(1)APPROXIMATION BY MODIFIED SZÁSZ-MIRAKYAN OPERATORS L

APPROXIMATION BY MODIFIED SZÁSZ-MIRAKYAN OPERATORS

L. REMPULSKA AND S. GRACZYK INSTITUTE OFMATHEMATICS

POZNANUNIVERSITY OFTECHNOLOGY UL. PIOTROWO3A, 60-965 POZNAN

POLAND

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

Received 23 December, 2008; accepted 28 May, 2009 Communicated by I. Gavrea

ABSTRACT. We introduce the modified Szász-Mirakyan operators Sn;r related to the Borel methodsBr of summability of sequences. We give theorems on approximation properties of these operators in the polynomial weight spaces.

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

2000 Mathematics Subject Classification. 41A36, 41A25.

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,

forx ∈ R0. TheC_p is the set of all functionsf : R0 → R(R = (−∞,∞))for whichf w_p is uniformly continuous and bounded onR0 and the norm is defined by

(1.3) kfk_p ≡ kf(·)k_p := sup

x∈R0

w_p(x)|f(x)|.

003-09

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

In [1] it was proved thatS_n is a positive linear operator acting from the spaceC_p toC_p for every p ∈ N0. Moreover, for a fixed p ∈ N0 there exist M_k(p) = const. > 0, k = 1,2, depending only onpsuch that for everyf ∈C_p there hold the inequalities:

(1.4) kS_n(f)k_p ≤M₁(p)kfk_p for n∈N, 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;C_p;·)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

A_r(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.

Clearly A₁(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 ∈ R⁰ and n ∈ N. (The operators S_n;2 were investigated in [9] for functions belonging to exponential weight spaces.)

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

k=0 x^rk

(rk)!akis convergent onRand

x→∞lim re^−x

∞

X

k=0

x^rk

(rk)!a_k =g.

In Section 2 we shall give some elementary properties ofS_n;r. The approximation theorems will be given in Section 3.

2. AUXILIARY RESULTS

It is known ([1]) that fore_k(x) = x^k,k = 0,1,2,there holds: S_n(e₀;x) = 1,S_n(e₁;x) = x andS_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 fixed q ∈ N, there exists a polynomial P_q(x) = Pq

k=0a_kx^k with 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 that S_n;r is a positive linear operator well defined on every spaceC_p,p∈N0, and

S_n;r(e₀;x) = 1, (2.3)

S_n;r(e₁;x) = x n

A⁰_r(nx) A_r(nx), (2.4)

S_n;r(e₂;x) = x² n²

A⁰⁰_r(nx) Ar(nx) + x

n²

A⁰_r(nx) Ar(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 ∈ N be a fixed number. Then Ar defined 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

# , 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 forA₁andA₂ 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) +A_r(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.

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⁻¹ transform 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_kcan be written as follows

(2.13) L⁻¹

P(s) Q(s)

=X

sk

∗P(s_k)e^skt

Q⁰(s_k) + 2reX

sk

∗∗P(s_k)e^skt Q⁰(s_k) , whereP∗

denotes the sum for all reals_kandP∗∗

denotes the sum for all complexs_k=x_k+iy_k with a positivey_k.

The functionY defined by (2.11) has the simple poless_k=√^r

1 = e^2kπi/rfork = 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^skt+ 2reX

sk

∗∗

e^skt

!

= 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 Lemma 2.1, we immediately obtain the following:

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

(2.14) 1≤ e^nx

A_r(nx) ≤M₃(r) for x∈R0, n∈N. 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) S_n;r (e₁(t)−e₁(x))^2q;x

≤M₃(r)S_n (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 A_2m(nx)

1

nA⁰_2m(nx)−A_2m(nx)

= 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_pto C_pfor everyp∈N0. Forf ∈C_p

kSn;r(f)kp ≤ kfkpkSn;r(1/wp)kp

≤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).

3. THEOREMS

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

Theorem 3.1. Letp ∈ N⁰ and 2 ≤ r ∈ N be fixed numbers. Then there exists M₅(p, r) = const. >0(depending only onpandr) such that for everyf ∈C_p¹there 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 kf⁰k_p

rx

n for x∈R0, n∈N.

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

Theorem 3.2. Letp ∈ N⁰ and2 ≤ r ∈ Nbe fixed. Then there existsM₆(p, r) = const. > 0 (depending only onpandr) such that for everyf ∈C_p,x∈R0 andn ∈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;Cp;·)is the modulus of continuity off ∈Cp, 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_p andx >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_h belongs 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)|, forn ∈Nandh >0. Next, by Lemma 2.4 and (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 Theorem 3.1 and (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

, for x > 0, n ∈ N and h > 0. Puttingh = p

x/n in (3.6) for given x andn, we obtain the

desired estimation (3.2).

Theorem 3.2 implies 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 operators S_n can be extended to S_n;r withr ≥2.

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

(3.7) lim

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

Proof. The statement (3.7) for x = 0 is obvious by (2.6). Choosing x > 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 toCp andlim

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 Lemma 2.3 implies 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 Corollary 3.3

(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(n²Sn;r((t−x)⁴;x))^∞₁ is bounded at every fixedx∈R⁰. From this and (3.9) and (3.10) we get

n→∞lim nSn;r ϕ(t)(t−x)²;x

= 0

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

4. REMARKS

Remark 1. We observe that the estimation (1.5) for the operatorsSnis better than (3.2) obtained forSn;r withr ≥ 2. It is generated by formulas (2.3) – (2.5) and Lemma 2.1 which show that the operatorsS_n;r,r ≥ 2, preserve only the functione₀(x) = 1. The operatorsS_npreserve the functione_k(x) =x^k,k = 0,1.

Remark 2. In the paper [2], the approximation properties of the Szász-Mirakyan operatorsS_n in the exponential weight spaces C_q^∗, q > 0, with the weight functionv_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 for m-times differentiable functions and showed that these operators have better approximation properties than classical Bernstein polynomials.

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

We mention that the Kirov method can be extended to the operatorsSn;rwithr ≥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 theS_n;1^∗ have better approximation properties forf ∈C_p^m,m ≥2, thanS_n;1.

REFERENCES

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

[2] M. BECKER, D. KUCHARSKIANDR.J. NESSEL, Global approximation theorems for the Szász- Mirakyan operators in exponential weight spaces, Linear Spaces and Approximation (Proc. Conf.

Oberwolfach, 1977), Birkhäuser Verlag, 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. DITZIANANDV. TOTIK, Moduli of Smoothness, Springer-Verlag, New-York, 1987.

[6] P. GUPTAANDV. GUPTA, Rate of convergence on Baskakov-Szász type operators, 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 ´SNIEWICZ AND L. 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 (Bal- canica), 18 (2004), 53–63.

[11] O. SZÁSZ, Generalizations of S. Bernstein’s polynomials to the infinite interval, 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.