Journal of Operators

Volume 2013 (2013), Article ID 324630, 7 pages

http://dx.doi.org/10.1155/2013/324630

Research Article

On Approximation by Algebraic Version of the Trigonometric Jackson Integrals in Weighted Integral Metric

Teodora Zapryanova

Economic University of Varna, Boulevard. “Knyaz Boris I” 77, 9002 Varna, Bulgaria

Received 20 June 2013; Accepted 21 November 2013

Academic Editor: Hagen Neidhardt

Copyright © 2013 Teodora Zapryanova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We characterize the errors of the algebraic version of trigonometric Jackson integrals in weighted integral metric. We prove direct and strong converse theorem in terms of the weighted -functional.

1. Introduction

We study linear approximation process together with the characterization of the rate of convergence of the algebraic version of the trigonometric Jackson integrals defined by

where

We established the equivalence in uniform norm for the approximation error of the operator , the values of proper -functional, and modulus of smoothness (see [1, 2]):

For every and the -functional is defined by

where the differential operator

The modulus

The expression ~ means that there exists a positive constant independent of such that . The equivalence ~ consists of a direct inequality and strong converse inequality of type in the sense of [3]. Ditzian and Ivanov show that a converse inequality follows from several inequalities of Bernstein and Voronovskaya type. We apply their method.

Let be the weighted space with the norm

The approximation error of in will be compared with the -functional, which for every and is defined by

Our main result states the following.

Theorem 1. For every , and , one has

In Section 2 we state and prove some auxiliary lemmas. The proof of Theorem 1 is given in Section 3.

2. Auxiliary Lemmas

Lemma 2. Let be summable on and —periodic function, . Then the following holds true:

Proof. Using the Minkowski inequality we have

Lemma 3. For periodic integrable on function and every one has

Proof. We consider the function . By the Fubini theorem we have

As we obtain that the integral on the left-hand side of (13) is equal to

The integral on the right-hand side of (13) is

From (13), (14), and (15) we obtain . If then the sign of is constant and coincides with the sign of . Therefore

which proves the lemma.

Lemma 4. For periodic function and every one has

Proof. For Lemma 4 is fulfilled as equality. Let . Using the Hölder inequality we get

Therefore

Then

We apply Lemma 3 to to obtain

Hence

Thus

Definition 5. Set

Lemma 6 ([2, page 402]). Let be the space from Definition 5, , and . Then and for .

Lemma 7. Let be the space from Definition 5. Then for every and , one has

Proof. From we see that . In order to prove it is sufficient to show (see Lemma  2 in [4, page 116]) that for every and there exists such that and . Let . We put and consider . Since (see Lemma 6). We use the Jackson integrals of the following type:

where , .

Since , it follows that is an even nonnegative trigonometric polynomial of degree is a trigonometric polynomial of degree , which is even as is even. From Jackson theorem (see [5, Chapter 7, Theorem 2.2]) we get

By the substitution in we obtain an algebraic polynomial, which is desired function from . We set

From and (27) we get

From (26) we get

Using the Jackson theorem we get

Since and (31) implies

For given we choose such that and to obtain

This completes the proof of the lemma.

3. Proof of Theorem 1

The theorem will be proved if we show that ~ (see Lemma 7). First we prove the converse result , which is a strong converse inequality of type in the terminology of [3]. We utilize Theorems 3.1 and   4.1 in [3] with , , , and for , for . The result needed for inequality from Theorem  3.1 in [3] with is given by

We set . To prove (34) we recall the representation

where

And the convolution is defined by

We have

Using Lemma 2 as and are even, periodic functions, we estimate

to complete the proof of (34).

We will show now that for the following Voronovskaya-type estimate holds true:

with , , and , Inequality (40) will serve for from Theorem  3.1 in [3]. We note that as is an algebraic polynomial. Let . From and Lemma 6  . We apply Lemma 6 for to obtain that . With we have

Expanding by Taylor’s formula and using that , , we now have

We recall that . As for , , and the sign of is constant and coincides with the sign of (if the integral is not zero); via Minkowski’s inequality, we get

We apply Lemma 4 (as is even periodic function) to obtain ≤ . This establishes (40).

To obtain results corresponding to and from Theorem  3.1 in [3] we need a weighted Bernstein-type inequality for the power of the operator like (6.10) in [3].

We use representations

where

We estimate the action of on the th degree of the operator. Using Lemma 2 we have

We have already proved earlier (see Assertion  1.2 in [1]) that

and therefore

As commutes with the operator using estimation (48) we now observe that

Estimations (49) and (50) correspond to and from Theorem  3.1 in [3]. To match the conditions of Theorem  4.1 in [3] we need

to be satisfied with or

which is true for large . From (34), (40), (49), and (50) we obtain

To prove the direct result we need inequality (34) and

to be satisfied (see Theorem  3.4 in [3]).

Let . We set . We have

Expanding by Taylor’s formula (by Lemma 6  )

and using that we have

We recall that . As for , , and the sign of is constant and coincides with the sign of (if the integral is not zero); we get

Using the Minkowski inequality we have

We applied Lemma 4 (as is even periodic function) to obtain ≤ . The proof of the direct result

is completed. This concludes the proof of Theorem 1.

References

T. Zapryanova, “Approximation by operators of Cao-Gonska type and . Direct and converse theorem,” Mathematics and Education in Mathematics, vol. 33, pp. 189–194, 2004.

T. D. Zapryanova, “A characterization of the -functional for the algebraic version of the trigonometric Jackson integrals and the -functionals for Cao-Gonska operators and ,” Results in Mathematics, vol. 54, no. 3-4, pp. 397–413, 2009.

Z. Ditzian and K. G. Ivanov, “Strong converse inequalities,” Journal d'Analyse Mathématique, vol. 61, pp. 61–111, 1993.

K. G. Ivanov, “A characterization theorem for the -functional for Kantorovich and Durrmeyer operators,” in Approximation Theory: A Volume Dedicated to Borislav Bojanov, pp. 112–125, Marin Drinov Academic Publishing House, Sofia, Bulgaria, 2004.

R. A. DeVore and G. G. Lorentz, Constructive Approximation, vol. 303 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1993.