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Revista de la Unión Matemática Argentina

Print version ISSN 0041-6932

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

 

Simultaneous approximation by a new sequence of Szãsz-beta type operators

Ali J. Mohammad and Amal K. Hassan

Abstract. In this paper, we study some direct results in simultaneous approximation for a new sequence of linear positive operators Mn (f(t);x) of Szãsz-Beta type operators. First, we establish the basic pointwise convergence theorem and then proceed to discuss the Voronovaskaja-type asymptotic formula. Finally, we obtain an error estimate in terms of modulus of continuity of the function being approximated.

Key words and phrases. Linear positive operators; Simultaneous approximation; Voronovaskaja-type asymptotic formula; Degree of approximation; Modulus of continuity.

1. INTRODUCTION

In [3] Gupta and others studied some direct results in simultaneous approximation for the sequence:

 ∑∞ ∞∫ B (f(t);x) = q (x) b (t)f (t)dt, n n,k n,k k=0 0

where x, t ∈ [0,∞ ) , q (x) = e-nx(nx)k n,k k! and b (t) = -Γ (n+k+1-)tk(1 + t)-(n+k+1) n,k Γ (n)Γ (k+1) . After that, Agrawal and Thamer [1] applied the technique of linear combination introduced by May [4] and Rathore [5] for the sequence Bn (f(t);x) . Recently, Gupta and Lupas [2] studied some direct results for a sequence of mixed Beta-Szãsz type operator defined as  ∞∑ ∞∫ Ln (f(t);x) = bn,k(x ) qn,k-1(t)f(t) dt + (1 + x)-n- 1f(0) k=1 0 .

In this paper, we introduce a new sequence of linear positive operators Mn (f (t);x ) of Szãsz-Beta type operators to approximate a function f (x) belongs to the space Cα[0,∞ ) = {f ∈ C [0,∞ ) : |f(t)| ≤ C (1 + t)α forsome C > 0,α > 0} , as follows:

 ∫∞ ∑∞ -nx Mn (f (t);x ) = qn,k(x ) bn,k-1(t)f(t) dt + e f (0 ), k=1 0
(1.1)

We may also write the operator (1.1) as  ∞∫ Mn (f (t);x) = Wn (t,x)f (t) dt 0 where  ∞∑ -nx Wn (t,x) = k=1qn,k(x )bn,k-1(t) + e δ(t) , δ (t) being the Dirac-delta function.

The space C [0,∞ ) α is normed by ∥f ∥ = sup |f (t) | (1 + t)- α Cα 0≤t<∞ .

There are many sequences of linear positive operators with are approximate the space C [0,∞ ) α . All of them (in general) have the same order of approximation  -1 O(n ) [6]. So, to know what is the different between our sequence and the other sequences, we need to check that by using the computer. This object is outside our study in this paper.

Throughout this paper, we assume that C denotes a positive constant not necessarily the same at all occurrences, and [β ] denotes the integer part of β .

2. PRELIMINARY RESULTS

For f ∈ C [0,∞ ) the Szãsz operators are defined as  ∞∑ (k) Sn(f ;x) = qn,k(x )f n k=1 , x ∈ [0,∞ ) and for m ∈ N 0 (the set of nonnegative integers), the m -th order moment of the Szãsz operators is defined as  ∞ ( ) μn,m (x ) = ∑ qn,k(x) k - x m k=0 n .

LEMMA 2.1. [3] For  0 m ∈ N , the function μn,m (x) defined above, has the following properties:

(i)    μn,0(x) = 1 , μn,1(x) = 0 , and the recurrence relation is

 ( ′ ) nμn,m+1 (x) = x μn,m(x ) + m μn,m- 1(x ) , m ≥ 1;

(ii)   μn,m(x ) is a polynomial in x of degree at most [m ∕2] ;

(iii)  For everyx ∈ [0,∞ ) , ( - [(m+1)∕2]) μn,m (x) = O n .

From above lemma, we get

pict

For m ∈ N 0 , the m -th order moment Tn,m (x) for the operators (1.1) is defined as:

 ∑∞ ∫∞ Tn,m(x) = Mn ((t - x)m; x) = qn,k(x) bn,k- 1(t)(t- x)mdt+ (- x)me- nx. k=1 0

LEMMA 2.2.For the function Tn,m (x) , we have Tn,0(x) = 1 ,Tn,1(x) = -x-- n- 1 ,  nx2+2nx+2x2 Tn,2(x) = (n-1)(n-2) and there holds the recurrence relation:

pict

Further, we have the following consequences of Tn,m(x) :

(i)    Tn,m (x) is a polynomial in x of degree exactly m ;

(ii)   For every x ∈ [0,∞ ) , T (x ) = O (n-[(m+1 )∕2]) n,m .

Proof: By direct computation, we have T (x ) = 1 n,0 , T (x) = --x- n,1 n- 1 and  nx2+2nx+2x2 Tn,2(x) = -(n-1)(n-2)- . Next, we prove (2.2). For x = 0 it clearly holds. For x ∈ (0,∞ ) , we have

 ∞∑ ∫∞ T′n,m(x) = q′n,k(x) bn,k-1(t)(t- x)mdt - n(- x)me -nx- mTn,m -1(x). k=1 0

Using the relations xq′ (x ) = (k - nx)qn,k(x) n,k and t(1 + t)b′ (t) = (k - (n + 1)t)× n,k b (t) n,k , we get:

pict

By using the identity t(1 + t) = (t - x)2 + (1 + 2x )(t - x) + x(1 + x) , we have

pict

Integrating by parts, we get

 ′ xTn,m(x ) = (n - m - 1) Tn,m+1 (x)- (m+x+2mx )Tn,m(x)- mx (x+2 )Tn,m -1(x)

from which (2.2) is immediate.

From the values of Tn,0(x) and Tn,1(x) , it is clear that the consequences (i) and (ii) hold for m = 0 and m = 1 . By using (2.2) and the induction on m the proof of consequences (i) and (ii) follows, hence the details are omitted.

From the above lemma, we have

pict

LEMMA 2.3. Let δ and γ be any two positive real numbers and [a,b] ⊂ (0,∞ ) . Then, for any s > 0 , we have

∥∥ ∥∥ ∥∥ ∫ ∥∥ ∥ Wn (t,x)tγdt ∥ = O (n- s). ∥∥ ∥∥ |t-x|≥δ C[a,b]

Making use of Schwarz inequality for integration and then for summation and (2.3), the proof of the lemma easily follows.

LEMMA 2.4. [3] There exist polynomials Qi,j,r(x) independent of n and k such that

 r r ∑ i j x D (qn,k(x)) = n (k - nx) Qi,j,r(x)qn,k(x ), 2i + j ≤ r i,j ≥ 0 where D = -d dx .

 

3. MAIN RESULTS

Firstly, we show that the derivative  (r) M n (f(t);x) is an approximation process for f (r)(x), r = 1 , 2 , . .. .

Theorem 3.1. If r ∈ N ,f ∈ C α[0, ∞ ) for some α > 0 and  (r) f exists at a point x ∈ (0,∞ ) , then

 (r) (r) nl→im∞ M n (f(t);x) = f (x ).
(3.1)

Further, if f (r) exists and is continuous on (a - η,b + η) ⊂ (0,∞ ) , η > 0 , then (3.1) holds uniformly in [a,b] .

Proof: By Taylor's expansion of f , we have

 ∑r (i) f (t) = f--(x-)(t - x )i + ε(t,x)(t - x )r, i! i=0

where, ε(t,x) → 0 as t → x . Hence

pict

Now, using Lemma 2.2 we get that Mn (tm; x) is a polynomial in x of degree exactly

, for all  0 m ∈ N . Further , we can write it as:

 m m- 1 Mn (tm; x) = (n---m---1)! n-xm+ (n---m----1)! n---m (m- 1) xm-1+O (n -2). (n- 1)! (n - 1)!
(3.2)

Therefore,

pict

Next, making use of Lemma 2.4 we have

pict

Since ε(t,x) → 0 as t → x , then for a given ε > 0 , there exists a δ > 0 such that |ε(t,x)| < ε , whenever 0 < |t - x| < δ . For |t - x| ≥ δ , there exists a constant C > 0 such that  γ |ε(t,x)(t - x)r| ≤ C |t - x| .

Now, since  sup |Qi,j,rr(x)|:= M (x) = C ∀ x ∈ (0,∞ ) 2i + j ≤ r x i,j ≥ 0 . Hence,

pict

Now, applying Schwartz inequality for integration and then for summation, (2.1) and (2.3) we led to

pict

Again using Schwarz inequality for integration and then for summation, in view of (2.1) and Lemma 2.3, we have

pict

Now, since ε > 0 is arbitrary, it follows that I = o (1 ) 3 . Also, I → 0 4 as n → ∞ and hence I2 = o(1) , combining the estimates of I1 and I2 , we obtain (3.1).

To prove the uniformity assertion, it sufficient to remark that δ(ε) in above proof can be chosen to be independent of x ∈ [a,b] and also that the other estimates holds uniformly in [a,b] .

Our next theorem is a Voronovaskaja-type asymptotic formula for the operators M (r)(f(t);x ), r = 1, 2 , . .. n .

THEOREM 3.2. Let f ∈ Cα[0,∞ ) for some α > 0 . If f(r+2 ) exists at a point x ∈ (0, ∞ ) , then

pict

Further, if f(r+2) exists and is continuous on the interval (a - η,b+ η) ⊂ (0,∞ ), η > 0 , then (3.3) holds uniformly on [a,b] .

Proof: By the Taylor's expansion of f (t) , we get

pict

where ε(t,x) → 0 as t → x .

By Lemma 2.2 and (3.2), we have

pict

Hence in order to prove (3.3) it suffices to show that nI2 → 0 as n → ∞ , which follows on proceeding along the lines of proof of I2 → 0 as n → ∞ in Theorem 3.1.

The uniformity assertion follows as in the proof of Theorem 3.1.

Finally, we present a theorem which gives as an estimate of the degree of approximation by  (r) M n (.;x ) for smooth functions.

THEOREM 3.3. Let f ∈ C [0,∞ ) α for some α > 0 and r ≤ q ≤ r + 2 . If  (q) f exists and is continuous on (a - η, b + η ) ⊂ (0,∞ ), η > 0 , then for sufficiently large n ,

∥ ∥ ∑q ∥ ∥ ( ) ∥∥M (nr)(f(t);x) - f(r)(x)∥∥ ≤ C1n -1 ∥∥f(i)∥∥ +C2 n-1∕2ωf(q) n-1∕2 +O (n-2) C[a,b] i=r C[a,b]

where C1 ,C2 are constants independent of f and n , ωf(δ) is the modulus of continuity of f on (a - η, b + η) , and ∥ .∥C[a,b] denotes the sup-norm on [a,b] .

Proof. By Taylor's expansion of f , we have

 ∑q (i) (q) (q) f (t) = f--(x)-(t- x)i+ f--(ξ)---f--(x-)(t- x)qχ(t)+h (t,x)(1- χ(t)), i=0 i! q!

where ξ lies between t,x , and χ (t) is the characteristic function of the interval (a - η,b + η) . Now,

pict

By using Lemma 2.2 and (3.2), we get

 q (i) i ( ) r ( j I = ∑ f--(x)∑ i (- x)i-j-d- (n---j---1)! n-xj 1 i=r i! j=r j dxr (n- 1)! j-1 ) + (n---j---1)!-n---j(j - 1)xj- 1 + O (n -2) - f(r)(x). (n - 1)!

Consequently,

 ( q ) - 1 ∑ ∥ (i)∥ -2 ∥I1∥C[a,b] ≤ C1n ∥f ∥C[a,b] + O (n ),uniformly on [a,b]. i=r

To estimate I2 we proceed as follows:

 ∫∞ { | | } || (r) || |f-(q)(ξ) --f-(q)(x)| q |I2| ≤ W n (t,x) q! |t - x | χ(t) dt 0

 ∫∞ ( ) ≤ ωf(q)(δ)- || W (r)(t,x)|| 1 + |t---x|- |t - x|q dt q! n δ 0

 [ ∑∞ | |∫∞ ≤ ωf(q)(δ)- ||q(r)(x)|| bn,k-1(t)( |t - x|q + δ-1|t - x |q+1 ) dt q! n,k k=1 0

 ] r - nx q - 1 q+1 + (- n )e (x + δ x ) , δ > 0.

Now, for s = 0, 1, 2, ... , using Schwartz inequality for integration and then for summation, (2.1) and (2.3), we have

pict

Therefore, by Lemma 2.4 and (3.4), we get

pict

(since  Q (x) sup sup |i,jx,rr--|:= M(x) 2i+ j ≤ r x∈[a,b] i,j ≥ 0 but fixed )

Choosing δ = n -1∕2 and applying (3.5), we are led to

pict

Since t ∈ [0,∞ )\ (a - η,b + η) , we can choose δ > 0 in such a way that |t - x| ≥ δ for all x ∈ [a,b] . Thus, by Lemmas 2.3 and 2.4 , we obtain

 ∑∞ ∑ j |Qi,j,r(x)| |I3| ≤ ni|k- nx| ---xr----qn,k(x ) k=1 2i+ j ≤ r i,j ≥ 0 ∫ bn,k-1(t)|h(t,x)| dt+ (- n)re-nx|h(0,x)|. |t-x|≥ δ

For |t - x | ≥ δ , we can find a constant C such that |h (t,x )| ≤ C |t - x|α . Hence, using Schwarz inequality for integration and then for summation ,(2.1), (2.3), it easily follows that  -s I3 = O (n ) for any s > 0 , uniformly on [a,b] .

Combining the estimates of I1, I2I3 , the required result is immediate.

References

[1]    P.N. Agrawal and Kareem J. Thamer. Linear combinations of Szãsz-Baskakov type operators, Demonstratio Math.,32(3) (1999), 575-580.        [ Links ]

[2]    Vijay Gupta and Alexandru Lupas. Direct results for mixed Beta-Szãsztype operators, General Mathematics 13(2), (2005), 83-94.        [ Links ]

[3]    Vijay Gupta, G.S. Servastava and A. Shahai. On simultaneous approximation by Szãsz-Beta operators, Soochow J. Math. 21, (1995), 1-11.        [ Links ]

[4]    C.P. May. Saturation and inverse theorems for combinations of a class of exponential- type operators, Canad. J. Math. 28 (1976), 1224-1250.        [ Links ]

[5]    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 ]

[6]    E. Voronovskaja. Détermination de la forme asymptotique d'ápproximation des fonctions par les polynômes de S.N. Bernstein, C.R. Adad. Sci. USSR (1932), 79-85.         [ Links ]

Ali J. Mohammad
University of Basrah,
College of Education,
Dept. of Mathematics,
Basrah, IRAQ.
alijasmoh@yahoo.com

Amal K. Hassan
University of Basrah,
College of Science,
Dept. of Mathematics,
Basrah, IRAQ.

Recibido: 8 de noviembre de 2006
Aceptado: 11 de marzo de 2008