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Strong converse result for uniform approximation by Meyer-König and Zeller operator
J. Math. Anal. Appl. 428 (2015) 32–42
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Strong converse result for uniform approximation by Meyer-König and Zeller operator Ivan Gadjev Department of Mathematics and Informatics, University of Transport, 158 Geo Milev str., 1574 Sofia, Bulgaria
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 4 October 2014 Available online 6 March 2015 Submitted by K. Driver Keywords: Meyer-König and Zeller operator K-functional Direct theorem Strong converse theorem
We characterize the approximation of functions in uniform norm by classical Meyer-König and Zeller operator. Using the closed connection between the approximation by Meyer-König and Zeller operator and the weighted approximation by Baskakov operator, we prove a strong converse inequality of type A in terms of the K-functional. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The Meyer-König and Zeller operator is defined for functions f ∈ C[0, 1) by the formula ∞ Mn (f, x) = f k=0
k n+k
Mn,k (x) where
Mn,k (x) =
n+k k x (1 − x)n+1 . k
(1.1)
The direct theorem for Meyer-König and Zeller operator can be found in [3]. A converse inequality of weaker type B in terminology of [2] is proved in [5]. In this paper we prove the strong converse inequality of type A for Meyer-König and Zeller operator. Before stating our main result, let us introduce some notations. The first derivative operator is denoted d by D = dx . Thus, Dg(x) = g (x) and D2 g(x) = g (x). By ϕ(x) = x(1 − x)2 we denote the weight which is naturally connected with the second derivative of Meyer-König and Zeller operator. By C[0, 1) we denote the space of all continuous on [0, 1) functions. The functions from C[0, 1) are not expected to be continuous or bounded at 1. By L∞ [0, 1) we denote the space of all Lebesgue measurable and essentially bounded in [0, 1) functions equipped with the uniform norm · and by CB[0, 1) = C[0, ∞) ∩ L∞ [0, 1) the space of all continuous and bounded in [0, 1) functions. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2015.03.004 0022-247X/© 2015 Elsevier Inc. All rights reserved.
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
33
Also, we define 2 W∞ (ϕ) = {g : Dg ∈ AC loc (0, 1) and ϕD2 g ∈ L∞ [0, 1)}
where AC loc (0, 1) consists of the functions which are absolutely continuous in [a, b] for every interval [a, b] ⊂ (0, 1). To estimate the approximation f ≈ Mn (f ) we will use the K-functional, defined by 2 Kϕ (f, t) = inf f − g + tϕD2 g : g ∈ W∞ (ϕ), f − g ∈ CB[0, 1) 2 for every function f ∈ CB[0, 1) + W∞ (ϕ), i.e. for every function f which can be represented as f = f1 + f2 2 where f1 ∈ CB[0, 1) and f2 ∈ W∞ (ϕ). It is known, that the K-functional Kϕ (f, t) is equivalent to the √ 2 modulus of smoothness ω√ ϕ (f, t ) [3], where
2√ 2 ω√ ϕ (f, δ) = sup Δh ϕ f 0
and Δ2hϕ(x) f (x)
=
f x + h ϕ(x) − 2f (x) + f x − h ϕ(x) , 0,
x±h
ϕ(x) ∈ [0, 1],
otherwise.
Our main result is the following theorem. Theorem 1.1. For Mn defined by (1.1) there exist absolute constants L, C > 0 such that for every natural n>L 1 ≤ CMn f − f Kϕ f, n 2 holds for all f ∈ CB[0, 1) + W∞ (ϕ).
It is proved in Section 2. Combining Theorem 1.1 with the direct theorem and the equivalency between the modulus of smoothness and the K-functional, we obtain 1 2 √ , f, Mn f − f ∼ ω√ ϕ n i.e. there exist constants C1 and C2 such that 1 2 √ ≤ C2 Mn f − f . f, C1 Mn f − f ≤ ω√ ϕ n For the rest of this paper the constant C will always be an absolute constant, which means it does not depend on f and n. It may be different on each occurrence. 2. Proof of the main result The proof is based on the closed connection between Meyer-König and Zeller and Baskakov operators. V. Totik was the first to use it [7].
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
34
For functions f ∈ C[0, ∞) (the space of all continuous in [0, ∞) functions) the Baskakov operator is given by (see [1]) Vn f (x) = (Vn f, x) = Vn (f, x) =
∞ k Vn,k (x) for 0 ≤ x < ∞, f n
(2.1)
k=0
where Vn,k (x) =
n+k−1 k x (1 + x)−n−k . k
(2.2)
In [6, pp. 150–160] the authors proved that the approximation by Meyer-König and Zeller operators is equivalent to the weighted (weight w(x) = (1 +x)−1 ) approximation by Baskakov operators (see Propositions 2.4, 2.5, 2.6 and 2.7). So, we define for ψ(x) = x(1 + x) and w(x) = (1 + x)−1 f w :=
sup |w(x)f (x)|, x∈[0,∞)
C(w) = {g ∈ C[0, ∞); wg ∈ L∞ [0, ∞)} , W 2 (wψ) = g, Dg ∈ AC loc (0, ∞) and wψD2 g ∈ L∞ [0, ∞) ,
W 3 (wψ 3/2 ) = g, Dg, D2 g ∈ AC loc (0, ∞) and wψ 3/2 D3 g ∈ L∞ [0, ∞) , and Kw (f, t) = inf f − gw + t ψD2 g w : g ∈ W 2 (wψ), f − g ∈ C(w) for every function f ∈ C(w) + W 2 (wψ) and every t > 0. Then Theorem 1.1 follows from the next theorem. Theorem 2.1. For Vn defined by (2.2) there exist absolute constants L, C > 0 such that for every natural n>L 1 ≤ CVn f − f w Kw f, n holds for all f ∈ C(w) + W 2 (wψ). 3. Proof of Theorem 2.1 We will mention some properties of Baskakov operator, which can be found in [3]. is a linear, positive operator with Vn f ≤ f , ψ(x) Vn (1, x) = 1, , Vn (t − x, x) = 0, Vn (t − x)2 , x = n k n DVn,k (x) = − x Vn,k (x). ψ(x) n Vn
(3.1) (3.2) (3.3)
The next inequality is valid for all integers m. The constant C depends only on m.
Vn (t − x)
2m
,x ≤ C
ψ(x) n
m for x ≥
1 . n
(3.4)
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
35
Some lemmas follow. Lemma 3.1. For every natural n, we have Vn f w ≤ f w
(3.5)
for every function f ∈ C(w). Proof. ∞ ∞ k k |Vn f (x)| = f Vn,k (x) ≤ f w Vn,k (x) = (1 + x)f w . 1+ n n k=0
2
k=0
Lemma 3.2. For x, t ∈ [0, ∞) t (t − u)2 u−3/2 (1 + u)−1/2 du ≤ x−3/2 |t − x|3 (1 + x)−1/2 + (1 + t)−1/2 .
(3.6)
x
Proof. We consider two cases. Case 1. t ≥ x. t t (t − u)2 u−3/2 (1 + u)−1/2 du ≤ x−3/2 (1 + x)−1/2 (t − u)2 du x
x
=
1 −3/2 x (1 + x)−1/2 (t − x)3 . 3
Case 2. t < x. Then t x 2 −3/2 −1/2 −1/2 (t − u) u (1 + u) du ≤ (1 + t) (u − t)2 u−3/2 du x
t
−1/2
x
= (1 + t)
t 1− u
3/2 (u − t)1/2 du
t −1/2
x 1−
≤ (1 + t)
t x
3/2 (u − t)1/2 du
t
=
2 −3/2 x (1 + t)−1/2 (x − t)3 . 3
2
Lemma 3.3. There exists a constant C such that for every natural n and every x ∈ [0, ∞)
k/n ∞
k=0 x
k −u n
2
−3/2 −1/2 u (1 + u) du Vn,k (x) ≤ C(1 + x)n−3/2 .
(3.7)
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
36
Proof. Case 1. x ≥ By (3.6) we get
1 n.
2 k/n k −3/2 −1/2 −u u (1 + u) du Vn,k (x) n k=0 x 3 −1/2 ∞ k k − x (1 + x)−1/2 + 1 + ≤ x−3/2 Vn,k (x). n n ∞
k=0
Using Cauchy’s inequality and (3.4), we have for the first sum on the right
x
−3/2
12 ∞ 3 6 ∞ k k −1/2 −3/2 −1/2 − x (1 + x) − x Vn,k (x) Vn,k (x) ≤ x (1 + x) n n
k=0
k=0
−3/2
≤ Cx
(1 + x)
−1/2
ψ(x) n
32
≤ C(1 + x)n−3/2 ,
and for the second one x
−3/2
3 −1/2 ∞ k − x 1 + k Vn,k (x) n n
k=0
≤x
−3/2
12 ∞ 12 6 −1 k − x Vn,k (x) 1+ Vn,k (x) n n
∞ k k=0
≤ Cx−3/2 (1 + x)−1/2 Case 2. x < We have
ψ(x) n
32
k=0
≤ C(1 + x)n−3/2 .
1 n.
I0 = (1 + x)
−n
x u
1/2
(1 + u)
−1/2
x du ≤
0
u1/2 du ≤
2 (1 + x)n−3/2 3
0
and by (3.2) ∞ k=1
Ik =
k/n ∞ k k=1 x
n
2 −u
u
−3/2
(1 + u)
−1/2
k/n ∞ 2 k Vn,k (x)du ≤ u−3/2 Vn,k (x)du n k=1
x
∞ 2 k ψ(x) −1/2 −1/2 2 ≤ 2x x + ≤ 6(1 + x)n−3/2 . Vn,k (x) = 2x n n
2
k=1
Lemma 3.4. There exists a constant C such that for every natural n, we have 3/2 3 Vn g − g − 1 ψD2 g ≤ C D g ψ 2n w n3/2 w for every function g ∈ W 3 (wψ 3/2 ).
(3.8)
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Proof. By Taylor’s formula, we have (t − x)2 2 1 D g(x) + g(t) = g(x) + (t − x)Dg(x) + 2 2
t (t − v)2 D3 g(v)dv. x
Multiplying both sides by Vn,k (x), summing with respect to k and using the identities (3.2) we get Vn g(x) − g(x) − 1 ψD2 g(x) 2n k/n ∞ 2 1 k 3 ≤ − v D g(v)dv Vn,k (x) 2 n k=0 x k/n ∞ 2 k 1 3/2 3 −3/2 −1/2 −v v (1 + v) dv Vn,k (x). ≤ ψ D g 2 n w k=0 x Now we can use (3.7) to complete the proof. 2 Lemma 3.5. For every natural n and for every function f ∈ C(w) ψD2 Vn f ≤ 40nf w . w
(3.9)
Proof. We consider two cases. Case 1. x ≤ n1 . We have [3] D2 Vn f (x) = n(n + 1)
∞
Δ21 f n
k=0
k Vn+2,k (x), n
where, as usual, Δ2h f (x) = f (x + 2h) − 2f (x + h) + f (x). Then, ∞ k + 2 k + 1 k 2 w(x)ψ(x) D Vn f (x) = n(n + 1)x f − 2f +f Vn+2,k (x) n n n k=0 ∞ k+2 k+1 k ≤ n(n + 1)xf w 1+ +2 1+ + 1+ Vn+2,k (x) n n n k=0
= 4(n + 1)x [n + 1 + (n + 2)x] f w ≤ 40nf w . Case 2. x > n1 . Differentiating (3.3) we get
n2 ψ(x)D2 Vn,k (x) = ψ(x) Then
2
k −x n
(1 + 2x)n − ψ(x)
k − x − n Vn,k (x). n
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
38
2 ∞ 2 n ψ(x) 1 + 2x x k k k w(x)ψ(x) D2 Vn f (x) = −x − −x − f Vn,k (x) ψ 2 (x) n n n n n k=0 2 ∞ ψ(x) k 1 + 2x k n2 x k + f w −x + − x 1 + Vn,k (x) ≤ 2 ψ (x) n n n n n k=0
= (I1 + I2 + I3 ) f w . Now we estimate Ii separately. Differentiating (3.2) we have 2 ∞ (1 + 2x)ψ(x) xψ(x) k k − x Vn,k (x) = + n n n2 n
k=0
and 2 ∞ k n2 x k 1 + − x Vn,k (x) ψ 2 (x) n n k=0 n2 x ψ(x) (1 + 2x)ψ(x) xψ(x) + ≤ 5n. + = 2 ψ (x) n n2 n
I1 =
For I2 , by Cauchy’s inequality and (3.2) ∞ nx(1 + 2x) k 1 + k Vn,k (x) − x I2 = n ψ 2 (x) n k=0
2n ≤ ψ(x) 2n = ψ(x)
∞ k k=0
ψ(x) n
n
1/2
2 −x
Vn,k (x)
∞ k=0
k 1+ n
1/2
2 Vn,k (x)
1/2 1/2 √ ψ(x) 2 (1 + x) + ≤ 2 3n. n
I3 =
∞ k nx 1+ Vn,k (x) = n. ψ(x) n k=0
The proof is complete. 2 The proof of Theorem 2.1 is based on the next theorem. Theorem 3.6. There exists an absolute constant L such that for n ≥ L √ 3/2 3 N ψ D Vn g ≤ K(N ) n ψD2 g w w
where
lim K(N ) = 0
N →∞
(3.10)
holds for all g ∈ W 2 (wψ). Proof. Inequality (4.12) of [4] gives the following estimate of the third derivative of the N -th power of Baskakov operator
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
39
n(n + 1) 3 N Vn+3,kN (x)P (k1 , ..kN ; n) Q2 D Vn g(x) ≤ N −1 N
⎡ ⎤2 1/n 1/n ⎢ k1 ⎥ + u1 + v1 du1 dv1 ⎦ Vn+3,kN (x)P (k1 , ..kN ; n) × D2 g ⎣ n N
0
0
where
∞
=
N
···
k1 =0
∞
,
P (k1 , ..kN ; n) =
N −1 "
T2,kj
j=1
kN =0
kj+1 n
and 1/n 1/n T2,k (x) = n(n + 1) Vn+2,k (x + t1 + t2 ) dt1 dt2 . 0
0
For the first factor, we use the estimate from Lemma 4.2 of [4] which for 2 ≤ N ≤ n and n ≥ 10 is
Vn+3,kN (x)P (k1 , ..kN ; n) Q2 ≤ CnN ψ −1 (x).
N
For the second factor, using (3.11) of Lemma 3.7 (below), we have ⎡
⎤2 1/n 1/n ⎢ k1 ⎥ + u1 + v1 du1 dv1 ⎦ Vn+3,kN (x)P (k1 , ..kN ; n) D2 g ⎣ n N
0
0
⎡ ⎤2 −1 1/n 1/n k1 2 ⎢ ⎥ + u1 + v1 ≤ ψD2 g w du1 dv1 ⎦ Vn+3,kN (x)P (k1 , ..kN ; n) ⎣ n N
−4
≤ 16n
0
0
−2 k1 + 1 ψD2 g 2 Vn+3,kN (x)P (k1 , ..kN ; n) w n N
2 ≤ 16n−4 K1 (N ) ψD2 g w x−2
where
K1 (N ) = 0. N →∞ N lim
Then, n(n + 1) 3 N CnN ψ −1 (x) 16n−4 K1 (N )x−2 ψD2 g w D Vn g(x) ≤ N −1 # √ K1 (N ) −1 w (x)ψ −3/2 (x) ψD2 g w ≤C n N √ −1 = K(N ) nw (x)ψ −3/2 (x) ψD2 g w
with # K(N ) = C
K1 (N ) . N
2
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
40
Lemma 3.7. There exists a constant L such that for every N ∈ N, n ∈ N, N ≤ the next inequality holds:
Vn+3,kN (x)P (k1 , ..kN ; n)
N
k1 + 1 n
−2
≤ K1 (N )x−2
where
n ≥ L and x ∈ (0, ∞),
n−2 2 ,
K1 (N ) = 0. N →∞ N lim
(3.11)
Proof. We have 1 1 1 . ≤2 2 + x2 x (1 + x)2 (1 + x)2 So, −2 k1 + 1 n N −2 −2 k1 + 1 k1 + 1 1+ ≤2 Vn+3,kN (x)P (k1 , ..kN ; n) n n N −2 k1 + 1 +2 Vn+3,kN (x)P (k1 , ..kN ; n) 1 + . n
Vn+3,kN (x)P (k1 , ..kN ; n)
N
For the first factor on the right, we have from Lemma 4.3 of [4]
Vn+3,kN (x)P (k1 , ..kN ; n)
N
k1 + 1 n
−2
−2
k1 + 1 1+ n
≤ CN 3/4 ln N ψ −2 (x).
(3.12)
For the second one, we have N
−2 k1 + 1 Vn+3,kN (x)P (k1 , ..kN ; n) 1 + n =
∞
...
kN =0
∞
Vn+3,kN (x)
N −1 "
T2,kj
j=2
k2 =0
kj+1 n
∞
1+
k1 =0
k1 + 1 n
−2 T2,k1
k2 n
Now, ∞
1+
k1 =0
k1 + 1 n
= n(n + 1)
−2 T2,k1
k2 n
1/n 1/n ∞ 0
0
k1 =0
k1 + 1 1+ n
−2
Vn+2,k1
k2 + t1 + t 2 n
But because of ∞ k=0
we obtain
1+
k+1 n
−2
Vn+2,k (z) ≤
1+
C n
(1 + z)−2
dt1 dt2 .
.
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42 ∞ k1 =0
1+
k1 + 1 n
So, inductively we obtain for N ≤ N
−2
T2,k1
k2 n
≤
1+
C n
1+
k2 + 1 n
41
−2 .
n−2 2
k1 + 1 Vn+3,kN (x)P (k1 , ..kN ; n) 1 + n
−2
≤
C 1+ n
N
(1 + x)−2 ≤ C(1 + x)−2 .
Consequently,
Vn+3,kN (x)P (k1 , ..kN ; n)
N
k1 + 1 n
−2
≤ 2CN 3/4 ln N x−2 + 2Cx−2 ≤ K1 (N )x−2
with K1 (N ) = CN 3/4 ln N.
2
Proof of Theorem 2.1. We have $ % 1 1 1 2 = inf f − gw + ψD g w ≤ f − Vn f w + ψD2 Vn f w Kw f, n n n which means that it is sufficient to show that for some constant C 1 ψD2 Vn f ≤ C f − Vn f . w w n Also, 1 ψD2 Vn f = 1 ψD2 Vn f − VnN +1 f + VnN +1 f w w n n 1 1 ≤ ψD2 Vn f − VnN f w + ψD2 VnN +1 f w . n n Applying (3.9) and (3.5) we obtain N −1 i 1 Vn f − Vni+1 f ψD2 Vn f − VnN f ≤ 40 f − VnN f ≤ 40 w w w n i=0
≤ 40N f − Vn f w . For the second term, using (3.8), (3.10) and (3.5) we obtain 1 ψD2 VnN +1 f ≤ VnN +2 f − VnN +1 f − ψ D2 VnN +1 f w 2n 2n w N +2 + Vn f − VnN +1 f w ≤ Cn−3/2 ψ 3/2 D3 VnN +1 f + f − Vn f w w ≤ Cn−1 K(N ) ψD2 Vn f w + f − Vn f w , i.e.
(3.13)
42
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
1 ψD2 Vn f ≤ (40N + 2) f − Vn f + 2Cn−1 K(N ) ψD2 Vn f . w w w n Because of limN →∞ K(N ) = 0 we can choose N such that 2CK(N ) ≤
1 2
and Theorem 2.1 follows. 2
Acknowledgment The author would like to thank Professor K.G. Ivanov for his valuable comments and suggestions which greatly help us to improve the presentation of this paper. References [1] V.A. Baskakov, An instance of a sequence of the linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR 113 (1957) 249–251. [2] Z. Ditzian, K.G. Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993) 61–111. [3] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer, Berlin, New York, 1987. [4] I. Gadjev, Strong converse result for Baskakov operator, Serdica Math. J. 40 (2014) 273–318. [5] Shunsheng Guo, Qiulan Qi, Cuixiang Li, Strong converse inequalities for Meyer-König and Zeller operators, J. Math. Anal. Appl. 337 (2008) 994–1001. [6] K.G. Ivanov, P.E. Parvanov, Weighted approximation by Meyer-König and Zeller type operators, in: Constructive Theory of Functions, Sozopol, 2010, 2011, pp. 150–160. [7] V. Totik, Uniform approximation by Baskakov and Meyer-König and Zeller-type operators, Period. Math. Hungar. 14 (3–4) (1983) 209–228.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Strong converse result for uniform approximation by Meyer-König and Zeller operator Ivan Gadjev Department of Mathematics and Informatics, University of Transport, 158 Geo Milev str., 1574 Sofia, Bulgaria
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 4 October 2014 Available online 6 March 2015 Submitted by K. Driver Keywords: Meyer-König and Zeller operator K-functional Direct theorem Strong converse theorem
We characterize the approximation of functions in uniform norm by classical Meyer-König and Zeller operator. Using the closed connection between the approximation by Meyer-König and Zeller operator and the weighted approximation by Baskakov operator, we prove a strong converse inequality of type A in terms of the K-functional. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The Meyer-König and Zeller operator is defined for functions f ∈ C[0, 1) by the formula ∞ Mn (f, x) = f k=0
k n+k
Mn,k (x) where
Mn,k (x) =
n+k k x (1 − x)n+1 . k
(1.1)
The direct theorem for Meyer-König and Zeller operator can be found in [3]. A converse inequality of weaker type B in terminology of [2] is proved in [5]. In this paper we prove the strong converse inequality of type A for Meyer-König and Zeller operator. Before stating our main result, let us introduce some notations. The first derivative operator is denoted d by D = dx . Thus, Dg(x) = g (x) and D2 g(x) = g (x). By ϕ(x) = x(1 − x)2 we denote the weight which is naturally connected with the second derivative of Meyer-König and Zeller operator. By C[0, 1) we denote the space of all continuous on [0, 1) functions. The functions from C[0, 1) are not expected to be continuous or bounded at 1. By L∞ [0, 1) we denote the space of all Lebesgue measurable and essentially bounded in [0, 1) functions equipped with the uniform norm · and by CB[0, 1) = C[0, ∞) ∩ L∞ [0, 1) the space of all continuous and bounded in [0, 1) functions. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2015.03.004 0022-247X/© 2015 Elsevier Inc. All rights reserved.
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
33
Also, we define 2 W∞ (ϕ) = {g : Dg ∈ AC loc (0, 1) and ϕD2 g ∈ L∞ [0, 1)}
where AC loc (0, 1) consists of the functions which are absolutely continuous in [a, b] for every interval [a, b] ⊂ (0, 1). To estimate the approximation f ≈ Mn (f ) we will use the K-functional, defined by 2 Kϕ (f, t) = inf f − g + tϕD2 g : g ∈ W∞ (ϕ), f − g ∈ CB[0, 1) 2 for every function f ∈ CB[0, 1) + W∞ (ϕ), i.e. for every function f which can be represented as f = f1 + f2 2 where f1 ∈ CB[0, 1) and f2 ∈ W∞ (ϕ). It is known, that the K-functional Kϕ (f, t) is equivalent to the √ 2 modulus of smoothness ω√ ϕ (f, t ) [3], where
2√ 2 ω√ ϕ (f, δ) = sup Δh ϕ f 0
and Δ2hϕ(x) f (x)
=
f x + h ϕ(x) − 2f (x) + f x − h ϕ(x) , 0,
x±h
ϕ(x) ∈ [0, 1],
otherwise.
Our main result is the following theorem. Theorem 1.1. For Mn defined by (1.1) there exist absolute constants L, C > 0 such that for every natural n>L 1 ≤ CMn f − f Kϕ f, n 2 holds for all f ∈ CB[0, 1) + W∞ (ϕ).
It is proved in Section 2. Combining Theorem 1.1 with the direct theorem and the equivalency between the modulus of smoothness and the K-functional, we obtain 1 2 √ , f, Mn f − f ∼ ω√ ϕ n i.e. there exist constants C1 and C2 such that 1 2 √ ≤ C2 Mn f − f . f, C1 Mn f − f ≤ ω√ ϕ n For the rest of this paper the constant C will always be an absolute constant, which means it does not depend on f and n. It may be different on each occurrence. 2. Proof of the main result The proof is based on the closed connection between Meyer-König and Zeller and Baskakov operators. V. Totik was the first to use it [7].
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
34
For functions f ∈ C[0, ∞) (the space of all continuous in [0, ∞) functions) the Baskakov operator is given by (see [1]) Vn f (x) = (Vn f, x) = Vn (f, x) =
∞ k Vn,k (x) for 0 ≤ x < ∞, f n
(2.1)
k=0
where Vn,k (x) =
n+k−1 k x (1 + x)−n−k . k
(2.2)
In [6, pp. 150–160] the authors proved that the approximation by Meyer-König and Zeller operators is equivalent to the weighted (weight w(x) = (1 +x)−1 ) approximation by Baskakov operators (see Propositions 2.4, 2.5, 2.6 and 2.7). So, we define for ψ(x) = x(1 + x) and w(x) = (1 + x)−1 f w :=
sup |w(x)f (x)|, x∈[0,∞)
C(w) = {g ∈ C[0, ∞); wg ∈ L∞ [0, ∞)} , W 2 (wψ) = g, Dg ∈ AC loc (0, ∞) and wψD2 g ∈ L∞ [0, ∞) ,
W 3 (wψ 3/2 ) = g, Dg, D2 g ∈ AC loc (0, ∞) and wψ 3/2 D3 g ∈ L∞ [0, ∞) , and Kw (f, t) = inf f − gw + t ψD2 g w : g ∈ W 2 (wψ), f − g ∈ C(w) for every function f ∈ C(w) + W 2 (wψ) and every t > 0. Then Theorem 1.1 follows from the next theorem. Theorem 2.1. For Vn defined by (2.2) there exist absolute constants L, C > 0 such that for every natural n>L 1 ≤ CVn f − f w Kw f, n holds for all f ∈ C(w) + W 2 (wψ). 3. Proof of Theorem 2.1 We will mention some properties of Baskakov operator, which can be found in [3]. is a linear, positive operator with Vn f ≤ f , ψ(x) Vn (1, x) = 1, , Vn (t − x, x) = 0, Vn (t − x)2 , x = n k n DVn,k (x) = − x Vn,k (x). ψ(x) n Vn
(3.1) (3.2) (3.3)
The next inequality is valid for all integers m. The constant C depends only on m.
Vn (t − x)
2m
,x ≤ C
ψ(x) n
m for x ≥
1 . n
(3.4)
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
35
Some lemmas follow. Lemma 3.1. For every natural n, we have Vn f w ≤ f w
(3.5)
for every function f ∈ C(w). Proof. ∞ ∞ k k |Vn f (x)| = f Vn,k (x) ≤ f w Vn,k (x) = (1 + x)f w . 1+ n n k=0
2
k=0
Lemma 3.2. For x, t ∈ [0, ∞) t (t − u)2 u−3/2 (1 + u)−1/2 du ≤ x−3/2 |t − x|3 (1 + x)−1/2 + (1 + t)−1/2 .
(3.6)
x
Proof. We consider two cases. Case 1. t ≥ x. t t (t − u)2 u−3/2 (1 + u)−1/2 du ≤ x−3/2 (1 + x)−1/2 (t − u)2 du x
x
=
1 −3/2 x (1 + x)−1/2 (t − x)3 . 3
Case 2. t < x. Then t x 2 −3/2 −1/2 −1/2 (t − u) u (1 + u) du ≤ (1 + t) (u − t)2 u−3/2 du x
t
−1/2
x
= (1 + t)
t 1− u
3/2 (u − t)1/2 du
t −1/2
x 1−
≤ (1 + t)
t x
3/2 (u − t)1/2 du
t
=
2 −3/2 x (1 + t)−1/2 (x − t)3 . 3
2
Lemma 3.3. There exists a constant C such that for every natural n and every x ∈ [0, ∞)
k/n ∞
k=0 x
k −u n
2
−3/2 −1/2 u (1 + u) du Vn,k (x) ≤ C(1 + x)n−3/2 .
(3.7)
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
36
Proof. Case 1. x ≥ By (3.6) we get
1 n.
2 k/n k −3/2 −1/2 −u u (1 + u) du Vn,k (x) n k=0 x 3 −1/2 ∞ k k − x (1 + x)−1/2 + 1 + ≤ x−3/2 Vn,k (x). n n ∞
k=0
Using Cauchy’s inequality and (3.4), we have for the first sum on the right
x
−3/2
12 ∞ 3 6 ∞ k k −1/2 −3/2 −1/2 − x (1 + x) − x Vn,k (x) Vn,k (x) ≤ x (1 + x) n n
k=0
k=0
−3/2
≤ Cx
(1 + x)
−1/2
ψ(x) n
32
≤ C(1 + x)n−3/2 ,
and for the second one x
−3/2
3 −1/2 ∞ k − x 1 + k Vn,k (x) n n
k=0
≤x
−3/2
12 ∞ 12 6 −1 k − x Vn,k (x) 1+ Vn,k (x) n n
∞ k k=0
≤ Cx−3/2 (1 + x)−1/2 Case 2. x < We have
ψ(x) n
32
k=0
≤ C(1 + x)n−3/2 .
1 n.
I0 = (1 + x)
−n
x u
1/2
(1 + u)
−1/2
x du ≤
0
u1/2 du ≤
2 (1 + x)n−3/2 3
0
and by (3.2) ∞ k=1
Ik =
k/n ∞ k k=1 x
n
2 −u
u
−3/2
(1 + u)
−1/2
k/n ∞ 2 k Vn,k (x)du ≤ u−3/2 Vn,k (x)du n k=1
x
∞ 2 k ψ(x) −1/2 −1/2 2 ≤ 2x x + ≤ 6(1 + x)n−3/2 . Vn,k (x) = 2x n n
2
k=1
Lemma 3.4. There exists a constant C such that for every natural n, we have 3/2 3 Vn g − g − 1 ψD2 g ≤ C D g ψ 2n w n3/2 w for every function g ∈ W 3 (wψ 3/2 ).
(3.8)
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
37
Proof. By Taylor’s formula, we have (t − x)2 2 1 D g(x) + g(t) = g(x) + (t − x)Dg(x) + 2 2
t (t − v)2 D3 g(v)dv. x
Multiplying both sides by Vn,k (x), summing with respect to k and using the identities (3.2) we get Vn g(x) − g(x) − 1 ψD2 g(x) 2n k/n ∞ 2 1 k 3 ≤ − v D g(v)dv Vn,k (x) 2 n k=0 x k/n ∞ 2 k 1 3/2 3 −3/2 −1/2 −v v (1 + v) dv Vn,k (x). ≤ ψ D g 2 n w k=0 x Now we can use (3.7) to complete the proof. 2 Lemma 3.5. For every natural n and for every function f ∈ C(w) ψD2 Vn f ≤ 40nf w . w
(3.9)
Proof. We consider two cases. Case 1. x ≤ n1 . We have [3] D2 Vn f (x) = n(n + 1)
∞
Δ21 f n
k=0
k Vn+2,k (x), n
where, as usual, Δ2h f (x) = f (x + 2h) − 2f (x + h) + f (x). Then, ∞ k + 2 k + 1 k 2 w(x)ψ(x) D Vn f (x) = n(n + 1)x f − 2f +f Vn+2,k (x) n n n k=0 ∞ k+2 k+1 k ≤ n(n + 1)xf w 1+ +2 1+ + 1+ Vn+2,k (x) n n n k=0
= 4(n + 1)x [n + 1 + (n + 2)x] f w ≤ 40nf w . Case 2. x > n1 . Differentiating (3.3) we get
n2 ψ(x)D2 Vn,k (x) = ψ(x) Then
2
k −x n
(1 + 2x)n − ψ(x)
k − x − n Vn,k (x). n
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
38
2 ∞ 2 n ψ(x) 1 + 2x x k k k w(x)ψ(x) D2 Vn f (x) = −x − −x − f Vn,k (x) ψ 2 (x) n n n n n k=0 2 ∞ ψ(x) k 1 + 2x k n2 x k + f w −x + − x 1 + Vn,k (x) ≤ 2 ψ (x) n n n n n k=0
= (I1 + I2 + I3 ) f w . Now we estimate Ii separately. Differentiating (3.2) we have 2 ∞ (1 + 2x)ψ(x) xψ(x) k k − x Vn,k (x) = + n n n2 n
k=0
and 2 ∞ k n2 x k 1 + − x Vn,k (x) ψ 2 (x) n n k=0 n2 x ψ(x) (1 + 2x)ψ(x) xψ(x) + ≤ 5n. + = 2 ψ (x) n n2 n
I1 =
For I2 , by Cauchy’s inequality and (3.2) ∞ nx(1 + 2x) k 1 + k Vn,k (x) − x I2 = n ψ 2 (x) n k=0
2n ≤ ψ(x) 2n = ψ(x)
∞ k k=0
ψ(x) n
n
1/2
2 −x
Vn,k (x)
∞ k=0
k 1+ n
1/2
2 Vn,k (x)
1/2 1/2 √ ψ(x) 2 (1 + x) + ≤ 2 3n. n
I3 =
∞ k nx 1+ Vn,k (x) = n. ψ(x) n k=0
The proof is complete. 2 The proof of Theorem 2.1 is based on the next theorem. Theorem 3.6. There exists an absolute constant L such that for n ≥ L √ 3/2 3 N ψ D Vn g ≤ K(N ) n ψD2 g w w
where
lim K(N ) = 0
N →∞
(3.10)
holds for all g ∈ W 2 (wψ). Proof. Inequality (4.12) of [4] gives the following estimate of the third derivative of the N -th power of Baskakov operator
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
39
n(n + 1) 3 N Vn+3,kN (x)P (k1 , ..kN ; n) Q2 D Vn g(x) ≤ N −1 N
⎡ ⎤2 1/n 1/n ⎢ k1 ⎥ + u1 + v1 du1 dv1 ⎦ Vn+3,kN (x)P (k1 , ..kN ; n) × D2 g ⎣ n N
0
0
where
∞
=
N
···
k1 =0
∞
,
P (k1 , ..kN ; n) =
N −1 "
T2,kj
j=1
kN =0
kj+1 n
and 1/n 1/n T2,k (x) = n(n + 1) Vn+2,k (x + t1 + t2 ) dt1 dt2 . 0
0
For the first factor, we use the estimate from Lemma 4.2 of [4] which for 2 ≤ N ≤ n and n ≥ 10 is
Vn+3,kN (x)P (k1 , ..kN ; n) Q2 ≤ CnN ψ −1 (x).
N
For the second factor, using (3.11) of Lemma 3.7 (below), we have ⎡
⎤2 1/n 1/n ⎢ k1 ⎥ + u1 + v1 du1 dv1 ⎦ Vn+3,kN (x)P (k1 , ..kN ; n) D2 g ⎣ n N
0
0
⎡ ⎤2 −1 1/n 1/n k1 2 ⎢ ⎥ + u1 + v1 ≤ ψD2 g w du1 dv1 ⎦ Vn+3,kN (x)P (k1 , ..kN ; n) ⎣ n N
−4
≤ 16n
0
0
−2 k1 + 1 ψD2 g 2 Vn+3,kN (x)P (k1 , ..kN ; n) w n N
2 ≤ 16n−4 K1 (N ) ψD2 g w x−2
where
K1 (N ) = 0. N →∞ N lim
Then, n(n + 1) 3 N CnN ψ −1 (x) 16n−4 K1 (N )x−2 ψD2 g w D Vn g(x) ≤ N −1 # √ K1 (N ) −1 w (x)ψ −3/2 (x) ψD2 g w ≤C n N √ −1 = K(N ) nw (x)ψ −3/2 (x) ψD2 g w
with # K(N ) = C
K1 (N ) . N
2
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
40
Lemma 3.7. There exists a constant L such that for every N ∈ N, n ∈ N, N ≤ the next inequality holds:
Vn+3,kN (x)P (k1 , ..kN ; n)
N
k1 + 1 n
−2
≤ K1 (N )x−2
where
n ≥ L and x ∈ (0, ∞),
n−2 2 ,
K1 (N ) = 0. N →∞ N lim
(3.11)
Proof. We have 1 1 1 . ≤2 2 + x2 x (1 + x)2 (1 + x)2 So, −2 k1 + 1 n N −2 −2 k1 + 1 k1 + 1 1+ ≤2 Vn+3,kN (x)P (k1 , ..kN ; n) n n N −2 k1 + 1 +2 Vn+3,kN (x)P (k1 , ..kN ; n) 1 + . n
Vn+3,kN (x)P (k1 , ..kN ; n)
N
For the first factor on the right, we have from Lemma 4.3 of [4]
Vn+3,kN (x)P (k1 , ..kN ; n)
N
k1 + 1 n
−2
−2
k1 + 1 1+ n
≤ CN 3/4 ln N ψ −2 (x).
(3.12)
For the second one, we have N
−2 k1 + 1 Vn+3,kN (x)P (k1 , ..kN ; n) 1 + n =
∞
...
kN =0
∞
Vn+3,kN (x)
N −1 "
T2,kj
j=2
k2 =0
kj+1 n
∞
1+
k1 =0
k1 + 1 n
−2 T2,k1
k2 n
Now, ∞
1+
k1 =0
k1 + 1 n
= n(n + 1)
−2 T2,k1
k2 n
1/n 1/n ∞ 0
0
k1 =0
k1 + 1 1+ n
−2
Vn+2,k1
k2 + t1 + t 2 n
But because of ∞ k=0
we obtain
1+
k+1 n
−2
Vn+2,k (z) ≤
1+
C n
(1 + z)−2
dt1 dt2 .
.
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42 ∞ k1 =0
1+
k1 + 1 n
So, inductively we obtain for N ≤ N
−2
T2,k1
k2 n
≤
1+
C n
1+
k2 + 1 n
41
−2 .
n−2 2
k1 + 1 Vn+3,kN (x)P (k1 , ..kN ; n) 1 + n
−2
≤
C 1+ n
N
(1 + x)−2 ≤ C(1 + x)−2 .
Consequently,
Vn+3,kN (x)P (k1 , ..kN ; n)
N
k1 + 1 n
−2
≤ 2CN 3/4 ln N x−2 + 2Cx−2 ≤ K1 (N )x−2
with K1 (N ) = CN 3/4 ln N.
2
Proof of Theorem 2.1. We have $ % 1 1 1 2 = inf f − gw + ψD g w ≤ f − Vn f w + ψD2 Vn f w Kw f, n n n which means that it is sufficient to show that for some constant C 1 ψD2 Vn f ≤ C f − Vn f . w w n Also, 1 ψD2 Vn f = 1 ψD2 Vn f − VnN +1 f + VnN +1 f w w n n 1 1 ≤ ψD2 Vn f − VnN f w + ψD2 VnN +1 f w . n n Applying (3.9) and (3.5) we obtain N −1 i 1 Vn f − Vni+1 f ψD2 Vn f − VnN f ≤ 40 f − VnN f ≤ 40 w w w n i=0
≤ 40N f − Vn f w . For the second term, using (3.8), (3.10) and (3.5) we obtain 1 ψD2 VnN +1 f ≤ VnN +2 f − VnN +1 f − ψ D2 VnN +1 f w 2n 2n w N +2 + Vn f − VnN +1 f w ≤ Cn−3/2 ψ 3/2 D3 VnN +1 f + f − Vn f w w ≤ Cn−1 K(N ) ψD2 Vn f w + f − Vn f w , i.e.
(3.13)
42
I. Gadjev / J. Math. Anal. Appl. 428 (2015) 32–42
1 ψD2 Vn f ≤ (40N + 2) f − Vn f + 2Cn−1 K(N ) ψD2 Vn f . w w w n Because of limN →∞ K(N ) = 0 we can choose N such that 2CK(N ) ≤
1 2
and Theorem 2.1 follows. 2
Acknowledgment The author would like to thank Professor K.G. Ivanov for his valuable comments and suggestions which greatly help us to improve the presentation of this paper. References [1] V.A. Baskakov, An instance of a sequence of the linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR 113 (1957) 249–251. [2] Z. Ditzian, K.G. Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993) 61–111. [3] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer, Berlin, New York, 1987. [4] I. Gadjev, Strong converse result for Baskakov operator, Serdica Math. J. 40 (2014) 273–318. [5] Shunsheng Guo, Qiulan Qi, Cuixiang Li, Strong converse inequalities for Meyer-König and Zeller operators, J. Math. Anal. Appl. 337 (2008) 994–1001. [6] K.G. Ivanov, P.E. Parvanov, Weighted approximation by Meyer-König and Zeller type operators, in: Constructive Theory of Functions, Sozopol, 2010, 2011, pp. 150–160. [7] V. Totik, Uniform approximation by Baskakov and Meyer-König and Zeller-type operators, Period. Math. Hungar. 14 (3–4) (1983) 209–228.