t,x) χ (t) ,](/img/revistas/ruma/v46n1/1a01164x.png)
where lies between
is the characteristic function of the set
.Operating on this equality by
and breaking the right hand side into three parts
say, corresponding to the three terms on the right hand side of (3.3) as in the proof of Theorem 3.1, we have
![∑p+r (i) ( ) I1 = f---(x) M (nr) (t - x)i ,k,x i=0 i! (r) ( -(k+1)) = f (x) + O n ,uniformly for all x ∈ [a,b].](/img/revistas/ruma/v46n1/1a01170x.png)
To estimate , we have for every
![∣∣f (p+r)(ξ) - f(p+r)(x)∣∣ ≤ ω (∣ξ - x∣) f(p+r)](/img/revistas/ruma/v46n1/1a01173x.png)
![≤ ωf(p+r) (∣t - x ∣)](/img/revistas/ruma/v46n1/1a01174x.png)
![( ) ∣t - x∣ (3.4) ≤ 1 + ---δ--- ωf(p+r) (δ) .](/img/revistas/ruma/v46n1/1a01175x.png)
Since
![∑ k ∑∞ I2 = C (j,k)(djn - 1) p(rd)n,υ (x) j=0 υ=0 j](/img/revistas/ruma/v46n1/1a01176x.png)
![∫∞ (f(p+r)(ξ) - f (p+r)(x)) × pdjn,υ-1(t) -----------------------(t - x)(p+r) (1 - χ (t)) dt. (p + r)! 0](/img/revistas/ruma/v46n1/1a01177x.png)
Using (3.4) and Lemma 2.3, we have
![∑k ∑∞ ∣ ∣ ∣I2∣ ≤ ----1--- ∣C (j,k)∣ ∣∣p(r) (x)∣∣ (p + r)! j=0 υ=0 djn,υ](/img/revistas/ruma/v46n1/1a01178x.png)
![∫∞ ( ) × pdjn,υ-1(t) 1 + ∣t --x∣ ∣t - x∣(p+r)ωf(p+r) (δ) dt δ 0](/img/revistas/ruma/v46n1/1a01179x.png)
![ωf(p+r) (δ) ∑k ∑ i ∣qi,s,r (x)∣ ≤ ---------- ∣C (j,k)∣ (djn) --r-------r (p + r) ! j=0 2i + s ≤ r x (1 + x) i,s ≥ 0](/img/revistas/ruma/v46n1/1a01180x.png)
![∑∞ ∫∞ ( ) × p (x) ∣(υ - d nx)∣s p (t) ∣t - x∣p+r + 1-∣t - x ∣p+r+1 dt. djn,υ j djn,υ- 1 δ υ=1 0](/img/revistas/ruma/v46n1/1a01181x.png)
Putting , then applying Schwarz inequality for summation and for integral and Lemmas 2.1 and 2.2 as in the proof of theorem 3.1, we get
![[ ]∣I ∣ ≤ K O (n-p∕2)+ 1-O (n- (p+1)∕2) ω (p+r) (δ). 2 δ f](/img/revistas/ruma/v46n1/1a01183x.png)
Choosing , it follows that
![( ) ( ) I2 = ωf(p+r) n-1∕2 O n-p∕2 ,](/img/revistas/ruma/v46n1/1a01185x.png)
where term holds uniformly in
.
For and
, we can choose a
in such a way that
. Hence
![∑k ∑ ∣q (x)∣ ∣I3∣ ≤ ∣C (j,k)∣ (djn - 1) (djn)i--ri,s,r----r j=0 x (1 + x) 2i + s ≤ r i,s ≥ 0](/img/revistas/ruma/v46n1/1a01192x.png)
![∞ ∫ × ∑ p (x) ∣(υ - d nx)∣s p (t) ∣h(t,x)∣ dt. djn,υ j djn,υ-1 υ=1 ∣t-x∣≥s](/img/revistas/ruma/v46n1/1a01193x.png)
Now, for we can find a positive constant
such that
, where
is any integer
.
Hence, by Schwarz inequality, Lemmas 2.1 and 2.2 we have
![∑k ∑ ∣q (x)∣ ∣I3∣ ≤ M ∣C (j,k)∣ (djn - 1) (djn)i -ri,s,r---r- j=0 x (1 + x) 2i + s ≤ r i,s ≥ 0](/img/revistas/ruma/v46n1/1a01199x.png)
![∞ ∫ ∑ s γ × pdjn,υ (x) ∣(υ - djnx) ∣ pdjn,υ-1(t) ∣t - x∣ dt. υ=1 ∣t-x∣≥s](/img/revistas/ruma/v46n1/1a01200x.png)
![( ) ( ) = O n(r-γ)∕2 = O n- (k+1) uniformly in x ∈ [a,b].](/img/revistas/ruma/v46n1/1a01201x.png)
The required result follows on combining the estimates of and
. e o
Acknowledgement. The authors are thankful to the referee for making substantial improvements in the paper.
References:
[1] P.N. Agrawal and Kareem J. Thamer, Approximation of unbounded functions by a new sequence of linear positive operators, J. Math. Anal. App. 225(1998), 660-672. [ Links ]
[2] P.N. Agrawal and Kareem J. Thamer, Degree of approximation by a new sequencf linear operators, Kyungpook Math. J., 41(1) (2001), 65-73. [ Links ]
[3] P.N. Agrawal and T.A.K. Sinha, A saturation theorem for a combination of modified Lupas operators in Lp-spaces, Bull. Inst. Math. Academia Sinica 24 (1996), 159-165. [ Links ]
[4] H.S. Kasana, P.N. Agrawal and V. Gupta, Inverse and Saturation theorems for linear combination of modified Baskakov operators, Approx. Theory Appl. 7(2)(1991), 65-82. [ Links ]
[5] H.S. Kasana, G. Prasad, P.N. Agrawal and A. Sahai, On modified Szasz operators, Proc. Int. Conf. Math. Anal. And its Appl. Kuwait (1985), 29-41, Pergamon Press, Oxford (1988). [ Links ]
[6] C.P. May, Saturation and Inverse theorem for combinations of a class of exponential type operators, Canad. J. Math. 28(1976), 1224-1250. [ Links ]
[7] R.K.S. Rathore, Linear Combinations of Linear Positive Operators and Generating Relations in Special Functions, Ph.D. Thesis I.I.T. Delhi (India) (1973). [ Links ]
[8] A. Sahai and G. Prasad, On simultaneous approximation by modified Lupas operators, J. Approx. Theory 45 (1985), 122-128. [ Links ]
Kareem J. Thamer
Department of Mathematics,
College of Education-Amran,
Sana'a University,
Maeen Post Office, Box (13475), Sana'a – Republic of Yemen.
k_alabdullah2005@yahoo.com
May A. Al-Shibeeb
Rayed, P.O.Box (46379),
Post Code 11532, Saudia Arabia King Dom.
maey9999@hotmail.com
A.I. Ibrahem
Department of Mathematics,
College of Science,
Basrah University,
Basrah – Iraq.
Recibido: 26 de diciembre de 2002
Aceptado: 25 de agosto de 2005