Some approximation results on Bernstein-Schurer operators defined

arXiv:1504.05876v3 [math.CA] 26 Nov 2015

Some approximation results on Bernstein-Schurer operators defined by (p, q)-integers (Revised) M. Mursaleen, Md. Nasiruzzaman and Ashirbayev Nurgali Department of Mathematics, Aligarh Muslim University, Aligarh–202002, India [email protected]; [email protected]

Abstract In the present article, we have given a corrigendum to our paper Some approximation results on Bernstein-Schurer operators defined by (p, q)-integers published in Journal of Inequalities and Applications (2015) 2015:249. Keywords and phrases: q-integers; (p, q)-integers; Bernstein operator; (p, q)-Bernstein operator; q-Bernstein-Schurer operator; (p, q)-Bernstein-Schurer operator; Modulus of continuity.

AMS Subject Classification (2010): 41A10, 41A25, 41A36, 40A30. 1. Introduction and Preliminaries In 1912, S.N Bernstein [4] introduced the following sequence of operators Bn : C[0, 1] → C[0, 1] defined for any n ∈ N and for any f ∈ C[0, 1] such as   n   X k n k n−k , x ∈ [0, 1]. (1.1) x (1 − x) f Bn (f ; x) = n k k=0 In 1987, Lupa [8] introduced the q-Bernstein operators by applying the idea of q-integers, and in 1997 another generalization of these operators introduced by Philip [19]. Later on, many authors introduced q-generalization of various operators and investigated several approximation properties. For instance, q-analogue of Stancu-Beta operators in [3] and [12]; q-analogue of Bernstein-Kantorovich operators in [20]; q- Baskakov-Kantorovich operators in [7]; q-Sz´asz-Mirakjan operators in [18]; q-Bleimann, Butzer and Hahn operators in [2] and in [6]; qanalogue of Baskakov and Baskakov-Kantorovich operators in [9]; q-analogue of Sz´asz-Kantorovich operators in [10]; and q-analogue of generalized BernsteinShurer operators in [13]. We recall certain notations on (p, q)-calculus. The (p, q)-integer was introduced in order to generalize or unify several forms of q-oscillator algebras well known in the earlier physics literature related to the representation theory of single parameter quantum algebras [5]. The (p, q)-integer [n]p,q is defined by [n]p,q =

pn − q n , n = 0, 1, 2, · · · , 0 < q < p ≤ 1. p−q 1

2

The (p, q)-Binomial expansion is by)np,q

(ax +

:=

n X

p

(n−k)(n−k−1) 2

q

k(k−1) 2

k=0



n k



an−k bk xn−k y k p,q

(x + y)np,q := (x + y)(px + qy)(p2x + q 2 y) · · · (pn−1 x + q n−1 y). Also, the (p, q)-binomial coefficients are defined by   [n]p,q ! n := . k p,q [k]p,q ![n − k]p,q ! Details on (p, q)-calculus can be found in [22]. For p = 1, all the notions of (p, q)-calculus are reduced to q-calculus [1]. In 1962, Schurer [21] introduced and studied the operators Sm,ℓ : C[0, ℓ + 1] → C[0, 1] defined for any m ∈ N and ℓ be fixed in N and any function f ∈ C[0, ℓ + 1] as follows   m+ℓ X m+ℓ  k k m+ℓ−k x (1 − x) f Sm,ℓ (f ; x) = , x ∈ [0, 1]. (1.1) k m k=0 For any m ∈ and f ∈ C[0, ℓ+1], ℓ is fixed, then q− analogue of Bernstein-Schurer operators in [11] defined as follows ˜m,ℓ (f ; q; x) = B

m+ℓ X k=0

m+ℓ k



k

x q

m+ℓ−k−1 Y

s

(1 − q x) f

s=0



[k]q [m]q



, x ∈ [0, 1]. (1.2)

Our aim is to introduce a (p, q)-analogue of these operators. We investigate the approximation properties of this class and we estimate the rate of convergence and some theorem by using the modulus of continuity. We study the approximation properties based on Korovkin’s type approximation theorem and also establish the some direct theorem. 2. Construction of (p, q)-Bernstein-Schurer operators (Revised) Mursaleen et. al [14] has defined (p, q)-analogue of Bernstein operators as:    n−k−1 n  Y X [k]p,q n s s k p,q (p − q x)f x Bn (f ; x) = , x ∈ [0, 1]. (2.1) k p,q [n]p,q k=0

s=0

p,q Bm,ℓ (f ; x)

But 6= 1, for all x ∈ [0, 1]. Hence, They re-introduced the (p, q) Bernstein operators [15] as follows:

p,q Bm,ℓ (f ; x)

1

= p

n(n−1) 2

   n−k−1 n  Y X k(k−1) [k]p,q n s s k (p −q x)f , x ∈ [0, 1]. p 2 x k p,q pk−n [n]p,q k=0

s=0

(2.2)

3

Mursaleen et. al [17] introduced the (p, q)-analogue of Bernstein Schurer operators as: p,q Bm,ℓ (f ; x)

=

m+ℓ X k=0

m+ℓ k



k

x

p,q

m+ℓ−k−1 Y

s

s

(p − q x)f

s=0



[k]p,q [m]p,q



, x ∈ [0, 1]. (2.3)

p,q But Bm,ℓ (f ; x) 6= 1, for all x ∈ [0, 1]. Hence, we re-define our operators as follows:

We consider 0 < q < p ≤ 1 and for any m ∈ N, f ∈ C[0, ℓ + 1], ℓ is fixed, we construct a revised generalized (p, q)-Bernstein Schurer operators: m+ℓ X

1



m+ℓ−k−1 Y



[k]p,q = (m+ℓ)(m+ℓ−1) x (p −q x)f p k−m−ℓ p [m]p,q 2 p p,q s=0 k=0 (2.4) Clearly, the operator defined by (2.4) is linear and positive. And if we put p = 1 in (2.4), then (p, q) Shurer operator given by (2.4) turn out the q- Bernstein Shurer operators [11].

p,q Bm,ℓ (f ; x)

m+ℓ k

k(k−1) 2

k

s

s

p,q Lemma 2.1. Let Bm,ℓ (.; .) be given by (2.4), then for any x ∈ [0, 1] and 0 < q < p ≤ 1 we have the following identities p,q (i) Bm,ℓ (e0 ; x) = 1 p,q p,q (ii) Bm,ℓ (e1 ; x) = [m+ℓ] x [m]p,q p,q (iii) Bm,ℓ (e2 ; x) =

pm+ℓ−1 [m+ℓ]p,q x [m]2p,q

+

q[m+ℓ]p,q [m+ℓ−1]p,q 2 x [m]2p,q

where ej (t) = tj , j = 0, 1, 2, · · · . Proof.

(i) For 0 < q < p ≤ 1 we use the known identity from [14]  n  n−k−1 X Y k(k−1) (n)(n−1) n k p 2 x (ps − q s x) = p 2 , x ∈ [0, 1] k p,q s=0

k=0

Suppose we choose n = m + ℓ. Since (1 − x)m+ℓ−k = p,q

m+ℓ−k−1 Y

(ps − q s x),

s=0

we get m+ℓ X k=0

m+ℓ k



p,q

p

k(k−1) 2

k

x

m+ℓ−k−1 Y

(ps − q s x) = p

s=0

p,q Consequently, which implies Bm,ℓ (e0 ; x) = 1.

(m+ℓ)(m+ℓ−1) 2



, x ∈ [0, 1].

4

(ii) Clearly we have p,q Bm,ℓ (e1 ; x)

1

= p

(m+ℓ)(m+ℓ−1) 2

1

= p =

m+ℓ X k=0

[m + ℓ]p,q [m]p,q

n(n−3) 2

x (m+ℓ−1)(m+ℓ−2) 2

p [m + ℓ]p,q = x. [m]p,q

m+ℓ k m+ℓ−1 X  k=0

[m + ℓ]p,q [m]p,q



p

k(k−1) 2

k

x

p,q

m+ℓ−k−1 Y

[k]p,q k−m−ℓ p [m]

(ps − q s x)

s=0

p,q

 m+ℓ−k−2 Y (k+1)(k−2) m+ℓ−1 k+1 2 p (ps − q s x) x k p,q s=0   m+ℓ−1 m+ℓ−k−2 X Y k(k−1) m+ℓ−1 p 2 xk (ps − q s x) k p,q s=0

k=0

p,q (iii) Bm,ℓ (e2 ; x)

1

= p

(m+ℓ)(m+ℓ−1) 2

1

= p

(m+ℓ)(m+ℓ−5) 2

1

= p

(m+ℓ)(m+ℓ−5) 2

1

m+ℓ X k=0

m+ℓ k

[m + ℓ]p,q [m]2p,q [m + ℓ]p,q [m]2p,q [m + ℓ]p,q [m]2p,q



p p,q

m+ℓ−1 X  k=0

m+ℓ−1 X  k=0

m+ℓ−1 X 

k(k−1) 2

xk

m+ℓ−k−1 Y

(ps − q s x)

s=0

m+ℓ−1 k



m+ℓ−1 k



p

(k+1)(k−4) 2

[k]2p,q p2k−2m−2ℓ [m]2p,q

k+1

x

p,q

m+ℓ−k−2 Y

(ps − q s x) [k + 1]p,q

s=0

p

(k+1)(k−4) 2

p,q



xk+1

m+ℓ−k−2 Y

(ps − q s x) (pk + q[k]p,q )

s=0

m+ℓ−k−2 Y

k2 −k−4 m+ℓ−1 p 2 xk+1 (ps − q s x) k p p,q s=0 k=0   m+ℓ−2 m+ℓ−k−3 Y (k+2)(k−3) q[m + ℓ − 1]p,q [m + ℓ]p,q X m+ℓ−2 k+2 2 p + (m+ℓ)(m+ℓ−5) x (ps − q s x) 2 k [m] 2 p,q p p,q s=0 k=0   m+ℓ−1 m+ℓ−k−2 Y k(k−1) pm+ℓ−1 x [m + ℓ]p,q X m+ℓ−1 k 2 p = (ps − q s x) x (m+ℓ−1)(m+ℓ−2) 2 k [m] 2 p,q p p,q s=0 k=0   m+ℓ−k−3 m+ℓ−2 Y X k(k−1) 1 q[m + ℓ − 1]p,q [m + ℓ]p,q x2 m+ℓ−2 k 2 + p x (ps − q s x) (m+ℓ−2)(m+ℓ−3) 2 k [m]p,q 2 p p,q s=0 k=0

=

=

(m+ℓ)(m+ℓ−5) 2

q[m + ℓ − 1]p,q [m + ℓ]p,q 2 pm+ℓ−1 [m + ℓ]p,q x+ x. 2 [m]p,q [m]2p,q

 p,q Lemma 2.2. Let Bm,ℓ (.; .) be given by lemma (2.1), then for any x ∈ [0, 1] and 0 < q < p ≤ 1 we have the following identities p,q p,q x−1 (i) Bm,ℓ (e1 − 1; x) = [m+ℓ] [m]p,q  p,q p,q (ii) Bm,ℓ (e1 − x; x) = [m+ℓ] − 1 x [m]p,q   m+ℓ−1 q[m+ℓ−1]p,q [m+ℓ]p,q p,q p,q p,q x + 1 − 2 [m+ℓ] + x2 . (iii) Bm,ℓ ((e1 − x)2 ; x) = p [m][m+ℓ] 2 2 [m]p,q [m] p,q

p,q

5

3. On the convergence of (p, q)-Bernstein-Schurer operators Let f ∈ C[0, γ], and the modulus of continuity of f denoted by ω(f, δ) gives the maximum oscillation of f in any interval of length not exceeding δ > 0 and it is given by the relation ω(f, δ) = sup | f (y) − f (x) |, x, y ∈ [0, γ]. |y−x|≤δ

It is known that limδ→0+ ω(f, δ) = 0 for f ∈ C[0, γ] and for any δ > 0 one has   | y−x | | f (y) − f (x) |≤ + 1 ω(f, δ). (3.1) δ 1 . In order to reach For q ∈ (0, 1) and p ∈ (q, 1] obviously have limm→∞ [m]p,q = p−q p,q to the convergence results of the operator Bm,ℓ , we take a sequence qm ∈ (0, 1) and pm ∈ (qm , 1] such that limm→∞ pm = 1 and limm→∞ qm = 1, so we get limm→∞ [m]pm ,qm = ∞.

Theorem 3.1. Let p = pm , q = qm satisfying 0 < qm < pm ≤ 1 such that limm→∞ pm = 1, limm→∞ qm = 1. Then for each f ∈ C[0, ℓ + 1], pm ,qm lim Bm,ℓ (f ; x) = f,

m→∞

(3.2)

is uniformly on [0, 1]. Proof. The proof is based on the well known Korovkin theorem regarding the convergence of a sequence of linear and positive operators, so it is enough to prove the conditions pm ,qm ((ej ; x) = xj , j = 0, 1, 2, {as m → ∞} Bm,ℓ

uniformly on [0, 1]. Clearly we have pm ,qm (e0 ; x) = 1. lim Bm,ℓ

m→∞

By taking the simple calculation we get [m + ℓ]pm ,qm = 1, as 0 < qm < pm ≤ 1. m→∞ [m]pm ,qm lim

Since as 0 < qm < pm ≤ 1, then we get, [m + ℓ]pm ,qm = 0. m→∞ [m]2pm ,qm lim

Hence we have pm ,qm (e1 ; x) = x lim Bm,ℓ

m→∞

pm ,qm (e2 ; x) = x2 lim Bm,ℓ

m→∞



6

Theorem 3.2. If f ∈ C[0, ℓ + 1], then p,q | Bm,ℓ (f ; x) − f (x) |≤ 2ωf (δm ),

where s s [m + ℓ]p,q [m + ℓ] (q[m + ℓ − 1]p,q − [m + ℓ]p,q ) x2 + p(m+ℓ−1) x p,q δm = x −1 + . . [m]p,q [m]p,q [m]p,q p,q Proof. | Bm,ℓ (f ; x) − f (x) |

≤

≤

1 p(m+ℓ)(m+ℓ−1) 1 p(m+ℓ)(m+ℓ−1)

m+ℓ X

m+ℓ−k−1 Y

  [k]p,q p − f (x) x (p − q x) f k−m−ℓ p [m]p,q p,q s=0 k=0   [k] p,q   − x m+ℓ m+ℓ−k−1 X m+ℓ Y   pk−m−ℓ [m]p,q k(k−1)  ω(f, δ). p 2 xk + 1 (ps − q s x)    k δ p,q s=0 k=0 m+ℓ k



k(k−1) 2

k

s

s

By using the Cauchy inequality and lemma (2.1) we have p,q | Bm,ℓ (f ; x) − f (x) |  ( ) 12    2 m+ℓ−k−1  m+ℓ Y X k(k−1) 1 [k]p,q 1 m+ℓ p 2 xk −x (ps − q s x)  ≤ 1 + (m+ℓ)(m+ℓ−1) k δ p [m] p,q p,q s=0 k=0
1 p,q Bm,ℓ (e0 ; x) 2 ω(f, δ)  
12 1 p,q p,q 2 p,q Bm,ℓ (e2 ; x) − 2xBm,ℓ (e1 ; x) + x Bm,ℓ (e0 ; x) + 1 ω(f, δ) = δ (  )    12 1 [m + ℓ]p,q pm+ℓ−1 [m + ℓ]p,q [m + ℓ]p,q [m + ℓ − 1]p,q = + 1 ω(f, δ) x+ q−2 + 1 x2 δ [m]2p,q [m]2p,q [m]p,q   2 ! 12 q     2 q [m+ℓ]p,q (q[m+ℓ−1]p,q −[m+ℓ]p,q )x2 +p(m+ℓ−1) x [m+ℓ]p,q 1 = δ + 1 ω(f, δ) x [m]p,q − 1 . + [m]p,q [m]p,q   q     q p,q + [m+ℓ]p,q . (q[m+ℓ−1]p,q −[m+ℓ]p,q )x2 +p(m+ℓ−1) x + 1 ω(f, δ). − 1 ≤ 1δ x [m+ℓ] [m]p,q [m]p,q [m]p,q ×

1

{by using (a2 + b2 ) 2 ≤ (| a | + | b |)}. Hence we obtain the desired result by choosing δ = δm .

4. Direct Theorems on (p, q)-Bernstein-Schurer operators The Peetre’s K-functional is defined by  K2 (f, δ) = inf (k f − g k +δ k g ′′ k) : g ∈ W 2 , where

W 2 = {g ∈ C[0, ℓ + 1] : g ′, g ′′ ∈ C[0, ℓ + 1]} .



7 1

Then there exits a positive constant C > 0 such that K2 (f, δ) ≤ Cω2 (f, δ 2 ), δ > 0, where the second order modulus of continuity is given by 1

ω2 (f, δ 2 ) = sup

1 0 0. x

Now by lemma (2.2), we have  Z t   [m + ℓ]p,q p,q p,q ′ ′′ Bm,ℓ (g; x) = g(x) + xg (x) (e1 − u)g (u)du; p, q; x − 1 + Bm,ℓ [m]p,q x   Z t   p,q [m + ℓ] p,q p,q ′′ ′ ≤ B B (g; x) − g(x) − xg (x) − 1 | (e − u) | | g (u) | du; p, q; x 1 m,ℓ m,ℓ [m]p,q x
p,q (e1 − x)2 ; p, q; x k g ′′ k ≤ Bm,ℓ

Hence we get   p,q [m+ℓ] p,q ′ B (g; x) − g(x) − xg (x) − 1 m,ℓ [m]p,q   2 [m+ℓ]p,q (m+ℓ−1) [m+ℓ]p,q ′′ ≤k g k ( [m]2 p x+ − 1 + [m]p,q p,q

[m+ℓ]p,q [m]2p,q

 (q[m + ℓ − 1]p,q − [m + ℓ]p,q ) x .

On the other hand we have   p,q B (f ; x) − f (x) − xg ′ (x) [m+ℓ]p,q − 1 ≤ | B p,q ((f − g); x) − (f − g)(x) | m,ℓ [m]p,q m,ℓ   p,q [m + ℓ] p,q ′ + Bm,ℓ (g; x) − g(x) − xg (x) − 1 . [m]p,q

Since we know the relation

p,q | Bm,ℓ (f ; x) |≤k f k .



2

8

Therefore   p,q [m+ℓ] p,q ′ B (f ; x) − f (x) − xg (x) − 1 ≤ k f − g k m,ℓ [m]p,q     2 [m+ℓ]p,q (m+ℓ−1) [m+ℓ]p,q [m+ℓ]p,q ′′ + k g k ( [m]2 p x+ − 1 + [m]2 (q[m + ℓ − 1]p,q − [m + ℓ]p,q ) x2 , [m]p,q p,q

p,q

Now taking the infimum on the right hand side over all g ∈ W 2 , we get   p,q
[m + ℓ] p,q 2 ′ ≤ CK2 f, δm B (f ; x) − f (x) − xg (x) − 1 (x) . m,ℓ [m]p,q

In the view of the property of K-functional, we get   p,q B (f ; x) − f (x) − xg ′ (x) [m + ℓ]p,q − 1 ≤ Cω2 (f, δm (x)) . m,ℓ [m]p,q This completes the proof.



Theorem 4.2. Let f ∈ C[0, ℓ + 1] be such that f ′ , f ′′ ∈ C[0, ℓ + 1], and the sequence {pm }, {qm } satisfying 0 < qm < pm ≤ 1 such that pm → 1, qm → 1 and m pm m → α, qm → β as m → ∞, where 0 ≤ α, β < 1. Then
x(λ − αx) ′′ pm ,qm (f ; x) − f (x) = lim [m]pm ,qm Bm,ℓ f (x), m→∞ 2 is uniformly on [0, ℓ + 1], where 0 < λ ≤ 1. Proof. From the Taylors formula we have 1 f (t) = f (x) + f ′ (x)(t − x) + f ′′ (x)(t − x)2 + r(t, x)(e1 − x)2 , 2 where r(t, x) is the remainder term and limt→x r(t, x) = 0, therefore we have
pm ,qm (f ; x) − f (x) [m]pm ,qm Bm,ℓ   ′′ pm ,qm pm ,qm pm ,qm ((e1 − x); x) + f 2(x) Bm,ℓ = [m]pm ,qm f ′ (x)Bm,ℓ (r(t, x)(t − x)2 ; x) . ((e1 − x)2 ; x) + Bm,ℓ Now by applying the Cauchy-Schwartz inequality, we have q
q pm ,qm pm ,qm pm ,qm 2 2 r(t, x)(t − x) ; x) ≤ Bm,ℓ (r (t, x); x)). Bm,ℓ Bm,ℓ ((t − x)4 ; x)).

Since r 2 (x, x) = 0, and r 2 (t, x) ∈ C[0, ℓ + 1], then for from the Theorem 3.1 we have
pm ,qm r 2 (t, x); x) = r 2 (x, x) = 0, Bm,ℓ which imply that


pm ,qm r(t, x)(t − x)2 ; x) = 0 Bm,ℓ  
[m + ℓ]pm ,qm pm ,qm −1 =0 lim [m]pm ,qm Bm,ℓ ((e1 − x); x)) = x lim [m]pm ,qm m→∞ m→∞ [m]pm ,qm
pm ,qm ((e1 − x)2 ; x)) limm→∞ [m]pm ,qm Bm,ℓ = x lim [m]pm ,qm m→∞

[m + ℓ]pm ,qm m+ℓ−1 p [m]2pm ,qm m

9

2

+x lim [m]pm ,qm m→∞



[m + ℓ]pm ,qm −1 [m]pm ,qm

2

[m + ℓ]pm ,qm (qm [m + ℓ − 1]pm ,qm − [m + ℓ]pm ,qm ) + [m]2pm ,qm

pm ,qm (e1 − x)2 ; x) lim [m]pm ,qm Bm,ℓ

m→∞





!

= λx − αx2 = x(λ − αx),

where λ ∈ (0, 1] depending on the sequence {pm }. Hence we have
x(λ − αx) ′′ pm ,qm f (x). (f ; x) − f (x) = lim [m]pm ,qm Bm,ℓ m→∞ 2 This completes the proof.



p,q Now we give the rate of convergence of the operators Bm,ℓ (f ; x) in terms of the elements of the usual Lipschitz class LipM (ν). Let f ∈ C[0, m + ℓ], M > 0 and 0 < ν ≤ 1. We recall that f belongs to the class LipM (ν) if the inequality

| f (t) − f (x) |≤ M | t − x |ν (t, x ∈ (0, 1]) is satisfied. Theorem 4.3. Let 0 < q < p ≤ 1. Then for each f ∈ LipM (ν) we have p,q ν | Bm,ℓ (f ; x) − f (x) |≤ Mδm (x) 2 where δm (x) =

+



[m+ℓ]p,q (m+ℓ−1) p x [m]2p,q

[m + ℓ]p,q −1 [m]p,q

2

! [m + ℓ]p,q (q[m + ℓ − 1]p,q − [m + ℓ]p,q ) x2 . + [m]2p,q

p,q Proof. By the monotonicity of the operators Bm,ℓ (f ; x), we can write p,q | Bm,ℓ (f ; x) − f (x) | p,q ≤ Bm,ℓ (| f (t) − f (x) |; p, q; x)   m+ℓ−k−1 m+ℓ Y X m +ℓ  k(k−1) [k] 1 p,q k s s (p − q x) f ≤ (m+ℓ)(m+ℓ−1) p 2 x − f (x) k−m−ℓ [m] k p p p,q p,q s=0 k=0 ν m+ℓ−k−1 m+ℓ Y X m+ℓ  k(k−1) [k]p,q 1 s s k 2 (p − q x) k−m−ℓ x p ≤ M (m+ℓ)(m+ℓ−1) − x k p p [m]p,q p,q s=0 k=0 ν 2 ! 2    2−ν m+ℓ X 2 1 [k]p,q 1 , − x P (x) P (x) = M m,ℓ,k m,ℓ,k p(m+ℓ)(m+ℓ−1) pk−m−ℓ [m]p,q p(m+ℓ)(m+ℓ−1) k=0

where Pm,ℓ,k (x) =



m+ℓ k



p,q

p

k(k−1) 2

xk

Qm+ℓ−k−1 s=0

(ps − q s x)

10

Now applying the H¨older’s inequality for the sum with p = p,q | Bm,ℓ (f ; x) − f (x) | ≤ M

1 p(m+ℓ)(m+ℓ−1)

m+ℓ X k=0

Pm,ℓ,k (x)



2 ! ν2 [k]p,q −x [m]p,q



ν (e1 − x) ; x 2 = M q p,q Choosing δ : δm (x) = Bm,ℓ ((e1 − x)2 ; x), we obtain p,q Bm,ℓ

2

2 ν

and q =

1 p(m+ℓ)(m+ℓ−1)

2 2−ν

m+ℓ X

Pm,ℓ,k (x)

k=0

p,q ν | Bm,ℓ (f ; x) − f (x) |≤ Mδm (x).

Hence, the desired result is obtained.



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! 2−ν 2

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