View of Convergence of Szász-Mirakyan type operators

Rev. Anal. Num´er. Th´eor. Approx., vol. 36 (2007) no. 1, pp. 107–113 ictp.acad.ro/jnaat

CONVERGENCE OF SZ ´ASZ-MIRAKYAN TYPE OPERATORS

ZBIGNIEW WALCZAK^∗

Abstract. We introduce certain modification of Sz´asz-Mirakyan operators and we study approximation properties of these operators. The result is in a form convenient for applications.

MSC 2000. 41A36.

Keywords. Sz´asz-Mirakyan operator, polynomial weighted spaces, order of ap- proximation.

1. INTRODUCTION

Approximation properties of Sz´asz-Mirakyan operators (1) S_n(f;x) := e^−nx

∞

X

k=0 (nx)^k

k! f_n^k,

x ∈ R0 := [0,+∞], n ∈ N := {1,2,· · · }, in polynomial weighted spaces C_p were examined in [1]. The space Cp,p∈N0 :={0,1,2, . . .}, considered in [1]

is associated with the weighted function

(2) w₀(x) := 1, w_p(x) := (1 +x^p)⁻¹, if p≥1,

and consists of all real-valued functions f, continuous on R0 and such that w_pf is uniformly continuous and bounded on R0. The norm on C_p is defined by the formula

(3) kfk_p≡ kf(·)k_p:= sup

x∈R0

wp(x)|f(x)|.

In this section we shall give some properties of the above operators, which we shall apply to the proof of the theorem. Most of these can be found in [1].

A. Snf is a positive linear operatorCp →Cp. B. S_n(1;x) = 1, S_nf preserves constants.

C. Forf ∈Cp,p∈N0,

(4) w_p(x)|S_n(f;x)−f(x)| ≤K₁(p)ω₂f;C_p;^qx_n, x∈R0, n∈N, whereω₂(f;·) is the modulus of smoothness of the order 2 andK₁(p) is a positive constant.

∗Institute of Mathematics, Pozna´n University of Technology, Piotrowo 3A, 60-965 Pozna´n, Poland, e-mail: [email protected].

D. For f ∈C_p²:={f ∈Cp:f⁰, f⁰⁰ ∈Cp},p∈N0,

n→∞lim n(S_n(f;x)−f(x)) = ^x₂f⁰⁰(x), x∈R0. E. For every 2≤q ∈Nwe have

(5) Sn((t−x)^q;x) =

[q/2]

X

j=1

cj,q

x^j

n^q−j, x∈R0, n∈N,

wherecj,q are positive numerical coefficients depending only onj and q ([y] denotes the integral part of y∈R0) (see [17, 20]).

From (4) it was deduced that

(6) lim

n→∞S_n(f;x) =f(x),

for every f ∈ C_p, p ∈ N0 and x ∈ R0. Moreover, the above convergence is uniform on every interval [x1, x2], x1 ≥0.

The Sz´asz-Mirakyan operators and their connections with different branches of analysis, such as convex and numerical analysis have been studied inten- sively. We refer the readers to P. Gupta and V. Gupta [8], N. Ispir and C.

Atakut [15], V. Gupta, V. Vasishtha and M. K. Gupta [12], G. Feng [6], [7], A. Ciupa [3], J. Grof [14]. Their results improve other related results in the literature.

The actual construction of the Sz´asz-Mirakyan operators requires estima- tions of infinite series which in a certain sense restrict their usefulness from the computational point of view. Thus the question arises, whether the Sz´asz- Mirakyan operators and their generalizations cannot be replaced by a finite sum, provided this will not change the order of approximation. In connection with this question in the paper [16] were considered certain positive linear operators for function of one variable

(7) S_n(f;a_n;x) := e^−nx

[n(x+an)]

X

k=0

(nx)^k

k! f^k_n, x∈R0, n∈N, where (an)^∞₁ is a sequence of positive numbers such that lim_n→∞√

nan=∞.

In [16] it was proved that if f ∈C_p,p∈N0, then

(8) lim

n→∞S_n(f;a_n;x) =f(x)

for every f ∈ Cp, p ∈ N0 and x ∈ R0. Moreover, the above convergence is uniform on every interval [x₁, x₂], x₁ ≥0.

Similar results in exponential weighted spaces can be found in [23].

The construction introduced the operators (7) not change the order of ap- proximation.

In the paper [21] it was examined similar approximation problems for the following operators

L_n(f;x) := _(1+(x+n¹−1)²)ⁿ n

X

k=0 n k

(x+n⁻¹)^2kf^k(1+(x+n_nx+1⁻¹⁾²⁾, f ∈C_p,p∈N0,x∈R0 andn∈N.

Thus the new question arises, whether the order of approximation given in C and D cannot be improved.

In connection in this question we propose a new family of linear operators.

This together with the form of the operator makes results, given in the present paper, more helpful from the computational point of view.

In this paper we shall denote the suitable positive constants depending only on α by K_i(α), i= 1,2, ....

2. MAIN RESULTS

Similarly as in the paper [20] (see also [22]) let Dp, p ∈ N, be the set of all real-valued functions f(x), continuous on R0 for which wp(x)x^kf^(k)(x), k= 0,1,2, . . . , p, are continuous and bounded onR0 and f^(p)(x) is uniformly continuous on R0. The norm on D_p,p∈N, is given by (3).

Approximation properties of modified Sz´asz-Mirakyan operators (9) A_n(f;p;x) := e^−nx

∞

X

k=0 (nx)^k

k!

p

X

j=0 f^(j)

_k

n

j!

x−^k_n^j, x∈R0, p∈N, inDp were examined in [20].

In [20] it was obtained that if f ∈D_p,p∈N, then (10) kA_n(f;p;·)−f(·)k_p =O(n^−p/2).

The assertion (10) for the operators An and f ∈ C₀^r := {f ∈ C0 : f⁰, . . . , f^(r)∈C₀} is given in [17].

We introduce the following

Definition 1. For functionsf ∈Dp,p∈N, we define the operators Bn

(11) Bn(f;an;p;x) := e^−nx

[n(x+an)]

X

k=0

(nx)^k k!

p

X

j=0 f^(j)

_k

n

j!

x− ^k_n^j, x∈R0, where (an)^∞₁ is a sequence of positive numbers such that lim_n→∞√

nan=∞.

In this paper we shall study a relation between the order of approximation by B_n and the smoothness of the functionf.

Now we shall give approximation theorem forBn.

The methods used in to prove Theorem are similar to those used in con- struction of modified Sz´asz-Mirakyan operators [14, 16, 23].

Theorem 2. Fix p∈N. Then for Bn defined by (11) we have

(12) lim

n→∞{B_n(f;a_n;p;x)−f(x)}= 0, f ∈D_p, uniformly on every interval [x₁, x₂], x₂ > x₁ ≥0.

Proof. We first suppose thatf ∈Dp,p∈N. From (11) and (9) we obtain Bn(f;an;p;x)−f(x) =

= e^−nx

[n(x+an)]

X

k=0

(nx)^k k!

p

X

j=0 f^(j)(^k_n)

j!

x−_n^kj−f(x)

= e^−nx

∞

X

k=0 (nx)^k

k!

p

X

j=0 f^(j)

_k

n

j!

x−_n^kj−f(x)

−e^−nx

∞

X

k=[n(x+an)]+1 (nx)^k

k!

p

X

j=0 f^(j)

_k

n

j!

x−^k_n^j

=A_n(f;p;x)−f(x)−M_n(f;x), x∈R0, n∈N.

By our assumption, using the elementary inequality (a+b)^k ≤2^k−1(a^k+b^k), a, b >0,k∈N0, we get

|f^(k)(t)| ≤K₂(1 +t^p−k) (13)

≤K₂(1 + (|t−x|+x)^p−k)

≤K₂(1 + 2^p−k−1(|t−x|^p−k+x^p−k)), k= 0,1,2. . . , p.

From this and by (1) we get

|M_n(f;x)| ≤

≤e^−nx

∞

X

k=[n(x+an)]+1 (nx)^k

k! K₂

p

X

j=0

1 + 2^p−j−1

k

n−x^p−j+x^p−j

k n−x^j

≤K₂

p

X

j=0



(1 + 2^p−j−1x^p−j)e^−nx

∞

X

k=[n(x+an)]+1 (nx)^k

k!

k

n−x^j+ +2^p−j−1e^−nx

∞

X

k=0 (nx)^k

k!

k n−x^p

!

=K₂

p

X

j=0

(1 + 2^p−j−1x^p−j)e^−nx

∞

X

k=[n(x+an)]+1 (nx)^k

k!

k n−x^j + 2^p−j−1Sn(|t−x|^p;x)

! .

We remark that e^−nx

∞

X

k=[n(x+an)]+1 (nx)^k

k!

k n−x^j

≤e^−nx

∞

X

an<|k/n−x|

(nx)^k k!

k

n−x^j ≤e^−nx

∞

X

an<|k/n−x|

(nx)^k k!

^k_n−x

p+j

a^pn

≤ ¹

a^pne^−nx

∞

X

k=0 (nx)^k

k!

k

n−x^p+j = ¹

a^pnSn

|t−x|^p+j;x.

From this and in view of (5), the H¨older inequality and the propertySn(1;x) = 1, we further have

|M_n(f;x)| ≤K3

^p X

j=0

1+2^p−j−1x^p−j a^pn

Sn

(t−x)^2p+2j;x^1/2 + 2^p−1Sn

(t−x)^2p;x^1/2

=K₃







p

X

j=0

1+2^p−j−1x^p−j a^pn







p+j

X

i=1

c_i,2p+2jn2p+2j−i^xi







1/2

+2^p−1 ( _p

X

i=1

ci,2p xⁱ n^2p−i

)1/2



≤ ^K4

n^p/2







1 a^pn

p

X

j=0

(1 + 2^p−j−1x^p−j)







p+j

X

i=1

c_i,2p+2jxⁱ







1/2

+2^p−1 ( _p

X

i=1

c_i,2pxⁱ )1/2

,

whereK3,K4are some positive constants depending only onp. The properties of a_n

n→∞lim

√nan=∞ imply that

n→∞lim M_n(f;x) = 0

uniformly on every interval [x₁, x₂], x₂ > x₁ ≥0. From this and by (10) we obtain

n→∞lim {B_n(f;a_n;p;x)−f(x)}= 0,

uniformly on every interval [x1, x2], x2 > x1 ≥ 0. This ends the proof of

(12).

3. REMARKS

Applying (10), the equality

Bn(f;an;p;x)−f(x) =An(f;p;x)−f(x)−Mn(f;x)

and arguing as in the proof of Theorem it is easy verified that operatorsB_n, n ∈ N, give better the order of approximation (O(n^−p/2)) function f ∈ D_p, p∈N, thanSn and the operators examined in [16, 21, 23] (O(n⁻¹)).

Observe that analogous approximation properties hold for the following operator

(14) Cn(f;p;x) := e^−nx

n

X

k=0 (nx)^k

k!

p

X

j=0 f^(j)

_k

n

j!

x− ^k_n^j, f ∈C_[0,1],x∈[0,1), n∈N.

We may remark here that the operator Cn,n∈ N, obtained from (11) for a_n= 1−x,x∈[0,1).

REFERENCES

[1] Becker, M.,Global approximation theorems for Szasz–Mirakyan and Baskakov opera- tors in polynomial weight spaces, Indiana Univ. Math. J.,27(1), pp. 127–142, 1978.

[2] Becker, M., Kucharski, D.andNessel, R. J.,Global approximation theorems for the Szasz-Mirakjan operators in exponential weight spaces, Linear Spaces and Approxima- tion (Proc. Conf. Oberwolfach, 1977), Birkhäuser Verlag, Basel ISNM,40, pp. 319–333, 1978.

[3] Ciupa, A., Approximation by a generalized Szasz type operators, J. Comput. Anal.

Appl.,5(4), pp. 413–424, 2003.

[4] De Vore, R. A. andLorentz, G. G.,Constructive Approximation, Springer-Verlag, Berlin, 1993.

[5] Ditzian, Z.and Totik, V.Moduli of Smoothness, Springer-Verlag, New York 1987.

[6] Feng, G.,Weighted approximation on Szasz-type operators, Anal. Theory Appl.,19(1), pp. 47–54, 2003.

[7] Feng, G., A characterization of pointwise approximation for linear combinations of Sz´asz-type operators, Chinese Quart. J. Math.,19(4), pp. 379–384, 2004.

[8] Gupta, P.andGupta, V.,Rate of convergence on Baskakov-Szasz type operators, Fasc.

Math.,31, pp. 37–44, 2001.

[9] Gupta, V., An estimate on the convergence of Baskakov-B´ezier operators, J. Math.

Anal. Appl., 312(1), pp. 280–288, 2005.

[10] Gupta, V.andMaheshwari, P.,On Baskakov-Szasz type operators, Kyungpook Math.

J.,43(3), pp. 315–325, 2003.

[11] Gupta, V. and Pant, R.P.,Rate of convergence of the modified Szasz-Mirakyan op- erators on functions of bounded variation, J. Math. Anal. Appl.,233(2), pp. 476–483, 1999.

[12] Gupta, V., Vasishtha, V. and Gupta, M.K. Rate of convergence of the Szasz- Kantorovitch-Bezier operators for bounded variation functions, Publ. Inst. Math.

(Beograd)(N.S.),72(86), pp. 137–143, 2002.

[13] Guo, S., Li, C., Sun, Y., Yand, G. andYue, S.,Pointwise estimate for Szasz-type operators, J. Approx. Th.,94, pp. 160–171, 1998.

[14] Grof, J.,Uber Approximation durch Polynome mit Belegfunktionen, Acta Math. Acad.¨ Sci. Hungar.,35, pp. 109–116, 1980.

[15] Ispir, N. and Atakut, C., Approximation by modified Szasz-Mirakyan operators on weighted spaces, Proc. Indian acad. Sci. Math. Sci.,112(4), pp. 571–578, 2002.

[16] Lehnhoff, H.G.,On a Modified Szasz-Mirakjan Operator, J. Approx. Th.,42, pp. 278–

282, 1984.

[17] Rempulska, L.andWalczak, Z.,Modified Szasz-Mirakyan operators, Mat. Balcanica (New Ser.),18(1–2), pp. 53–63, 2004.

[18] Totik, V., Uniform approximation by Szasz-Mirakyan type operators, Acta Math.

Hung.,41(3–4), pp. 291–307, 1983.

[19] Walczak, Z.,On certain positive linear operators in weighted polynomial spaces, Acta Math. Hungar.,1013, pp. 179–191, 2003.

[20] Walczak, Z., On the rate of convergence for modified Szasz-Mirakyan operators, Analele Universit˘at¸ii din Timi¸soara,61(2), pp. 141–149, 2003.

[21] Walczak, Z.,Approximation by some linear positive operators in polynomial weighted spaces, Publ. Math. Debrecen,64(3–4), pp. 353–367, 2004.

[22] Walczak, Z.,On the rate of convergence for some linear operators, Hiroshima Math.

J.,35(1), pp. 115–124, 2005.

[23] Xiehua, S.,On the convergence of the modified Szasz-Mirakjan operator, Approx. The- ory Appl., 10(1), pp. 20–25, 1994.

Received by the editors: July 11, 2006.