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.