APPROXIMATION BY MODIFIED SZÁSZ-MIRAKYAN OPERATORS

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Szász-Mirakyan Operators L. Rempulska and S. Graczyk

vol. 10, iss. 3, art. 61, 2009

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APPROXIMATION BY MODIFIED SZÁSZ-MIRAKYAN OPERATORS

L. REMPULSKA AND S. GRACZYK

Institute of Mathematics Poznan University of Technology ul. Piotrowo 3a, 60-965 Poznan Poland

EMail:lucyna.rempulska@put.poznan.pl szymon.graczyk@pisop.org.pl

Received: 23 December, 2008

Accepted: 28 May, 2009

Communicated by: I. Gavrea 2000 AMS Sub. Class.: 41A36, 41A25.

Key words: Szász-Mirakyan operator, Polynomial weight space, Order of approximation, Voronovskaya type theorem.

Abstract: We introduce the modified Szász-Mirakyan operatorsSn;rrelated to the Borel methodsBrof summability of sequences. We give theorems on approximation properties of these operators in the polynomial weight spaces.

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1. Introduction

The approximation of functions by Szász-Mirakyan operators (1.1) S_n(f;x) :=e^−nx

∞

X

k=0

(nx)^k k! f

k n

, x∈R0, n∈N,

(R0 = [0,∞),N ={1,2, ...})has been examined in many papers and monographs (e.g. [11], [1], [2], [4], [5]).

The above operators were modified by several authors (e.g. [3], [6], [9], [10], [12]) which showed that new operators have similar or better approximation properties thanS_n. M. Becker in the paper [1] studied approximation problems for the operatorsS_nin the polynomial weight spaceC_p,p∈N0 =N∪ {0}, connected with the weight functionw_p,

(1.2) w₀(x) := 1, w_p(x) := (1 +x^p)⁻¹ if p∈N,

for x ∈ R0. The C_p is the set of all functionsf : R0 → R(R = (−∞,∞)) for whichf w_p is uniformly continuous and bounded onR0and the norm is defined by (1.3) kfk_p ≡ kf(·)k_p := sup

x∈R0

w_p(x)|f(x)|.

The space C_p^m, m ∈ N, p ∈ N0, of m-times differentiable functions f ∈ C_p with derivativesf^(k)∈C_p,1≤k≤m, and the norm (1.3) was considered also in [1].

In [1] it was proved thatSnis a positive linear operator acting from the spaceCpto Cp for everyp∈N⁰. Moreover, for a fixedp∈N⁰ there existMk(p) =const. >0, k = 1,2, depending only onpsuch that for everyf ∈C_p there hold the inequalities:

(1.4) kSn(f)kp ≤M1(p)kfkp for n∈N,

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and

(1.5) w_p(x)|S_n(f;x)−f(x)| ≤M₂(p)ω₂

f;C_p; rx

n

, x∈R0, n∈N, whereω2(f;Cp;·)is the second modulus of continuity off.

In this paper we introduce the following modified Szász-Mirakyan operators (1.6) S_n;r(f;x) := 1

Ar(nx)

∞

X

k=0

(nx)^rk (rk)! f

rk n

, x∈R0, n∈N, forf ∈C_p and every fixedr∈N, where

(1.7) A_r(t) :=

∞

X

k=0

t^rk

(rk)! for t∈R0.

ClearlyA₁(t) = e^t, A₂(t) = cosht ≡ ¹₂(e^t+e^−t)and S_n;1(f;x) ≡ S_n(f;x) for f ∈C_p,x∈R0andn∈N. (The operatorsS_n;2were investigated in [9] for functions belonging to exponential weight spaces.)

We mention that the definition ofS_n;ris related to the Borel method of summability of sequences. It is well known ([7]) that a sequence(a_n)^∞₀ ,a_n ∈R, is summable tog by the Borel methodB_r, r ∈ N, if the seriesP∞

k=0 x^rk

(rk)!a_k is convergent on R and

x→∞lim re^−x

∞

X

k=0

x^rk

(rk)!a_k =g.

In Section2we shall give some elementary properties ofS_n;r. The approximation theorems will be given in Section3.

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2. Auxiliary Results

It is known ([1]) that for e_k(x) = x^k, k = 0,1,2, there holds: S_n(e₀;x) = 1, S_n(e₁;x) =xandS_n(e₂;x) = x²+_n^x, which imply that

(2.1) S_n (e₁(t)−e₁(x))²;x

= x

n for x∈R0, n∈N. Moreover, for every fixedq∈N,there exists a polynomialP_q(x) = Pq

k=0a_kx^kwith real coefficientsa_k,a_q 6= 0, depending only onqsuch that

(2.2) S_n (e₁(t)−e₁(x))^2q;x

≤ P_q(x)n^−q for x∈R0, n ∈N.

From (1.1) – (1.4), (1.6) and (1.7) we deduce thatS_n;ris a positive linear operator well defined on every spaceC_p,p∈N0, and

S_n;r(e₀;x) = 1, (2.3)

Sn;r(e1;x) = x n

A⁰_r(nx) A_r(nx), (2.4)

S_n;r(e₂;x) = x² n²

A⁰⁰_r(nx) A_r(nx) + x

n²

A⁰_r(nx) A_r(nx), (2.5)

forx∈R0 andn, r ∈N, and

(2.6) S_n;r(f; 0) =f(0) for f ∈C_p, n, r ∈N. Here we derive a simpler formula forA_r.

Lemma 2.1. Letr ∈Nbe a fixed number. ThenA_rdefined by (1.7) can be rewritten in the form:A₁(t) = e^t,A₂(t) = cosht,

(2.7) A_2m(t) = 1 m

"

cosht+

m−1

X

k=1

exp

tcoskπ m

cos

tsinkπ m

# ,

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for2≤m∈N, and (2.8) A_2m+1(t)

= 1

2m+ 1

"

e^t+ 2

m

X

k=1

exp

tcos 2kπ 2m+ 1

cos

tsin 2kπ 2m+ 1

# , form ∈Nandt ∈R⁰.

Proof. The formulas forA₁ andA₂ are obvious by (1.7). Forr ≥3andt ∈ R⁰ we have

e^t=

∞

X

k=0

t^rk (rk)! +

∞

X

k=0

t^rk+1

(rk+ 1)! +· · ·+

∞

X

k=0

t^rk+r−1 (rk+r−1)!

which by (1.7) can be written in the form e^t=A_r(t) +

Z t

0

A_r(u)du+ Z t

0

Z v1

0

A_r(u)du dv₁ +· · ·+

Z t

0

Z v1

0

. . . Z vr−2

0

A_r(u)du dvr−2. . . dv₁. By(r−1)-times differentiation we get the equality

e^t=A^(r−1)_r (t) +A^(r−2)_r (t) +· · ·+A⁰_r(t) +Ar(t) for t ∈R⁰, which shows thaty =A_r(t)is the solution of the differential equation (2.9) y^(r−1)+y^(r−2)+· · ·+y⁰+y=e^t

satisfying the initial conditions

(2.10) y(0) = 1, y⁰(0) =y⁰⁰(0) =· · ·=y^(r−2)(0) = 0.

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Using now the Laplace transformation L[y(t)] = Y(s) :=

Z ∞

0

y(t)e^−stdt, s=x+iy, we have by (2.10)

L

y^(k)(t)

=s^kY(s)−s^k−1 for k = 1, . . . , r−1, and consequently we get from (2.7)

s^r−1 +s^r−2 +· · ·+s+ 1

Y(s) = 1

s−1 +s^r−2+s^r−3+· · ·+s+ 1, and

(2.11) Y(s) = s^r−1

s^r−1. By the inverse Laplace transformation we get

(2.12) y(t) = L⁻¹

s^r−1 s^r−1

for t ∈R0, and thisL⁻¹transform can be calculated by the residues ofY.

It is known that the inverse transform of a rational function ^P_Q(s)^(s) with the simple poless_k can be written as follows

(2.13) L⁻¹

P(s) Q(s)

=X

sk

∗P(s_k)e^s^k^t

Q⁰(s_k) + 2reX

sk

∗∗P(s_k)e^s^k^t Q⁰(s_k) , whereP∗

denotes the sum for all reals_k andP∗∗

denotes the sum for all complex sk =xk+iykwith a positiveyk.

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The function Y defined by (2.11) has the simple poles s_k = √^r

1 = e^2kπi/r for k = 0,1, . . . , r−1. From this and (2.12) and (2.13) forr = 2m,2≤m∈N, we get

y(t) = 1 2m

X

sk

∗e^s^k^t+ 2reX

sk

∗∗e^s^k^t

!

= 1 m

"

cosht+

m−1

X

k=1

exp

tcoskπ m

cos

tsinkπ m

# . This shows that the formula (2.7) is proved.

Analogously by (2.12) and (2.13) we obtain (2.8).

From (2.7) and (2.8) we have that A₃(t) = 1

3 e^t+ 2e^−t/2cos

√3 2 t

!!

, A₄(t) = 1

2(cosht+ cost), A₆(t) = 1

3 cosht+ 2cosht 2cos

√3 2 t

!!

, for t∈R0.

Applying the formula (1.7) and Lemma2.1, we immediately obtain the following:

Lemma 2.2. For every fixedr ∈Nthere exists a positive constantM₃(r)depending only onrsuch that

(2.14) 1≤ e^nx

A_r(nx) ≤M3(r) for x∈R⁰, n∈N.

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Lemma 2.3. Letr∈N. Then fore₁(x) =xthere holds

(2.15) lim

n→∞nS_n;r(e₁(t)−e₁(x);x) = 0 and

n→∞lim nS_n;r (e₁(t)−e₁(x))²;x

=x, at everyx∈R0. Moreover, we have

(2.16) Sn;r (e1(t)−e1(x))^2q;x

≤M3(r)Sn (e1(t)−e1(x))^2q;x forx∈R⁰,n ∈Nand every fixedq∈N.

Proof. The inequality (2.16) is obvious by (1.1), (1.6) and (2.14).

We shall prove only (2.15) forr= 2m,m∈N. Ifr= 2thenA₂(t) = coshtand by (2.4) we have

S_n;2(e₁(t)−e₁(x);x) =x

sinhnx coshnx −1

= −2x

e^2nx+ 1 for x∈R0, n ∈N, which implies (2.15).

Ifr= 2mwith2≤m ∈N, then by (2.4), (2.7) and (2.14) we get

|S_n;2m(e₁(t)−e₁(x);x)|= x A2m(nx)

1

nA⁰_2m(nx)−A_2m(nx)

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= x

mA_2m(nx)

sinhnx−coshnx +

m−1

X

k=1

exp

nxcoskπ

m coskπ m cos

nxsinkπ m

−sinkπ m sin

nxsinkπ m

−cos

nxsinkπ m

≤M₃(2m)x m

"

e^−2nx+ 3

m−1

X

k=1

exp

−2nxsin² kπ m

#

and from this we immediately obtain (2.15).

From (1.6), (1.1) – (1.4) and (2.14) the following lemma results.

Lemma 2.4. The operatorS_n;r, n, r ∈ N, is linear and positive, and acts from the spaceC_ptoC_p for everyp∈N0. Forf ∈C_p

kS_n;r(f)k_p ≤ kfk_pkS_n;r(1/w_p)k_p

≤M₃(r)kfk_p· kS_n(1/w_p)k_p ≤M₄(p, r)kfk_p f or n, r,∈N, where M₄(p, r) = M₁(p)M₃(r)and M₁(p), M₃(r) are positive constants given in (1.4) and (2.14).

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3. Theorems

First we shall prove two theorems on the order of approximation off ∈C_p byS_n;r, r≥2.

Theorem 3.1. Let p ∈ N0 and 2 ≤ r ∈ N be fixed numbers. Then there exists M₅(p, r) = const. > 0 (depending only onp and r) such that for every f ∈ C_p¹ there holds the inequality

(3.1) w_p(x)|S_n;r(f;x)−f(x)| ≤M₅(p, r)kf⁰k_p rx

n , forx∈R⁰ andn∈N.

Proof. Letf ∈C_p¹. Then by (1.6), (1.7) and (2.14) it follows that

|Sn;r(f;x)−f(x)| ≤Sn;r(|f(t)−f(x)|;x)

≤M₃(r)S_n(|f(t)−f(x)|;x) for x∈R0, n∈N, and fort, x∈R⁰

|f(t)−f(x)|=

Z t

x

f⁰(u)du

≤ kf⁰k_p 1

w_p(t)+ 1 w_p(x)

|t−x|.

Using now the operatorS_n, (1.1) – (1.4) and (2.1), we get w_p(x)S_n(|f(t)−f(x)|;x)≤ kf⁰k_p

w_p(x)S_n

|t−x|

wp(t) ;x

+S_n(|t−x|;x)

≤ kf⁰k_p S_n (t−x)²;x1/2n

2kS_n(1/w_2p)k^1/2_2p + 1o

≤ 2p

M1(2p) + 1

kf⁰kp

rx

n for x∈R⁰, n∈N.

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Combining the above, we obtain the estimation (3.1).

Theorem 3.2. Letp ∈ N⁰ and2 ≤ r ∈ N be fixed. Then there existsM₆(p, r) = const. > 0(depending only on pand r) such that for every f ∈ C_p, x ∈ R0 and n∈Nthere holds

(3.2) w_p(x)|S_n;r(f;x)−f(x)| ≤M₆(p, r)ω₁

f;C_p; rx

n

, whereω₁(f;C_p;·)is the modulus of continuity off ∈C_p, i.e.

(3.3) ω₁(f;C_p;t) := sup

0≤u≤t

k∆_uf(·)k_p for t≥0, and∆_uf(x) = f(x+u)−f(x).

Proof. The inequality (3.2) forx= 0follows by (1.2), (2.6) and (3.3).

Letf ∈C_pandx >0. We use the Steklov functionf_h, f_h(x) := 1

h Z h

0

f(x+t)dt for x∈R0, h >0.

Thisf_hbelongs to the spaceC_p¹and by (3.3) it follows that (3.4) kf −fhkp ≤ω1(f;Cp;h) and

(3.5) kf_h⁰k_p ≤h⁻¹ω₁(f;C_p;h), for h >0.

By the above properties off_hand (2.3) we can write S_n;r f(t);x

−f(x)

≤ |S_n;r(f(t)−f_h(t);x)|+

S_n;r f_h(t);x

−f_h(x)

+|f_h(x)−f(x)|,

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forn ∈Nandh >0. Next, by Lemma2.4and (3.4) we get w_p(x)

S_n;r f(t)−f_h(t);x

≤M₄(p, r)kf −f_hk_p ≤M₄(p, r)ω₁(f;C_p;h).

In view of Theorem3.1and (3.5) we have w_p(x)

S_n;r f_h;x

−f_h(x)

≤M₅(p, r)kf_h⁰k_p rx

n ≤M₅(p, r)h⁻¹ rx

nω₁(f;C_p;h).

Consequently,

(3.6) w_p(x)|S_n;r(f;x)−f(x)|

≤ω₁(f;C_p;h)

M₄(p, r) +M₅(p, r)h⁻¹ rx

n + 1

, forx > 0, n ∈ Nand h > 0. Puttingh = p

x/n in (3.6) for given xand n, we obtain the desired estimation (3.2).

Theorem3.2implies the following:

Corollary 3.3. Iff ∈C_p,p∈N0, and2≤r ∈N, then

n→∞lim S_n;r(f;x) = f(x) at every x∈R0. This convergence is uniform on every interval[x₁, x₂],x₁ ≥0.

The Voronovskaya type theorem given in [1] for the operatorsS_ncan be extended toS_n;r withr≥2.

Theorem 3.4. Suppose thatf ∈C_p²,p∈N0, and2≤r ∈N. Then

(3.7) lim

n→∞n(S_n;r(f;x)−f(x)) = x 2f⁰⁰(x) at everyx∈R⁰.

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Proof. The statement (3.7) forx = 0is obvious by (2.6). Choosingx > 0, we can write the Taylor formula forf ∈C_p²:

f(t) = f(x) +f⁰(x) + 1

2f⁰⁰(x)(t−x)²+ϕ(t, x)(t−x)² for t∈R⁰, whereϕ(t)≡ϕ(t, x)is a function belonging toC_pandlim

t→xϕ(t) =ϕ(x) = 0.

Using now the operatorS_n;rand (2.3), we get S_n;r(f(t);x) =f(x) +f⁰(x)S_n;r(t−x;x)

+ 1

2f⁰⁰(x)S_n;r((t−x)²;x) +S_n;r ϕ(t)(t−x)²;x , forn ∈N, which by Lemma2.3implies that

(3.8) lim

n→∞n(S_n;r(f(t);x)−f(x)) = x

2f⁰⁰(x) + lim

n→∞nS_n;r ϕ(t)(t−x)²;x . It is clear that

(3.9)

S_n;r ϕ(t)(t−x)²;x

≤ S_n;r(ϕ²(t);x)S_n;r((t−x)⁴;x)1/2

, and by Corollary3.3

(3.10) lim

n→∞S_n;r(ϕ²(t);x) = ϕ²(x) = 0.

Moreover, by (2.16) and (2.2) we deduce that the sequence (n²S_n;r((t−x)⁴;x))^∞₁ is bounded at every fixedx∈R0. From this and (3.9) and (3.10) we get

n→∞lim nS_n;r ϕ(t)(t−x)²;x

= 0 which with (3.8) yields the statement (3.7).

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4. Remarks

Remark 1. We observe that the estimation (1.5) for the operators S_n is better than (3.2) obtained for S_n;r with r ≥ 2. It is generated by formulas (2.3) – (2.5) and Lemma2.1 which show that the operators S_n;r, r ≥ 2, preserve only the function e0(x) = 1. The operatorsSnpreserve the functionek(x) =x^k,k= 0,1.

Remark 2. In the paper [2], the approximation properties of the Szász-Mirakyan operatorsS_nin the exponential weight spaces C_q^∗, q > 0, with the weight function v_q(x) = e^−qx, x ∈R⁰ were examined. Obviously the operatorsS_n;r,r ≥ 2, can be investigated also in these spaces.

Remark 3. G. Kirov in [8] defined the new Bernstein polynomials form-times differentiable functions and showed that these operators have better approximation properties than classical Bernstein polynomials.

The Kirov idea was applied to the operatorsS_nin [10].

We mention that the Kirov method can be extended to the operators S_n;r with r ≥ 2, i.e. for functionsf ∈ C_p^m, m ∈ N, p ∈ N0, and a fixed2 ≤ r ∈ Nwe can consider the operators

S_n;r^∗ (f;x) := 1 A_r(nx)

∞

X

k=0

(nx)^rk (rk)!

m

X

j=0

f^(j) ^rk_n j!

rk n −x

j

, forx∈R0 andn∈N.

In [10] it was proved that the S_n;1^∗ have better approximation properties for f ∈ C_p^m,m≥2, thanS_n;1.

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References

[1] M. BECKER, Global approximation theorems for Szász-Mirakyan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27(1) (1978), 127–142.

[2] M. BECKER, D. KUCHARSKIANDR.J. NESSEL, Global approximation the- orems for the Szász-Mirakyan operators in exponential weight spaces, Linear Spaces and Approximation (Proc. Conf. Oberwolfach, 1977), Birkhäuser Ver- lag, Basel, Internat. Series of Num. Math., 40 (1978), 319–333.

[3] A. CIUPA, Approximation by a generalized Szász type operator, J. Comput.

Anal. and Applic., 5(4) (2003), 413–424.

[4] R.A. DE VOREANDG.G. LORENTZ, Constructive Approximation, Springer- Verlag, Berlin, New York, 1993.

[5] Z. DITZIAN AND V. TOTIK, Moduli of Smoothness, Springer-Verlag, New- York, 1987.

[6] P. GUPTAANDV. GUPTA, Rate of convergence on Baskakov-Szász type op- erators, Fasc. Math., 31 (2001), 37–44.

[7] G.H. HARDY, Divergent Series, Oxford Univ. Press, Oxford, 1949.

[8] G.H. KIROV, A generalization of the Bernstein polynomials, Mathematica (Balcanica), 2(2) (1992), 147–153.

[9] M. LE ´SNIEWICZANDL. REMPULSKA, Approximation by some operators of the Szász-Mirakyan type in exponential weight spaces, Glas. Mat. Ser. III, 32(52)(1) (1997), 57–69.

[10] L. REMPULSKA AND Z. WALCZAK, Modified Szász-Mirakyan operators, Mathematica (Balcanica), 18 (2004), 53–63.

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[11] O. SZÁSZ, Generalizations of S. Bernstein’s polynomials to the infinite inter- val, J. Res. Nat. Bur. Standards, Sect. B, 45 (1950), 239–245.

[12] Z. WALCZAK, On the convergence of the modified Szász-Mirakyan operators, Yokohama Math. J., 51(1) (2004), 11–18.