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volume 6, issue 1, article 4, 2005.

Received 25 November, 2004;

accepted 29 December, 2004.

Communicated by:A. Lupa¸s

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Journal of Inequalities in Pure and Applied Mathematics

ON THE APPROXIMATION OF LOCALLY BOUNDED FUNCTIONS BY OPERATORS OF BLEIMANN, BUTZER AND HAHN

JESÚS DE LA CAL AND VIJAY GUPTA

Departamento de Matemática Aplicada y Estadística e Investigación Operativa Facultad de Ciencias

Universidad del País Vasco Apartado 644, 48080 Bilbao, Spain EMail:mepcaagj@lg.ehu.es School of Applied Sciences

Netaji Subhas Institute of Technology Sector 3 Dwarka, New Delhi-110045, India EMail:vijay@nsit.ac.in

c

2000Victoria University ISSN (electronic): 1443-5756 228-04

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On The Approximation Of Locally Bounded Functions By Operators Of Bleimann, Butzer

And Hahn

Jesús de la Cal and Vijay Gupta

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J. Ineq. Pure and Appl. Math. 6(1) Art. 4, 2005

Abstract

We estimate the rate of the pointwise approximation by operators of Bleimann, Butzer and Hahn of locally bounded functions, and of functions having a locally bounded derivative.

2000 Mathematics Subject Classification:41A20, 41A25, 41A36.

Key words: Operators of Bleimann, Butzer and Hahn, Locally bounded function, Function of bounded variation, Total variation, Rate of convergence, Bi- nomial distribution.

J. de la Cal was supported by the Spanish MCYT, Proyecto BFM2002-04163-C02- 02, and by FEDER.

1. Introduction and Main Results

Bleimann, Butzer and Hahn [1] introduced the Bernstein type operatorL_nover the interval[0,∞)given by

L_n(f, x) :=

n

X

k=0

f

k n−k+ 1

b_n,k(x), x≥0, n= 1,2, . . . ,

wheref is a real function on[0,∞), and (1.1) bn,k(x) :=

n k

p^k_xq^n−k_x , px := x

1 +x, qx := 1−px = 1 1 +x. The approximation of uniformly continuous functions by these operators has been considered in [1] – [4]. For other properties ofL_n(preservation of global smoothness, preservation ofφ-variation, behavior of the iterates, etc.) we refer, for instance, to [4] – [10]. In some of the mentioned works, the results are achieved by using probabilistic methods. This comes from the fact thatL_nis an operator of probabilistic type. We can actually write

L_n(f, x) =Ef(Z_n,x),

where E denotes mathematical expectation, and Z_n,x is the random variable given by

(1.2) Z_n,x := Sn,x

n−S_n,x+ 1, S_n,x :=ξ_1,x+· · ·+ξ_n,x,

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whereξ_1,x, ξ_2,x, . . . are independent random variables having the same Bernoulli distribution with parameterpx, i.e.,

P(ξ_k,x= 1) =p_x = 1−P(ξ_k,x = 0)

(so that S_n,x has the binomial distribution with parameters n, p_x). This probabilistic representation also plays a significant role in the present paper (for a more refined representation useful for other purposes, see [5,6]).

Here, we discuss the approximation of real functions f on the semi axis which are locally bounded, i.e., bounded on each finite subinterval of[0,∞). In such a case, we set, forx >0andh≥0,

ω_x⁺(f;h) := sup

x≤t≤x+h

|f(t)−f(x)|, ω⁻_x(f;h) := sup

(x−h)⁺≤t≤x

|f(t)−f(x)|, ω_x(f;h) :=ω_x⁺(f;h) +ω_x⁻(f;h),

where(x−h)⁺ := max(x−h,0), and we observe that these functions are (non- negative and) nondecreasing on[0,∞). In particular, every continuous function is locally bounded. Also, iff is locally of bounded variation, i.e., such that

b

_

a

(f)<∞, 0≤a < b <∞,

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where Wb

a(f) stands for the total variation off on the interval [a, b], thenf is locally bounded, and we obviously have

ω_x(f;h)≤

x+h

_

x−h

(f), 0≤h ≤x.

This kind of problem has been already considered for other Bernstein-type operators (see, for instance, [11] – [14] and the references therein). Our main results are stated as follows.

Theorem 1.1. Let g be a real locally bounded function on [0,∞) such that g(t) = O(t^r) (t → ∞), for somer = 1,2, . . .. If g is continuous at x > 0, then, fornlarge enough, we have

(1.3) |Ln(g, x)−g(x)| ≤ 7(1 +x)² (n+ 2)x

n

X

k=1

ωx

g; x

√k

+Or,x

1 n

. In the following statements (and throughout the paper), we use the notations:

f^∗(x) :=f(x+)−f(x−) f˜(x) := f(x+) +f(x−)

2 ,

f_x := (f −f(x−))1_[0,x)+ (f −f(x+))1_(x,∞)

(1A being the indicator function of the set A), provided that the lateral lim- its f(x+) and f(x−) exist (such a condition is fulfilled when f is locally of bounded variation). We also use the symbolbacto indicate the integral part of the real numbera.

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Theorem 1.2. Let f be a real locally bounded function on [0,∞) such that f(t) = O(t^r) (t → ∞), for some r = 1,2, . . .. If x > 0, and f(x+) and f(x−)exist, then we have fornlarge enough

L_n(f, x)−f(x)˜

≤∆_n,x(f_x) + 1.6 +x+ 2.6x²

√nx(1 +x) ·|f^∗(x)|

2 +_n,x(1 +x)

√2enx |f(x)−f(x−)|, where∆_n,x(f_x)is the right-hand side of (1.3) withgreplaced byf_x, and

_n,x :=

( 1 if (n+ 1)p_x∈ {1,2, . . . , n}

0 otherwise.

Theorem 1.3. Letgbe a real function on[0,∞)such thatg(t) = O(t^r) (t→

∞), for somer= 1,2, . . ., and having the form g(t) = c+

Z t 0

f(u)du, t≥0,

where c is a constant andf is measurable and locally bounded on[0,∞). If x >0, andf(x+)andf(x−)exist, then we have fornlarge enough

L_n(g, x)−g(x)−

√x(1 +x)

√2πn f^∗(x)

≤ 5(1 +x)² n+ 2

b√ nc

X

k=1

ωx

fx;x

k

+|f^∗(x)|ox n^−1/2

+Or,x(n⁻¹).

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The proofs of the preceding theorems are given in Sections3–5. In Section 2, we collect the necessary auxiliary results. Some remarks on moments close the paper.

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2. Auxiliary Results

In the following lemma, Φ denotes the standard normal distribution function, andF_n,x^∗ stands for the distribution function ofS_n,x^∗ := (S_n,x−np_x)√

np_xq_x ,, whereS_n,xis the same as in (1.2). Such a lemma is nothing but the application of the well-known Berry-Esseen theorem (cf. [15]) to the situation at hand.

Lemma 2.1. We have, forx >0andn ≥1, sup

−∞<t<∞

|F_n,x^∗ (t)−Φ(t)| ≤ 0.8(p³_xq_x+p_xq_x³)

√n(pxqx)^3/2 = 0.8(1 +x²)

√nx(1 +x). Lemma 2.2. Letx >0andn≥1. Then, we have:

(a)

Ln((· −x)², x) =E(Zn,x−x)² ≤ 3x(1 +x)² n+ 2 . (b)

P(Zn,x ≤x−h) +P(Zn,x ≥x+h)≤ 3x(1 +x)²

(n+ 2)h² , h >0.

(c)

|P(Zn,x > x)−P(Zn,x ≤x)| ≤ rx

n + 1.6(1 +x²)

√nx(1 +x). (d)

Ln((· −x), x) =E(Zn,x−x) =−xpⁿ_x =ox(n⁻¹), (n → ∞).

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(e)

L_n(| · −x|, x) =E|Z_n,x−x|=

√2x(1 +x)

√πn +o_x(n^−1/2), (n → ∞).

Proof. Part (a) was shown in [10]. Part (b) follows from (a) and the fact that, by Markov’s inequality,

P(Z_n,x ≤x−h) +P(Z_n,x ≥x+h) = P(|Z_n,x−x| ≥h)≤ E(Z_n,x−x)² h² . To show (c), observe that

|P(Z_n,x > x)−P(Z_n,x ≤x)|

=|1−2P(Z_n,x ≤x)|

=|1−2P(S_n,x ≤(n+ 1)p_x)|

=

1−2F_n,x^∗ rx

n

≤2

Φ rx

n

−F_n,x^∗ rx

n

+

1−2Φ rx

n

.

Thus, the conclusion in part (c) follows from Lemma 2.1and the fact that (cf.

[16])

0<2Φ(t)−1≤

1−e^−t²1/2

≤t, (t >0).

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Part (d) is immediate. Finally, to show (e), letm :=b(n+ 1)p_xc. We have L_n(| · −x|, x)−L_n((· −x), x)

= 2

m

X

k=0

x− k

n−k+ 1

b_n,k(x)

= 2x

m

X

k=0

b_n,k(x)−2

m

X

k=1

n!

(k−1)!(n−k+ 1)!p^k_xq_x^n−k

= 2x

m

X

k=0

b_n,k(x)−2x

m−1

X

k=0

b_n,k(x)

= 2x b_n,m(x)

=

√2x(1 +x)

√πn +o_x(n^−1/2), (n→ ∞),

the last equality by [13, Lemma 1], and the conclusion follows from (d).

Lemma 2.3. Letx >0andr= 1,2, . . .. Then, we have for all integersnsuch that(n+ 1)(p2x−p3x/2)≥r,

X

k∈K

k^r

(n−k+ 1)^rb_n,k(x)≤12r!

r

X

s=1

nr s

ox^s−1(1 +x)^r−s+2

n+r−s+ 2 · n!

(n+r−s)!

=O_r,x(n⁻¹), (n→ ∞), where the_r

s are the Stirling numbers of the second kind, andK is the set of all integersksuch thatn≥k >(n−k+ 1)2x(i.e.,n ≥k >(n+ 1)p2x).

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Proof. Using the well known identity

a^r =

r

X

s=1

nr s

o

a(a−1)· · ·(a−s+ 1), we can write

(2.1) X

k∈K

k^r

(n−k+ 1)^r b_n,k(x) =

r

X

s=1

nr s

o A_s, where

As:=X

k∈K

k(k−1)· · ·(k−s+ 1)

(n−k+ 1)^r bn,k(x)

=X

k∈K

1

(n−k+ 1)^r· n!

(k−s)!(n−k)!p^k_xq^n−k_x . Since

1

(n−k+ 1)^r=

r

Y

i=1

1 n−k+i

n−k+i n−k+ 1

=

r

Y

i=1

1 n−k+i

1 + i−1 n−k+ 1

≤

r

Y

i=1

i

n−k+i = r!(n−k)!

(n−k+r)!,

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we have

A_s≤r!X

k∈K

n!

(k−s)!(n−k+r)!p^k_xq_x^n−k

=r!X

l∈Ks

n!

l!(n+r−s−l)!p^l+s_x q_x^n−l−s

= r!n!p^s_xq_x^−r (n+r−s)!

X

l∈K_s

n+r−s l

x^l (1 +x)^n+r−s

≤ r!n!p^s_xq_x^−r (n+r−s)!

X

l∈K⁰

n+r−s l

x^l (1 +x)^n+r−s,

whereK_s :={k−s :k ∈ K}, and K⁰ stands for the set of all integersl such thatn≥l >(n−l+ 1)(3x/2)(observe that, by the assumption onn, we have K_s ⊂K⁰). The probabilistic interpretation of the last sum together with Lemma 2.2(b) yield

A_s≤ r!n!x^s(1 +x)^r−s (n+r−s)! P

Zn+r−s,x>3x 2

≤ 12r!n!x^s−1(1 +x)^r−s+2 (n+r−s)!(n+r−s+ 2), (2.2)

and the conclusion follows from (2.1) and (2.2).

Remark 1. The same procedure as in the preceding proof leads to the following

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upper bound for the integral moments ofL_n(orZ_n,x):

L_n(t^r, x) = E(Z_n,x)^r

=

n

X

k=0

k^r

(n−k+ 1)^r b_n,k(x)

≤r!

r

X

s=1

nr s

on!x^s(1 +x)^r−s (n+r−s)! .

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3. Proof of Theorem 1.1

Without loss of generality, we assume that g(x) = 0. Denote by K_n,x the distribution function ofZn,x, i.e.,

K_n,x(t) := P(Z_n,x ≤t) = X

k≤(n−k+1)t

b_n,k(x) t≥0.

We can writeL_n(g, x)as the Lebesgue-Stieltjes integral L_n(g, x) = Eg(Z_n,x) =

Z

[0,∞)

g(t)dK_n,x(t) =

4

X

j=1

Z

Ij

g(t)dK_n,x(t), where

I₁ :=

0, x− x

√n

, I₂ :=

x− x

√n, x+ x

√n

, I₃ :=

x+ x

√n,2x

and I₄ := (2x,∞).

We obviously have Z

I2

|g(t)|dK_n,x(t)≤ω_x

g; x

√n Z

I2

dK_n,x(t)

≤ω_x

g; x

√n

≤ 1 n

n

X

k=1

ωx

g; x

√k

. (3.1)

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On the other hand, from the asymptotic assumption ong, we have

|g(t)| ≤M t^r, t≥α, for some constantsM > 0andα ≥2x. Therefore,

Z

I4

|g(t)|dK_n,x(t)

= Z

(2x,α]

+ Z

(α,∞)

|g(t)|dKn,x(t)

≤ω_x⁺(g;α−x)P(Z_n,x >2x) +M X

k>(n−k+1)α

k^r

(n−k+ 1)^rb_n,k(x).

By Lemma2.2(b) and Lemma2.3, this shows that (3.2)

Z

I4

|g(t)|dK_n,x(t) =O_r,x(n⁻¹) (n→ ∞).

Finally, using Lemma 2.2(b) and integration by parts (follow the same procedure as in the proof of Theorem 1 in [13]), we obtain

Z

I1

|g(t)|dKn,x(t)≤

Z

I1

ω⁻_x(g;x−t)dKn,x(t)

≤ 3x(1 +x)² (n+ 2)

"

ω_x⁻(g;x) x² + 2

Z x−x/√ n 0

ω_x⁻(g;x−t) (x−t)³ dt

#

≤ 6(1 +x)² (n+ 2)x

n

X

k=1

ω⁻_x

g; x

√k

, (3.3)

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and, analogously, (3.4)

Z

I3

|g(t)|dKn,x(t)≤ 6(1 +x)² (n+ 2)x

n

X

k=1

ω_x⁺

g; x

√k

. The conclusion follows from (3.1) – (3.4).

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4. Proof of Theorem 1.2

We can write, fort≥0,

(4.1) f(t)−f(x) =˜ f_x(t) + f^∗(x)

2 σ_x(t) + (f(x)−f˜(x))δ_x(t),

whereσ_x :=−1_[0,x)+ 1(x,∞), andδ_x := 1{x}is Dirac’s delta atx(this is the so called Bojanic-Vuilleumier-Cheng decomposition).

By Theorem1.1, we have

(4.2) |L_n(f_x, x)| ≤∆_n,x(f_x),

where∆_n,x(f_x)is the right-hand side of (1.2) withg replaced byf_x. Moreover, Ln(σx, x)= P(Zn,x > x)−P(Zn,x < x)

= (P(Z_n,x > x)−P(Z_n,x ≤x)) +P(Z_n,x =x), (4.3)

and

(4.4) L_n(δ_x, x) =P(Z_n,x =x).

Using Lemma2.2(c) and the fact that (cf. [17, Theorem 1]) P(Zn,x =x) =







n k

p^k_xq^n−k_x ≤ ^√^(1+x)

2enx if (n+ 1)p_x =k∈ {1,2, . . . , n}

0 otherwise,

the conclusion readily follows from (4.1) – (4.4).

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5. Proof of Theorem 1.3

Using the decomposition (4.1), it is easily checked that

(5.1) L_n(g, x)−g(x) =

4

X

i=1

A_i(n, x), where

A₁(n, x) := ˜f(x)L_n((· −x), x) + f^∗(x)

2 L_n(| · −x|, x), A₂(n, x) :=

Z

[0,x]

Z x t

f_x(u)du

dK_n,x(t), A3(n, x) :=

Z

(x,2x]

Z t x

fx(u)du

dKn,x(t), A₄(n, x) :=

Z

(2x,∞)

Z t x

f_x(u)du

dK_n,x(t), andK_n,x(t)is the same as in the preceding proofs.

From Lemma2.2(d,e), we have (5.2) A₁(n, x) =

√x(1 +x)

√2πn f^∗(x)+f^∗(x)o_x(n^−1/2)+o_x(n⁻¹), (n → ∞).

Next, we estimateA₂(n, x). By Fubini’s theorem, A2(n, x) =

Z x 0

Kn,x(u)fx(u)du=

Z x−x/√ n 0

+ Z x

x−x/√ n

!

Kn,x(u)fx(u)du.

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It is clear that

Z x x−x/√

n

K_n,x(u)f_x(u)du

≤ Z x

x−x/√ n

|f_x(u)|du

≤ Z x

x−x/√ n

ω_x⁻(f_x;x−u)du

≤ x

√n ω⁻_x

fx; x

√n

≤ 2x n

b√ nc

X

k=1

ω⁻_x f_x;x

k

, and, using Lemma2.2(b),

Z x−x/√ n 0

K_n,x(u)f_x(u)du

≤ 3x(1 +x)² (n+ 2)

Z x−x/√ n 0

|f_x(u)|

(x−u)² du

≤ 3x(1 +x)² (n+ 2)

Z x−x/√ n 0

ω_x⁻(f_x;x−u) (x−u)² du

≤ 3(1 +x)² (n+ 2)

Z

√n

1

ω_x⁻ f_x;x

t

dt

≤ 3(1 +x)² n+ 2

b√ nc

X

k=1

ω⁻_x f_x;x

k

.

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We therefore conclude that

(5.3) |A₂(n, x)| ≤ 5(1 +x)² n+ 2

b√ nc

X

k=1

ω_x⁻ f_x;x

k

. Similarly,

(5.4) |A₃(n, x)| ≤ 5(1 +x)² n+ 2

b√ nc

X

k=1

ω⁺_x f_x;x

k

.

Finally, A₄(n, x) =

Z

(2x,∞)

g(t)dK_n,x(t)− Z

(2x,∞)

[g(x) +f(x+)(t−x)]dK_n,x(t), and, by the asymptotic assumption on g, Lemma 2.2(b) and Lemma 2.3, we obtain

(5.5) |A₄(n, x)|=O_r,x(n⁻¹), (n→ ∞).

The conclusion follows from (5.1) – (5.5).

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6. Remarks on Moments

Fixx >0, and letg(·) :=| · −x|^β, withβ >2. Since ωx(g, h) = 2h^β, 0≤h ≤x,

and n

X

k=1

k^−β/2 =O(1), (n→ ∞), we conclude from Theorem1.1that

L_n(| · −x|^β, x) =O_r,x(n⁻¹), (n → ∞).

In the case that 0 < β ≤ 2, we have, by Jensen’s inequality (or Hölder’s inequality) and Lemma2.2(a),

Ln(| · −x|^β, x) =E|Zn,x−x|^β ≤ E(Zn,x−x)²β/2

≤

3x(1 +x)² n+ 2

^β₂ , for alln ≥1.

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References

[1] G. BLEIMANN, P.L. BUTZERANDL. HAHN, A Bernstein-type operator approximating continuous functions on the semi axis, Indag. Math., 42 (1980), 255–262.

[2] V. TOTIK, Uniform approximation by Bernstein-type operators, Indag.

Math., 46 (1984), 87–93.

[3] R.A. KHAN, A note on a Bernstein type operator of Bleimann, Butzer and Hahn, J. Approx. Theory, 53 (1988), 295–303.

[4] R.A. KHAN, Some properties of a Bernstein type operator of Bleimann, Butzer and Hahn, In Progress in Approximation Theory, (Edited by P.

Nevai and A. Pinkus), pp. 497–504, Academic Press, New York (1991).

[5] J.A. ADELL AND J. DE LA CAL, Preservation of moduli of continuity for Bernstein-type operators, In Approximation, Probability, and Related Fields, (Edited by G. Anastassiou and S.T. Rachev), pp. 1–18, Plenum Press, New York (1994).

[6] J.A. ADELL AND J. DE LA CAL, Bernstein-type operators diminish the φ-variation, Constr. Approx., 12 (1996), 489–507.

[7] J.A. ADELL, F.G. BADíA ANDJ. DE LA CAL, On the iterates of some Bernstein-type operators, J. Math. Anal. Appl., 209 (1997), 529–541.

[8] B. DELLA VECCHIA, Some properties of a rational operator of Bernstein type, In Progress in Approximation Theory, (Edited by P. Nevai and A.

Pinkus), pp. 177–185, Academic Press, New York (1991).

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