Polynomial operators for one-sided -approximation to Riemann integrable functions
© Adell et al.; licensee Springer. 2014
Received: 1 July 2014
Accepted: 25 November 2014
Published: 12 December 2014
We present some operators for one-sided approximation of Riemann integrable functions on by algebraic polynomials in -spaces. The estimates for the error of approximation are given with an explicit constant.
and let be the set of all continuous functions with the sup norm . We simply write . Let be the set of all Riemann integrable functions on (recall that Riemann integrable functions on are bounded). As usual, we denote by the space of all absolutely continuous functions such that . We also denote by the family of all algebraic polynomials of degree not greater than n.
This modulus is well defined whenever g is a bounded measurable function.
In this paper, we present some sequences of polynomial operators for one-sided -approximation which realize the rate of convergence given in (2). Our construction also provides a specific constant. We point out that a one-sided approximation cannot be realized with polynomial linear operators.
The existence of such sequences of polynomials and satisfying (4) is well known (see, for example, ). Probably, the first construction of the optimal solution for (4) is due to Markoff or Stieltjes (cf. Szegö [, Section 3.411, p.50]).
and present upper estimates for the error and in in terms of the average modulus of continuity.
In the last years, there has been interest in studying open problems related to one-sided approximations (see [4, 6–8] and ). We point out that other operators for the one-sided approximation have been constructed in [10, 11] and . In particular, the operators presented in  yield the non-optimal rate whereas the ones considered in [10, 11] give the optimal rate, but without an explicit constant.
The paper is organized as follows. In Section 2 we present some properties of the Steklov type functions (10) and (11). Finally, in Section 3 we consider an approximation by means of the operators defined in (12).
2 Properties of Steklov type functions
We start with the following auxiliary results.
- (i)The functions and are absolutely continuous. Moreover, if and , then(13)
- (ii)For each ,(14)
- (iii)For one has(15)
- (ii)Observe that
- (iii)We present a proof for a fixed (the case follows analogously). As usual, take q such that . Using (14) and Hölder inequality, we obtain
For the proof follows analogously.
From (13), (18), and (19), we obtain (17). The proof is complete. □
3 Approximation of Riemann integrable functions
The estimate for follows analogously. Finally, (21) follows immediately from (20). □
for and for .
The following result was proved in .
From Theorem 1 and Proposition 2 we can state our main results.
This completes the proof. □
Finally, from Theorem 2 we have immediately the following.
where is the best one-sided approximation defined in (1).
The authors are partially supported by Research Project MTM2011-23998 and by Junta de Andalucía Research Group FQM268. This work was written during the stay of JAA and JB at the Department of Mathematics of the University of Jaén in May 2014.
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