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Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135]
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Corrigendum
Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135] Borislav R. Draganov ∗ Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria Received 16 October 2019; received in revised form 23 October 2019; accepted 27 October 2019 Available online xxxx Communicated by Dany Leviatan
Abstract We correct a mistake in the statements of Corollaries 4.11 and 4.12. The mistake has no bearing to their proofs or other results in the paper. c 2019 Elsevier Inc. All rights reserved. ⃝ Keywords: Bernstein polynomial; Kantorovich polynomial; Boolean sum; Simultaneous approximation; Rate of convergence
There is a mistake in the statements of Corollaries 4.11 and 4.12 in [1]. It does not affect 1 r their proofs or the proofs of the other results in the paper. The factor (−1)r −1 of (2n) r D f was omitted. Thus the correct statements are: Corollary 4.11. Let 1 < p ≤ ∞, r, s ∈ N and w = w(γ0 , γ1 ) be given by (1.3). Set s ′′ = 2r +s+1 max{3, s}. If −1/ p < γ0 , γ1 ≤ s + 1, then for all f ∈ C[0, 1] such that f ∈ ACloc (0, 1) DOI of original article: https://doi.org/10.1016/j.jat.2015.07.006.
∗ Correspondence to: Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd.,
1164 Sofia, Bulgaria. E-mail address: [email protected]. https://doi.org/10.1016/j.jat.2019.105321 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝
Please cite this article as: B.R. Draganov, Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135], Journal of Approximation Theory (2019) 105321, https://doi.org/10.1016/j.jat.2019.105321.
2
B.R. Draganov / Journal of Approximation Theory xxx (xxxx) xxx ′′
and w f (s ) , wϕ 2r +2 f (2r +s+2) ∈ L p [0, 1], and all n ∈ N there holds ( )(s) (−1)r −1 r D f w Br,n f − f − (2n)r p ( ) c (s ′′ ) 2r +2 (2r +s+2) ≤ r +1 ∥w f ∥ p + ∥wϕ f ∥p . n For p = ∞ we may allow γ0 γ1 = 0, while still assuming 0 ≤ γ0 , γ1 ≤ s + 1, and have ( )(s) (−1)r −1 r D f w Br,n f − f − (2n)r ∞ ) c ( ′′ (s ) (r +s+1) ≤ r +1 ∥w f ∥∞ + ∥w f ∥∞ + ∥wϕ 2r +2 f (2r +s+2) ∥∞ n provided that w f (r +s+1) ∈ L ∞ [0, 1] too. Corollary 4.12. Let 1 < p ≤ ∞, r, s ∈ N and w = w(γ0 , γ1 ) be given by (1.3) as −1/ p <γ0 , γ1 < s − 1/ p if 1 < p < ∞, 0 ≤γ0 , γ1 < s if p = ∞. Then for all f ∈ AC 2r +s+1 [0, 1] such that wϕ 2r +2 f (2r +s+2) ∈ L p [0, 1], and all n ∈ N there holds ( )(s) c (−1)r −1 r D f ≤ r +1 ∥w(Dr +1 f )(s) ∥ p . w Br,n f − f − (2n)r n p
This is what was actually established in the proofs of [1, Corollaries 4.11 and 4.12]. Only [1, (4.60)] should be ( )(s) (−1)r −1 r D f w Br,n f − f − (2n)r p ( ) c (s ′′ ) (r +s+1) ≤ r +1 ∥w f ∥ p + ∥w f ∥ p + ∥wϕ 2r +2 f (2r +s+2) ∥ p (4.60) n and the auxiliary operator Vr,n should be defined on p. 122 in [1] by (−1)r −1 r D f. (*) (2n)r We prove (4.60), and hence Corollary 4.11, by induction on r . The case r = 1 is established in [1, Proposition 4.9]. Then we use the following recurrence relation, not explicitly mentioned in [1], 1 Vr +1,n f = (−1)r V1,n Fr,n − DVr,n f 2n to get [1, (4.62)]. It is valid with Vr,n defined in (∗), not as in [1, p. 122]. Let us note that V1,n f is still equal to the quantity whose weighted semi-norm is estimated in [1, Proposition 4.9]. Then the proofs of Corollaries 4.11 and 4.12 are completed just as in [1]. Also, the inequality in [1, p. 129, l. −10] should be ( )(s) (−1)r −1 r c D f w Q n f − f − ≤ r +1 Φ( f ), r (2n) n Vr,n f = Br,n f − f −
p
Please cite this article as: B.R. Draganov, Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135], Journal of Approximation Theory (2019) 105321, https://doi.org/10.1016/j.jat.2019.105321.
B.R. Draganov / Journal of Approximation Theory xxx (xxxx) xxx
3
and the phrase “with Dr in place of D” in [1, p. 129, l.−9], should be “with (−1)r −1 Dr in place of D”. I take the opportunity to add the bibliographic details to Ref. [22], which were inadvertently omitted: K. Kopotun, D. Leviatan, I.A. Shevchuk, New moduli of smoothness: weighted DT moduli revisited and applied, Constr. Approx. 42 (2015) 129–159. References [1] B.R. Draganov, Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums, J. Approx. Theory 200 (2015) 92–135.
Please cite this article as: B.R. Draganov, Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135], Journal of Approximation Theory (2019) 105321, https://doi.org/10.1016/j.jat.2019.105321.
ScienceDirect Journal of Approximation Theory xxx (xxxx) xxx www.elsevier.com/locate/jat
Corrigendum
Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135] Borislav R. Draganov ∗ Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria Received 16 October 2019; received in revised form 23 October 2019; accepted 27 October 2019 Available online xxxx Communicated by Dany Leviatan
Abstract We correct a mistake in the statements of Corollaries 4.11 and 4.12. The mistake has no bearing to their proofs or other results in the paper. c 2019 Elsevier Inc. All rights reserved. ⃝ Keywords: Bernstein polynomial; Kantorovich polynomial; Boolean sum; Simultaneous approximation; Rate of convergence
There is a mistake in the statements of Corollaries 4.11 and 4.12 in [1]. It does not affect 1 r their proofs or the proofs of the other results in the paper. The factor (−1)r −1 of (2n) r D f was omitted. Thus the correct statements are: Corollary 4.11. Let 1 < p ≤ ∞, r, s ∈ N and w = w(γ0 , γ1 ) be given by (1.3). Set s ′′ = 2r +s+1 max{3, s}. If −1/ p < γ0 , γ1 ≤ s + 1, then for all f ∈ C[0, 1] such that f ∈ ACloc (0, 1) DOI of original article: https://doi.org/10.1016/j.jat.2015.07.006.
∗ Correspondence to: Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd.,
1164 Sofia, Bulgaria. E-mail address: [email protected]. https://doi.org/10.1016/j.jat.2019.105321 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝
Please cite this article as: B.R. Draganov, Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135], Journal of Approximation Theory (2019) 105321, https://doi.org/10.1016/j.jat.2019.105321.
2
B.R. Draganov / Journal of Approximation Theory xxx (xxxx) xxx ′′
and w f (s ) , wϕ 2r +2 f (2r +s+2) ∈ L p [0, 1], and all n ∈ N there holds ( )(s) (−1)r −1 r D f w Br,n f − f − (2n)r p ( ) c (s ′′ ) 2r +2 (2r +s+2) ≤ r +1 ∥w f ∥ p + ∥wϕ f ∥p . n For p = ∞ we may allow γ0 γ1 = 0, while still assuming 0 ≤ γ0 , γ1 ≤ s + 1, and have ( )(s) (−1)r −1 r D f w Br,n f − f − (2n)r ∞ ) c ( ′′ (s ) (r +s+1) ≤ r +1 ∥w f ∥∞ + ∥w f ∥∞ + ∥wϕ 2r +2 f (2r +s+2) ∥∞ n provided that w f (r +s+1) ∈ L ∞ [0, 1] too. Corollary 4.12. Let 1 < p ≤ ∞, r, s ∈ N and w = w(γ0 , γ1 ) be given by (1.3) as −1/ p <γ0 , γ1 < s − 1/ p if 1 < p < ∞, 0 ≤γ0 , γ1 < s if p = ∞. Then for all f ∈ AC 2r +s+1 [0, 1] such that wϕ 2r +2 f (2r +s+2) ∈ L p [0, 1], and all n ∈ N there holds ( )(s) c (−1)r −1 r D f ≤ r +1 ∥w(Dr +1 f )(s) ∥ p . w Br,n f − f − (2n)r n p
This is what was actually established in the proofs of [1, Corollaries 4.11 and 4.12]. Only [1, (4.60)] should be ( )(s) (−1)r −1 r D f w Br,n f − f − (2n)r p ( ) c (s ′′ ) (r +s+1) ≤ r +1 ∥w f ∥ p + ∥w f ∥ p + ∥wϕ 2r +2 f (2r +s+2) ∥ p (4.60) n and the auxiliary operator Vr,n should be defined on p. 122 in [1] by (−1)r −1 r D f. (*) (2n)r We prove (4.60), and hence Corollary 4.11, by induction on r . The case r = 1 is established in [1, Proposition 4.9]. Then we use the following recurrence relation, not explicitly mentioned in [1], 1 Vr +1,n f = (−1)r V1,n Fr,n − DVr,n f 2n to get [1, (4.62)]. It is valid with Vr,n defined in (∗), not as in [1, p. 122]. Let us note that V1,n f is still equal to the quantity whose weighted semi-norm is estimated in [1, Proposition 4.9]. Then the proofs of Corollaries 4.11 and 4.12 are completed just as in [1]. Also, the inequality in [1, p. 129, l. −10] should be ( )(s) (−1)r −1 r c D f w Q n f − f − ≤ r +1 Φ( f ), r (2n) n Vr,n f = Br,n f − f −
p
Please cite this article as: B.R. Draganov, Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135], Journal of Approximation Theory (2019) 105321, https://doi.org/10.1016/j.jat.2019.105321.
B.R. Draganov / Journal of Approximation Theory xxx (xxxx) xxx
3
and the phrase “with Dr in place of D” in [1, p. 129, l.−9], should be “with (−1)r −1 Dr in place of D”. I take the opportunity to add the bibliographic details to Ref. [22], which were inadvertently omitted: K. Kopotun, D. Leviatan, I.A. Shevchuk, New moduli of smoothness: weighted DT moduli revisited and applied, Constr. Approx. 42 (2015) 129–159. References [1] B.R. Draganov, Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums, J. Approx. Theory 200 (2015) 92–135.
Please cite this article as: B.R. Draganov, Corrigendum to “Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums” [J. Approx. Theory 200 (2015) 92–135], Journal of Approximation Theory (2019) 105321, https://doi.org/10.1016/j.jat.2019.105321.