On the rate of convergence for some linear operators

35 (2005), 115–124

On the rate of convergence for some linear operators

Zbigniew Walczak

(Received February 4, 2004) (Revised September 21, 2004)

Abstract. We consider certain linear operators Ln in polynomial weighted spaces of functions of one variable and study approximation properties of these operators, including theorems on the degree of approximation.

1. Introduction

Approximation properties of Sza´sz-Mirakyan operators ð1Þ

Snðf;xÞ:¼e^nxX^y

k¼0

ðnxÞ^k k! f k

n ; xAR0¼ ½0;þyÞ;nAN:¼ f1;2. . .g;

in polynomial weighted spaces C_p were examined in [1]. The space C_p, pA N0:¼ f0;1;2;. . .g, is associated with the weighted function

w0ðxÞ:¼1; wpðxÞ:¼ ð1þx^pÞ¹ if pb1;

ð2Þ

and consists of all real-valued continuous functions f on R₀ for which w_pf is uniformly continuous and bounded on R0. The norm on Cp is defined by

kfk_p1kfðÞk_p:¼ sup

xAR0

wpðxÞjfðxÞj:

ð3Þ

In [1] there were theorems on the degree of approximation of f AC_p by operators Sn defined by (1). From these theorems it was deduced that

n!ylim Snðf;xÞ ¼ fðxÞ

for every f AC_p, pAN₀, and xAR₀. Moreover, the above convergence is uniform on every interval ½x1;x2, x2>x1b0.

In this paper by M_kða;bÞ, we shall denote suitable positive constants depending only on indicated parameters a and b.

2000 Mathematics Subject Classification. 41A36

Key words and phrases. Linear operator, degree of approximation.

The Sza´sz-Mirakyan operators are important in approximation theory.

They have been studied intensively, in connection with di¤erent branches of analysis, such as numerical analysis. Recently in many papers various modifications of operators S_n were introduced. Approximation properties of modified Sza´sz-Mirakyan operators

B_nðf;r;xÞ:¼ 1 gððnxþ1Þ²;rÞ

X^y

k¼0

ðnxþ1Þ^2k

ðkþrÞ! f kþr nðnxþ1Þ

; xAR₀;n;rAN;

where

gðt;rÞ ¼X^y

k¼0

t^k

ðkþrÞ!; tAR₀; in polynomial weighted spaces were examined in [10].

In [10] it was proved that if f AC_p, pAN₀, then kBnðf;r;Þ fðÞk_paM0o1 f;Cp;1

n

; n;rAN;

where M0¼const: >0 and o1ðf;Cp;Þ is the modulus of continuity of f defined by

o1ðf;Cp;tÞ:¼ sup

0ahat

kDhfðÞk_p; tAR0;

where DhfðxÞ ¼ fðxþhÞ fðxÞ for h;xAR0. In particular, if f AC_p¹, pAN₀, then

kBnðf;r;Þ fðÞk_paM₁

n ; n;rAN;

where M1¼const: >0. The above inequalities estimate the rate of uniform convergence offBnðf;r;Þg. Similar results in exponential weighted spaces can be found in [9, 11].

The Sza´sz-Mirakyan operators Sn are defined in terms of a sample of the given function f on the points k=n, called knots, for kAN₀, nAN. For the operators introduced in [9–11] the knots are the numbers ðkþrÞ=ðnðnxþ1ÞÞ for kAN₀, nAN and xAR₀ (r being fixed).

Thus the question arises, whether the knots ðkþrÞ=ðnðnxþ1ÞÞ cannot be replaced by a given subset of points, which are independent of x, provided this will not change essentially the degree of convergence. In connection with this question we introduce the operators (4).

Let Bp, pAN, be the set of all real-valued continuous functions fðxÞ on R₀ for whichw_pðxÞx^kf^ðkÞðxÞ, k¼0;1;2. . .;p, are continuous and bounded on R0 and f^ðpÞðxÞ is uniformly continuous on R0. The norm on Bp is given by (3).

We introduce the following class of operators in Bp, pAN.

Definition. Let rAN, pAN, s>0 be fixed numbers. For functions f ABp we define the operators

L_nðf;p;r;s;xÞ ð4Þ

:¼ 1

gðn^sx;rÞ X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

X^p

j¼0

f^ðjÞ kþr n^s

xkþr n^s j

j! ; xAR₀;nAN;

where

gðt;rÞ ¼X^y

k¼0

t^k

ðkþrÞ!; tAR0: ð5Þ

Observe that

gð0;rÞ ¼1

r!; gðt;rÞ ¼ 1

t^r e^tX^r1

j¼0

t^j j!

!

if t>0:

In this paper we shall state some estimates of the rate of uniform con- vergence of the operators Ln, nAN.

Theorems given in [1, 4–7] are concerned with pointwise approximation.

We give theorems on the degree of approximation of functions in B_p by L_n with respect to norm estimates.

Let us introduce the notation

A_nðf;r;s;xÞ:¼ 1 gðn^sx;rÞ

X^y

k¼0

ðn^sxÞ^k

ðkþrÞ!f kþr n^s ð6Þ

for f ABp, pAN0, xAR0, n;rAN and s>0.

We shall apply the method used in [1, 3–11].

2. Auxiliary results

In this section we shall give some properties of the above operators, which we shall apply to the proofs of the main theorems.

From (5) and (6) we easily derive the following formulas A_nð1;r;s;xÞ ¼1;

ð7Þ

Anðt;r;s;xÞ ¼xþ 1

n^sðr1Þ!gðn^sx;rÞ; Anðt²;r;s;xÞ ¼x²þx

n^s 1þ 1

ðr1Þ!gðn^sx;rÞ

þ r

n^2sðr1Þ!gðn^sx;rÞ for every fixed rAN and for all nAN and xAR0.

Using (5), (6) and mathematical induction on qAN we can prove the following lemma (see [10]).

Lemma 1. Fix qAN0, rAN and s>0. Then there exist positive numbers aq;j depending only on j, q, and b_q;jðrÞ depending only on r, j and q, 0ajaq such that

A_nðt^q;r;s;xÞ ¼X^q

j¼0

x^j

n^sðqjÞ a_q;jþ b_q;jðrÞ gðn^sx;rÞ

ð8Þ

for all nAN and xAR₀. Moreovera_0;0¼1,b_0;0ðrÞ ¼0 and a_q;0¼b_q;qðrÞ ¼0, a_q;q¼1, b_q;0ðrÞ ¼ r^q1

ðr1Þ! for qAN.

Next we shall prove

Lemma 2. Let pAN₀, rAN and s>0 be fixed numbers. Then there exists a positive constant M21M2ðp;rÞ such that

kAnð1=wpðtÞ;r;s;Þk_paM2; nAN.

ð9Þ

Proof. The inequality (9) is obvious for p¼0 by (2), (3) and (7).

Let pAN. From (5) we get 1

gðt;rÞar! for tAR₀: ð10Þ

From (10) and by (2) and (6)–(8) we have

wpðxÞAnð1=wpðtÞ;r;s;xÞ ¼wpðxÞf1þAnðt^p;r;s;xÞg

¼ 1

1þx^pþX^p

j¼0

x^j

n^sðpjÞð1þx^pÞ ap;jþ b_p;jðrÞ gðn^sx;rÞ

a1þX^p

j¼0

x^j

1þx^pðap;jþr!b_p;jðrÞÞaM₂ðp;rÞ

for xAR0, nAN, s>0 and rAN, where M2ðp;rÞ is a positive constant depending only p and r. This completes the proof of Lemma 2. 9

Similarly we can prove the following lemma.

Lemma 3. Let pAN₀, rAN and s>0 be fixed numbers. Then there exists a positive constant M31M3ðp;rÞ such that

sup

xAR0

w_pðxÞx^kA_nð1=wpkðtÞ;r;s;xÞaM₃; nAN;k¼0;1;2;. . .;p:

Lemma 4. Fix pAN, rAN and s>0. Then for all xAR₀ and nAN we have

AnððxtÞ^p;r;s;xÞ ¼X^p

j¼1

x^j1

n^sðpjþ1Þ ap;jþ b_p;j gðn^sx;rÞ

ð11Þ ;

where a_p;j, b_p;j are numbers depending only on the paremeters r, j and p.

Proof. Let pb1. Applying Lemma 1, we get A_nððxtÞ^p;r;s;xÞ ¼ 1

gðn^sx;rÞ X^y

k¼0

ðn^sxÞ^k

ðkþrÞ! xkþr n^s p

¼ 1

gðn^sx;rÞ X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

X^p

i¼0

p

i ð1Þⁱx^pi kþr n^s i

¼X^p

i¼0

p

i ð1Þⁱx^piAnðtⁱ;r;s;xÞ

¼X^p

i¼0

p

i ð1Þⁱx^piXⁱ

j¼0

x^j

n^sðijÞ ai;jþ b_i;jðrÞ gðn^sx;rÞ

¼x^pþX^p

i¼1

p

i ð1Þⁱx^pi Xⁱ¹

j¼0

x^j

n^sðijÞ a_i;jþ b_i;jðrÞ gðn^sx;rÞ

þxⁱ

" #

¼x^pX^p

i¼0

p

i ð1ÞⁱþX^p

i¼1

p i ð1Þⁱ Xⁱ¹

j¼0

x^jþpi

n^sðijÞ ai;jþ b_i;jðrÞ gðn^sx;rÞ

:

Using the elementary identity

X^p

i¼0

p

i ð1Þⁱ¼ ð11Þ^p¼0; pAN;

we receive the representation AnððxtÞ^p;r;s;xÞ ¼X^p

i¼1

p

i ð1ÞⁱXⁱ

k¼1

x^kþpi1

n^sðikþ1Þ ai;k1þb_i;k1ðrÞ gðn^sx;rÞ

¼X^p

j¼1

x^j1 n^sðpjþ1Þ

X^p

i¼pjþ1

p

i ð1Þⁱ ai;iþjp1þb_i;iþjp1ðrÞ gðn^sx;rÞ

¼X^p

j¼1

x^j1

n^sðpjþ1Þ a_p;jþ b_p;j gðn^sx;rÞ

: 9

Lemma 5. Fix p;rAN and s>0. Then there exists a positive constant M₄1M₄ðf;p;rÞ such that

kLnðf;p;r;s;Þk_paM₄ ð12Þ

for all f ABp.

The formulas (4), (5) and (12) show that L_nðf;p;r;sÞ is well-defined on the space Bp, pAN.

Proof. First we suppose that f ABp, pAN. From this, using the ele- mentary inequality ðaþbÞ^ka2^k1ða^kþb^kÞ, a;b>0, kAN₀, we get

jxtj^kjf^ðkÞðtÞja2^k1jf^ðkÞðtÞjfx^kþt^kg aM5ðf;p;kÞ 1

w_pðtÞþ x^k wpkðtÞ

; k¼0;1;2;. . .;p; x;tAR0: This implies that

w_pðxÞjLnðf;p;r;s;xÞj aM6ðf;pÞ wpðxÞ

gðn^sx;rÞ X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

1

w_pððkþrÞ=n^sÞþX^p

j¼0

x^j wpjððkþrÞ=n^sÞ

( )

¼M6ðf;pÞwpðxÞ Anð1=wpðtÞ;r;s;xÞ þX^p

j¼0

x^jAnð1=wpjðtÞ;r;s;xÞ

( )

:

From this and in view of Lemmas 2 and 3 we get wpðxÞjLnðf;p;r;s;xÞjaM4ðf;p;rÞ:

This ends the proof of (12). 9

3. Rate of convergence

In this section we shall study properties of Lnðf;p;r;sÞ. We shall give theorems on the degree of approximation of f AB_p, pAN, by these operators.

We can state now the main results of this paper.

Theorem 1. Fix pAN0, rAN and s>0. Then there exists a positive constant M71M7ðp;r;sÞ such that for every f AB2pþ1 we have

kLnðf;2pþ1;r;s;Þ fðÞk_2pþ1aM₇o₁ f^ð2pþ1Þ;C₀;1 n^s

; nAN:

ð13Þ

Proof. First we suppose that f AB2pþ1. This implies that f^ð2pþ1ÞAC0. Let pAN₀. Using the modified Taylor formula

fðxÞ ¼^2pþ1X

j¼0

f^ðjÞ kþr n^s

xkþr n^s j

j! þ

xkþr n^s 2pþ1

ð2pÞ!

ð1

0

ð1tÞ

2p

f^ð2pþ1Þ kþr

n^s þt xkþr n^s

f^ð2pþ1Þ kþr n^s

dt and the definition of modulus of continuity and (4), (5), we obtain

w2pþ1ðxÞjLnðf;2pþ1;r;s;xÞ fðxÞj

aw2pþ1ðxÞ gðn^sx;rÞ

X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

2pþ1X

j¼0

f^ðjÞ kþr n^s

xkþr n^s j

j! fðxÞ

aw2pþ1ðxÞ gðn^sx;rÞ

X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

xkþr n^s

2pþ1

ð2pÞ!

ð1

0

ð1tÞ

2p

f^ð2pþ1Þ kþr

n^s þt xkþr n^s

f^ð2pþ1Þ kþr n^s

dt

aw2pþ1ðxÞ gðn^sx;rÞ

X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

xkþr n^s

2pþ1

ð2pÞ!

ð1

0

ð1tÞ^2po₁ðf^ð2pþ1Þ;C₀;tjx ðkþrÞ=n^sjÞdt:

Observe that

o1ðf^ð2pþ1Þ;C0;tjx ðkþrÞ=n^sjÞað1þtjx ðkþrÞ=n^sjn^sÞo1ðf^ð2pþ1Þ;C0;1=n^sÞ:

From this, using the elementary inequality ðaþbÞ^ka2^k1ða^kþb^kÞ, a;b>0, kAN₀, and (6) and (7), we get

w_2pþ1ðxÞjLnðf;2pþ1;r;s;xÞ fðxÞj aw2pþ1ðxÞ

gðn^sx;rÞ X^y

k¼0

ðn^sxÞ^k

ðkþrÞ! xkþr n^s 2pþ2

n^sþ xkþr n^s

" 2pþ1#

o1ðf^ð2pþ1Þ;C₀;1=n^sÞ

aw_2pþ1ðxÞfn^sA_nððxtÞ^2pþ2;r;s;xÞ þA_nððtþxÞ^2pþ1;r;s;xÞg o₁ðf^ð2pþ1Þ;C₀;1=n^sÞ

aw2pþ1ðxÞfn^sA_nððtxÞ^2pþ2;r;s;xÞ þ2^2pA_nðt^2pþ1;r;s;xÞ þ2^2px^2pþ1g o1ðf^ð2pþ1Þ;C0;1=n^sÞ:

Applying Lemmas 1 and 4, (10) and (2), we immediately obtain w2pþ1ðxÞjLnðf;2pþ1;r;s;xÞ fðxÞj

aw2pþ1ðxÞ (^2pþ2X

j¼1

x^j1

n^sð2pjþ2Þ a2pþ2;jþ b2pþ2;j

gðn^sx;rÞ

þ2^2p2pþ1X

j¼0

x^j

n^sð2pþ1jÞ a2pþ1;jþb_2pþ1;jðrÞ gðn^sx;rÞ

þ2^2px^2pþ1 )

o1ðf^ð2pþ1Þ;C0;1=n^sÞ

aM8ðp;sÞ ^2pþ2X

j¼1

ðja2pþ2;jj þr!jb2pþ2;jjÞ þ2^2p2pþ1X

j¼1

ða2pþ1þr!b_2pþ1ðrÞÞ þ2^2p

( )

o1ðf^ð2pþ1Þ;C0;1=n^sÞaM7ðp;r;sÞo1ðf^ð2pþ1Þ;C0;1=n^sÞ

for xAR₀, n;rAN, s>0. This completes the proof of Theorem 1. 9 Theorem 2. Fix pAN₀, rAN and s>0. Then there exists a positive constant M91M9ðp;r;sÞ such that for every f AB2pþ2 we have

kLnðf;2pþ2;r;s;Þ fðÞk_2pþ2aM9ðp;r;sÞ

n^s kf^ð2pþ2Þk₀; nAN:

ð14Þ

Proof. First we suppose that f AB2pþ2. This implies that f^ð2pþ2ÞAC0. Arguing as in the first part of the proof of Theorem 1 we obtain w2pþ2ðxÞjLnðf;2pþ2;r;s;xÞ fðxÞj

aw2pþ2ðxÞ gðn^sx;rÞ

X^y

k¼0

ðn^sxÞ^k ðkþrÞ!

xkþr n^s 2pþ2

ð2pþ1Þ!

ð1

0

ð1tÞ

2pþ1

f^ð2pþ2Þ kþr

n^s þt xkþr n^s

þ f^ð2pþ2Þ kþr n^s

dt:

From this and by our assumption we get w2pþ2ðxÞjLnðf;2pþ2;r;s;xÞ fðxÞj

a2kf^ð2pþ2Þk₀w2pþ2ðxÞ gðn^sx;rÞ

X^y

k¼0

ðn^sxÞ^k

ðkþrÞ! xkþr n^s 2pþ2

¼2kf^ð2pþ2Þk₀w2pþ2ðxÞAnððxtÞ^2pþ2;r;s;xÞ:

From this, applying Lemma 4, (10) and (2), we immediately obtain w2pþ2ðxÞjLnðf;2pþ2;r;s;xÞ fðxÞj

a 2

n^skf^ð2pþ2Þk₀w2pþ2ðxÞ^2pþ2X

j¼1

x^j1

n^sð2pjþ2Þ japþ2;jj þ jbpþ2;jj gðn^sx;rÞ

aM9ðp;r;sÞ

n^s kf^ð2pþ2Þk₀

for xAR0, n;rAN, s>0. This ends the proof of Theorem 2. 9 From above Theorems we obtain

Corollary. For every fixed rAN, pAN, s>0 and f AB_p we have

n!ylim kLnðf;p;r;s;Þ fðÞk_p¼0:

Remark. In [1] it was proved that if f AC_p, pAN₀, then w_pðxÞjSnðf;xÞ fðxÞjaM₉o₂ f;C_p;

ffiffiffix n r

; xAR₀;nAN;

where M10 ¼const: >0 and o2ðf;Þ is the modulus of smoothness defined by o2ðf;Cp;tÞ:¼ sup

0ahat

kD²_hfðÞk_p; tAR0;

withD_h²fðxÞ:¼ fðxÞ 2fðxþhÞ þfðxþ2hÞ. In particular, if f AC_p¹, pAN0, then

wpðxÞjSnðf;xÞ fðxÞjaM10

ffiffiffix n r

; xAR0;nAN

where M10¼const: >0.

Theorem 1, Theorem 2 and Corollary in this paper show that operators Lnðf;p;r;1;Þ, nAN, give better the degree of approximation of functions

f AB_p, pAN, than S_n and the operators examined in [1, 4, 6, 7].

Acknowledgment

The author would like to thank the referee for his valuable comments and proposals.

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Institute of Mathematics Poznan´ University of Technology

Piotrowo 3A 60-965 Poznan´, Poland E-mail: zwalczak@math.put.poznan.pl