Modified Bernstein–Kantorovich operators for functions of one and two variables
Arun Kajla & Meenu Goyal
Rendiconti del Circolo Matematico di Palermo Series 2 ISSN 0009-725X Rend. Circ. Mat. Palermo, II. Ser DOI 10.1007/s12215-017-0320-z
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Author's personal copy Rend. Circ. Mat. Palermo, II. Ser DOI 10.1007/s12215-017-0320-z
Modified Bernstein–Kantorovich operators for functions of one and two variables Arun Kajla1 · Meenu Goyal2
Received: 16 February 2017 / Accepted: 27 August 2017 © Springer-Verlag Italia S.r.l. 2017
Abstract The purpose of this note is to obtain some direct results for the modified Bernstein– Kantorovich operators for functions of one variable proposed by Özarslan and Duman (Numer Funct Anal Optim 37:92–105, 2016). We also introduce the bivariate extension of these operators and compute the Voronovskaja type asymptotic formula, the degree of approximation for the Lipschitz class of elements and order of approximation by means of the Peetre’s K -functional. We illustrate the convergence of the operators to a certain function with the help of Mathematica software. Keywords Bernstein–Kantorovich operators · Lipschitz space · Bounded variation · Degree of approximation Mathematics Subject Classification 26A15 · 41A25 · 41A28
1 Introduction For f ∈ C(I ), with I = [0, 1], the classical Bernstein polynomials defined as follows: n k pn,k (x) f Bn ( f ; x) = , x ∈ I, (1.1) n k=0
n k x (1 − x)(n−k) . where pn,k (x) = k
B
Arun Kajla
[email protected] Meenu Goyal
[email protected]
1
Department of Mathematics, Central University of Haryana, Haryana 123031, India
2
Department of Mathematics, DIT University, Dehradun 248009, India
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Also for f : [0, 1] → R an integrable function, the classical Bernstein–Kantorovich operators are defined by K n ( f ; x) = (n + 1)
n
pn,k (x)
(k+1)/(n+1)
f (t)dt, x ∈ [0, 1], n ∈ N.
k/(n+1)
k=0
The above operators K n can also be written as follows: K n ( f ; x) =
n
1
pn,k (x)
f 0
k=0
k+t dt. n+1
(1.2)
P˘alt˘anea [22] proposed a generalization of the Bernstein–Kantorovich operators depending on a parameter and proved the differential inequality. In [14], Gonska and P˘alt˘anea introduced one parameter genuine Bernstein–Durrmeyer operators and obtained simultaneous approximation properties.Very recently, Agrawal et al. [3] proposed Lupa¸s–Kantorovich operators involving Polya distribution and studied local and global approximation properties of these operators. They also considered the bivariate case of these operators and discussed the degree of approximation by means of the second order Ditzian–Totik modulus of continuity and the corresponding Peetre’s K -functional. Acu and Muraru [2] proposed a bivariate extension of the Bernstein–Schurer–Kantorovich operators involving q-integers and studied approximation properties of these operators. In [25,26] Wafi and Khatoon investigated bivariate case for the generalized Baskakov operators in polynomial and exponential weighted spaces. They obtained the degree of approximation, direct theorems, simultaneous approximation for first order partial derivatives. Doˇgru and Gupta [13] constructed and established Korovkin-type approximation properties of bivariate Meyer-König and Zeller operators based on q-integers. For α > 0, although replacing t in (1.2) with t α does not effect the positivity or linearity of K n , it does generate a new sequence of positive linear operators i.e. modified Bernstein– Kantorovich operators given by Özarslan and Duman [20] as follows: K n,α ( f ; x) =
n k=0
1
pn,k (x)
f 0
k + tα n+1
dt, x ∈ I,
(1.3)
where pn,k (x) is defined in (1.1) and we discuss some direct approximation theorems. Bojanic and Cheng [7] obtained the rate of approximation of the Bernstein polynomials for an absolutely continuous function whose derivative is of bounded variation. In [6], Bai and Shaw gave the rate of convergence for functions with derivatives of bounded variation and they obtained some applications for some important operators. The rate of approximation of summation integral type operators is considered in [15]. In [17], the rate of convergence was estimated for Kantorovich-type operators. In the literature, several researchers have made significant contributions in this direction. We refer to some of the related papers (cf. [1,4,8–12,16,18,21,24] etc.). The aim of the present note is to study Lipschitz type space and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation for the operators (1.3). We also introduce the bivariate generalization of these operators and study Voronovskaja type asymptotic theorem and the degree of approximation for the Lipschitz class of functions for the bivariate case.
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Lemma 1 [20] For α > 0, n ∈ N and x ∈ I, we have i nm 1 i K n,α (ei ; x) = Bn (em ; x), (n + 1)i (α(i − m) + 1) m m=0
where Bn denotes the classical Bernstein polynomial and ei (t) = t i (i = 0, 1, · · · ). Corollary 1 From Lemma 1, we get (i) K n,α (e0 ; x) = 1; nx (ii) K n,α (e1 ; x) = n+1 + (iii) (iv) (v)
1 (1+α)(n+1) ; 2nx 1 K n,α (e2 ; x) = (n+1)2 + nx(1−x) + (1+α)(n+1) 2 + (1+2α)(n+1)2 ; (n+1)2 3 2 n(2α 2 +12α+7)x 1 K n,α (e3 ; x) = n(n−1)(n−2)x + 3n(n−1)(α+2)x + (1+α)(1+2α)(n+1) 3 + (1+3α)(n+1)3 ; (n+1)3 (1+α)(n+1)3 4 3 2 +51α+25)x 2 K n,α (e4 ; x) = n(n−1)(n−2)(n−3)x + 2n(n−1)(n−2)(5+3α)x + n(n−1)(14α (n+1)4 (1+α)(n+1)4 (1+α)(1+2α)(n+1)4 n(6α 3 +61α 2 +62α+15)x 1 + (1+4α)(n+1) 4. (1+α)(1+2α)(1+3α)(n+1)4 n2 x 2
Corollary 2 [20] For x ∈ [0, 1], we have 1 1 −x + 1+α ; (i) K n,α ((t − x); x) = (n+1) 1 2x x 2 − 1+α (ii) K n,α ((t − x)2 ; x) = (n+1) + 2
1 1+2α
+
+ nx(1 − x) .
2 Main results 2.1 Degree of approximation Let us define the Lipschitz-type space with two parameters [19]: For a1 ≥ 0, a2 > 0 and η ∈ (0, 1], we consider
(a ,a ) Li p M1 2 (η)
:=
f ∈ C[0, 1] : | f (t) − f (x)| ≤ M
|t − x|η
η
(t + a1 x 2 + a2 x) 2
; t ∈ [0, 1], x ∈ (0, 1] ,
where M is a positive constant. (0,1) We note that the space Li p M (η) coincides with the space Li p ∗M (η) considered by Szász [23]. (a ,a )
Theorem 1 If f ∈ Li p M1 2 (η) and x ∈ (0, 1], then we have η/2
μ∗n,α (x)
K n,α ( f ; x) − f (x) ≤ M , where μ∗n,α (x) = K n,α ((t − x)2 ; x). a1 x 2 + a2 x Proof Let we prove the theorem for the case 0 < η ≤ 1, applying Holder’s inequality with 2 p = η2 , q = 2−η n
K n,α ( f ; x) − f (x) ≤ pn,k (x) 0
k=0
≤
n k=0
α
dt
f k+t − f (x)
n+1
1
pn,k (x) 0
2 η2
α
η k + t
f − f (x)
dt
n+1
1
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≤
n
pn,k (x) 0
k=0
× =
2 η2
α
η k + t
f − f (x)
dt
n+1
1
n
2−η 2
pn,k (x)
k=0 n
2 η2 α
η k + t
f − f (x)
dt
n+1
1
pn,k (x) 0
k=0
⎛ n ⎜ ≤ M⎝ pn,k (x) k=0
M
1
0
k+t α n+1
n
k+t α n+1
−x
+ a1
x2
2
+ a2 x
1 k
⎞ η2 ⎟ dt ⎠
+ tα −x n+1
2
η 2
≤ pn,k (x) dt η 0 a1 x 2 + a2 x 2 k=0 η η M M 2 ∗ = η K n,α ((t − x) ; x) 2 = η (μn,α (x)) 2 . a1 x 2 + a2 x 2 a1 x 2 + a2 x 2
Therefore, the proof is completed.
2.2 Rate of convergence In this section, D BV (I ) denotes the class of all absolutely continuous functions f defined on I , having on I a derivative f equivalent with a function of bounded variation on I . We notice that the functions f ∈ D BV (I ) possess a representation x f (x) = g(t)dt + f (0) 0
where g ∈ BV (I ), i.e., g is a function of bounded variation on I . The operators K n,α ( f ; x) also admit the integral representation 1 Bn,α (x, t) f (t)dt, K n,α ( f ; x) =
(2.1)
0
where the kernel Bn,α (x, t) is given by Bn,α (x, t) =
n
α pn,k (x)(x)χn,k (t),
k=0 α (t) is the characteristic function of the interval [k/(n + 1), (k + 1)/(n + 1)] with where χn,k respect to I.
Lemma 2 For a fixed x ∈ (0, 1) and sufficiently large n, we obtain q 1 α2 nx(1−x)+ , 0 ≤ q < x, (i) ϑn,α (x, q) = 0 Bn,α (x, t)dt ≤ (n+1)(x−q) 2 (1+α)2 (1+2α) 1 1 α2 (ii) 1−ϑn,α (x, z) = z Bn,α (x, t)dt ≤ (n+1)(z−x) 2 nx(1−x)+ (1+α)2 (1+2α) , x < z < 1.
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Proof (i) Using ([20], Lemma 2.5) we get q ϑn,α (x, q) = Bn,α (x, t)dt ≤ 0
q
0
x −t x −u
2 Bn,α (x, t)dt
= K n,α ((t − x)2 ; x)(x − q)−2 α2 1 nx(1 − x) + ≤ . (n + 1)(x − q)2 (1 + α)2 (1 + 2α) The proof of (ii) is similar. Hence the details are omitted.
Theorem 2 Let f ∈ D BV (I ). Then for every x ∈ (0, 1) and sufficiently large n, we have
| f (x+) + f (x−)|
1 1
−x + |K n,α ( f ; x) − f (x)| ≤
(n + 1) 1 + α1
2 2 α1 1 | f (x+) − f (x−)| nx(1 − x) + + 2 n+1 2 (1 + α1 ) (1 + 2α1 ) √
[ n] α12 1 nx(1 − x) + + x(n + 1) (1 + α1 )2 (1 + 2α1 )
x
( f x )
k=1 x−(x/k)
x +√ n
x √ x−(x/ n)
( f x ) √
[ n] x+((1−x)/k) α12 1 nx(1 − x) + ( f x ) + 2 (1 − x)(n + 1) (1 + α1 ) (1 + 2α1 ) x +
where
b
a ( fx )
(1 − x) √ n
k=1
√ x+((1−x)/ n)
( f x ),
x
denotes the total variation of f x on [a, b] and f x is defined by ⎧ ⎨ f (t) − f (x−), 0 ≤ t < x t=x f x (t) = 0, ⎩ f (t) − f (x+), x < t < 1.
(2.2)
Proof Since K n,α (1; x) = 1, by using (2.1), for every x ∈ (0, 1) we get 1 K n,α ( f ; x) − f (x) = Bn,α (x, t)( f (t) − f (x))dt 0
=
1
t
Bn,α (x, t)
0
f (u)dudt.
(2.3)
x
For any f ∈ D BV (I ), by (2.2) we may write 1 1 f (u) = f x (u) + ( f (x+) + f (x−)) + ( f (x+) − f (x−))sgn(u − x) 2 2 1 + δx (u)[ f (u) − ( f (x+) + f (x−))], 2 where
δx (u) =
(2.4)
1, u = x . 0, u = x
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Obviously, 0
1 t
f (u) −
x
1 f (x+) + f (x−) δx (u)du Bn,α (x, t)dt = 0. 2
By (2.1) and simple calculations we have 1 t 1 1 f (x+) + f (x−) du Bn,α (x, t)dt = 2 0 x 2 1 = 2
f (x+) + f (x−)
1 0
(t − x)Bn,α (x, t)dt
f (x+) + f (x−) K n,α ((t − x); x)
and
0
1 f (x+) − f (x−) sgn(u − x)du dt
2 x 1 1 ≤ | f (x+) − f (x−) | |t − x|Bn,α (x, t)dt 2 0 1 ≤ | f (x+) − f (x−) | K n,α (|t − x|; x) 2 1/2 1 ∗(1/n) 2 . ≤ | f (x+) − f (x−) | Dn (t − x) ; x 2
1
Bn,α (x, t)
t
Considering Corollary 2(i), ([20], Lemma 2.5) and using (2.3), (2.4) we get the following estimate |K n,α ( f ; x) − f (x)| ≤
1 1 1
| f (x+) + f (x−)|
−x + 2 (n + 1) 1 + α1
α12 1 1 + | f (x+) − f (x−)| nx(1 − x) + 2 2 n+1 (1 + α1 ) (1 + 2α1 )
x t
1 t
+
f x (u)du Bn,α (x, t)dt + f x (u)du Bn,α (x, t)dt
. 0
x
x
x
(2.5) Let E n,α f x , x =
Fn,α f x , x =
x 0
x
t
x 1 t x
f x (u)du Bn,α (x, t)dt, f x (u)du Bn,α (x, t)dt.
To complete the proof, it is sufficient to evaluate the terms E n ( f x , x) and Fn ( f x , x). Since b a dt ϑn,α (x, t) ≤√1 for all [a, b] ⊆ [0, 1], using integration by parts and from Lemma 2 with q = x − (x/ n), we get
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|E n,α ( f x , x)|
x t
=
f x (u)du dt ϑn,α (x, t)
x
0 x
=
ϑn,α (x, t) f x (t)dt
0 q x ≤ + | f x (t)| |ϑn,α (x, t)|dt 0
q
α12 1 ≤ nx(1 − x) + (n + 1) (1 + α1 )2 (1 + 2α1 ) x x + ( f x )dt q
0
x q
( f x )(x − t)−2 dt
t
t
q x α12 1 nx(1 − x) + ≤ ( f )(x − t)−2 dt (n + 1) (1 + α1 )2 (1 + 2α1 ) 0 t x x +√ n
x
( f x ).
√ x−(x/ n)
By the substitution of u = x/(x − t), we have x−(x/√n ) x α12 1 −2 f x dt (x − t) nx(1 − x) + (n + 1) (1 + α1 )2 (1 + 2α1 ) 0 t √n x α12 1 f x du = nx(1 − x) + x(n + 1) (1 + α1 )2 (1 + 2α1 ) 1 ≤
α12 1 nx(1 − x) + x(n + 1) (1 + α1 )2 (1 + 2α1 )
1 nx(1 − x) + ≤ x(n + 1)
x−(x/u) √ [ n] k+1 x
k=1 k x−(x/k) √ [ n] x α12 ( f x ). 2 (1 + α1 ) (1 + 2α1 ) k=1 x−(x/k)
f x du
Thus, |E n,α
f x , x
√ [ n] x α12 1 fx |≤ nx(1 − x) + 2 x(n + 1) (1 + α1 ) (1 + 2α1 ) k=1 x−(x/k)
x +√ n
x √ x−(x/ n )
f x .
(2.6)
√ Using integration by parts and applying Lemma 2 with z = x + ((1 − x)/ n), we obtain
1 t
|Fn,α ( f x , x)| =
f x (u)du Bn,α (x, t)dt
x x
z t
=
f x (u)du dt 1 − ϑn,α (x, t) x x
1 t
+ f x (u)du dt 1 − ϑn,α (x, t)
z
x
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t z z
=
f x (u) 1 − ϑn,α (x, t) du − f x (t) 1 − ϑn,α (x, t) dt x x x
1 t
+ f x (u)du dt 1 − ϑn,α (x, t)
x
zz z
=
f x (u)du 1 − ϑn,α (x, z) − f x (t) 1 − ϑn,α (x, t) dt x
x
t
+
=
≤
x z x
f x (u)du 1 − ϑn,α (x, t)
f x (t) 1 − ϑn,α (x, t) dt +
1 nx(1 − x) + (n + 1) z t + f x dt x
1
1
− z 1
z
z
f x (t)
1 − ϑn,α (x, t) dt
f x (t) 1 − ϑn,α (x, t) dt
1 t α12 f x (t (1 + α1 )2 (1 + 2α1 ) z x
− x)−2 dt
x
1 = nx(1 − x) (1 − x)(n + 1) 1 t α12 f x (t − x)−2 dt + √ (1 + α1 )2 (1 + 2α1 ) x+((1−x)/ n ) x (1 − x) + √ n
√ x+((1−x)/ n )
f x
x
By the substitution of v = (1 − x)/(t − x), we have √n x+((1−x)/v) α12 1 nx(1 − x) + f x (1 − x)−1 dv |Fn,α f x , x | ≤ 2 (n + 1) (1 + α1 ) (1 + 2α1 ) 1 x (1 − x) + √ n
√ x+ (1−x)/ n
x
1 ≤ (1 − x)(n + 1) +
(1 − x) √ n
(1 − x) √ n
nx(1 − x) +
√ x+ (1−x)/ n
α12 (1 + α1 )2 (1 + 2α1 )
nx(1 − x) +
√ x+((1−x))/ n
√
[ n] k+1 x+((1−x)/v) k=1 k
f x dv
x
fx
x
1 = (1 − x)(n + 1) +
fx
α12 (1 + α1 )2 (1 + 2α1 )
√
[ n] x+((1−x)/k) k=1
fx .
fx
x
(2.7)
x
Collecting the estimates (2.5)–(2.7), we obtain the required result.
Example 1 The comparison of convergence of K n,α ( f ; x) (red) and the Bernstein– Kantorovich K n ( f ; x)(green) operators for f (x) = x 3 + 3x 2 − 5x (blue), α = 5, n = 10 and α = 10, n = 100 is illustrated in Figs. 1 and 2.
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Fig. 1 The convergence of K 10,5 ( f ; x) and K 10 ( f ; x) to f (x)
Fig. 2 The convergence of K 100,10 ( f ; x) and K 100 ( f ; x) to f (x)
3 Construction of the bivariate operators For I 2 = [0, 1] × [0, 1], let C(I 2 ) be the space of all ordinary continuous functions on I 2 , endowed with the norm given by || f ||C(I 2 ) = sup | f (x, y)|. For f ∈ C(I 2 ), the bivariate (x,y)∈I 2
extension of the operator (1.3) is defined by K n 1 ,n 2 ,α1 ,α2 ( f ; x, y) =
n2 n1
pn 1 ,n 2 ,k1 ,k2 (x, y)
k1 =0 k2 =0
1 1
f 0
0
k1 + t α1 k2 + s α2 , n1 + 1 n2 + 1
dtds. (3.1)
Lemma 3 Let ei, j (x, y) = x i y j , (i, j) ∈ N0 × N0 , with i + j ≤ 4 and N0 = N ∪ {0} be the bivariate test functions. Then (i) K n 1 ,n 2 ,α1 ,α2 (e00 ; x, y) = 1; x (ii) K n 1 ,n 2 ,α1 ,α2 (e10 ; x, y) = nn11+1 + n2 y (iii) K n 1 ,n 2 ,α1 ,α2 (e01 ; x, y) = n 2 +1 +
1 (1+α1 )(n 1 +1) ; 1 (1+α2 )(n 2 +1) ;
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(iv) K n 1 ,n 2 ,α1 ,α2 (e20 ; x, y) = +
1 ; (1+2α1 )(n 1 +1)2
(v) K n 1 ,n 2 ,α1 ,α2 (e02 ; x, y) = +
1 , (1+2α2 )(n 2 +1)2
n 21 x 2 (n 1 +1)2
+
n 1 x(1−x) (n 1 +1)2
+
2n 1 x (1+α1 )(n 1 +1)2
n 22 y 2 (n 2 +1)2
+
n 2 y(1−y) (n 2 +1)2
+
2n 2 y (1+α2 )(n 2 +1)2
2 n 1 (n 1 −1)(n 1 −2)x 3 1 (n 1 −1)(α1 +2)x + 3n(1+α 3 (n 1 +1)3 1 )(n 1 +1) n (2α12 +12α1 +7)x 1 + (1+α1 )(1+2α 3 + (1+3α )(n +1)3 ; 1 1 )(n 1 +1) 1 1 2 −1)(n 2 −2)y 3 2 (n 2 −1)(α2 +2)y K n 1 ,n 2 ,α1 ,α2 (e03 ; x, y) = n 2 (n 2(n + 3n (1+α 3 3 2 +1) 2 )(n 2 +1) n (2α22 +12α2 +7)y 1 + (1+α2 )(1+2α 3 + (1+3α )(n +1)3 ; 2 2 )(n 2 +1) 2 2 4 1 −2)(n 1 −3)x K n 1 ,n 2 ,α1 ,α2 (e40 ; x, y) = n 1 (n 1 −1)(n (n 1 +1)4 n (n 1 −1)(14α12 +51α1 +25)x 2 −1)(n 1 −2)(5+3α1 )x 3 + 2n 1 (n 1(1+α + 1 (1+α 4 4 1 )(n 1 +1) 1 )(1+2α1 )(n 1 +1) n 1 (6α13 +61α12 +62α1 +15)x 1 + (1+α )(1+2α )(1+3α )(n +1)4 + (1+4α )(n +1)4 ; 1 1 1 1 1 1 4 2 −2)(n 2 −3)y K n 1 ,n 2 ,α1 ,α2 (e04 ; x, y) = n 2 (n 2 −1)(n 4 (n 2 +1) n (n 2 −1)(14α22 +51α2 +25)y 2 −1)(n 2 −2)(5+3α2 )y 3 + 2n 2 (n 2(1+α + 2 (1+α 4 4 )(n +1) 2 2 2 )(1+2α2 )(n 2 +1) n 2 (6α23 +61α22 +62α2 +15)y 1 + (1+α )(1+2α )(1+3α )(n +1)4 + (1+4α )(n +1)4 . 2 2 2 1 2 2
(vi) K n 1 ,n 2 ,α1 ,α2 (e30 ; x, y) =
(vii)
(viii)
(ix)
Proof This lemma follows easily by using the Lemma 1 and the definition of the bivariate Bernstein–Kantorovich operators (3.1). Hence the details are omitted. Lemma 4 From Lemma 3, we have
1 −x + 1+α ; 1 1 1 (ii) K n 1 ,n 2 ,α1 ,α2 ((s − y); x, y) = (n 2 +1) −y + 1+α2 ; 1 2x 1 (iii) K n 1 ,n 2 ,α1 ,α2 ((t − x)2 ; x, y) = (n +1) x 2 − 1+α + + nx(1 − x) ; 2 1+2α 1 1 1 2y 1 1 2 2 (iv) K n 1 ,n 2 ,α1 ,α2 ((s − y) ; x, y) = (n +1)2 y − 1+α2 + 1+2α2 + ny(1 − y) ; (i) K n 1 ,n 2 ,α1 ,α2 ((t − x); x, y) =
1 (n 1 +1)
2
(v)
(vi)
(3n 2 −20n 1 +1)x 4 (−6n 21 (1+α1 )+10n 1 (5+3α1 )−4)x 3 y) = 1 (n +1) + K n 1 ,n 2 ,α1 ,α2 ((t − 4 (1+α1 )(n 1 +1)4 1 (n 1 (3n 1 (1+α1 )(1+2α1 )−α1 (22α1 +87)−41)+6(1+α1 ))x 2 + (1+α1 )(1+2α1 )(n 1 +1)4 n 1 (1+3α1 )(2α12 +17α1 +11)−4(1+α1 )(1+2α1 ) x 1 + + (1+4α )(n 4 4; (1+α1 )(1+2α1 )(1+3α1 )(n 1 +1) 1 1 +1) 2 4 (3n 2 −20n 2 +1)y (−6n 22 (1+α2 )+10n 2 (5+3α2 )−4)y 3 4 + K n 1 ,n 2 ,α1 ,α2 ((s − y) ; x, y) = (n 2 +1)4 (1+α2 )(n 2 +1)4 (n 2 (3n 2 (1+α2 )(1+2α2 )−α2 (22α2 +87)−41)+6(1+α2 ))y 2 + (1+α2 )(1+2α2 )(n 2 +1)4 n 2 (1+3α2 )(2α22 +17α2 +11)−4(1+α2 )(1+2α2 ) y 1 + + (1+4α )(n 4. (1+α2 )(1+2α2 )(1+3α2 )(n 2 +1)4 2 2 +1)
x)4 ; x,
Remark 1 From Lemma 4, we get
α12 n 1 x(1 − x) + (1+α )2 (1+2α ; 1 1) α22 (ii) K n 1 ,n 2 ,α1 ,α2 ((s − y)2 ; x, y) ≤ n 21+1 n 2 y(1 − y) + (1+α )2 (1+2α . ) (i) K n 1 ,n 2 ,α1 ,α2 ((t − x)2 ; x, y) ≤
1 n 1 +1
2
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Theorem 3 For any f ∈ C(I 2 ), we have lim
n 1 ,n 2 →∞
||K n 1 ,n 2 ,α1 ,α2 ( f ) − f || = 0.
Proof From Lemma 3, we have lim
n 1 ,n 2 →∞
K n 1 ,n 2 ,α1 ,α2 (ei j ) = ei j , (i, j) ∈ {(0, 0), (0, 1), (1, 0)}
and lim
n 1 ,n 2 →∞
K n 1 ,n 2 ,α1 ,α2 (e20 + e02 ) = e20 + e02 ,
uniformly on I 2 . The result follows applying Theorem 2.1 of [5].
3.1 Voronovskaja type theorem Theorem 4 Let f ∈ C 2 (I 2 ). Then, we have lim n K n,n,α,α ( f ; x, y) − f (x, y) =
n→∞
1 1 f x (x, y) + −y + f y (x, y) 1+α 1+α x(1 − x) y(1 − y) f x x (x, y) + f yy (x, y), + 2 2
−x +
uniformly on I 2 . Proof Let (x, y) ∈ I 2 be arbitrary. By the Taylor’s expansion, we get 1 f (t, s) = f (x, y) + f x (x, y)(t − x) + f y (x, y)(s − y) + { f x x (x, y)(t − x)2 2 + 2 f x y (x, y)(t − x)(s − y) (3.2) + f yy (x, y)(s − y)2 } + ε(t, s; x, y) (t − x)4 + (s − y)4 , for (t, s) ∈ I 2 , where ε(t, s; x, y) ∈ C(I 2 ) and ε(t, s; x, y) → 0 as (t, s) → (x, y). Applying K n,n,α,α ( f ; x, y) on both sides of (3.2), we have K n,n,α,α ( f ; x, y) = f (x, y) + f x (x, y)K n,α ((t − x); x) + f y (x, y)K n,α ((s − y); y) 1 + { f x x (x, y)K n,α ((t − x)2 ; x) + f yy (x, y)K n,α ((s − y)2 ; y) 2 + 2 f x y (x, y)K n,n,α,α ((t − x)(s − y); x, y)} (3.3) + K n,n,α,α ε(t, s; x, y) (t − x)4 + (s − y)4 ; x, y . Now, using the Hölder’s inequality, we obtain
K n,n,α,α ε(t, s; x, y) (t − x)4 + (s − y)4 ; x, y
! !1/2 1/2 ≤ K n,n,α,α (ε 2 (t, s; x, y); x, y) K n,n,α,α ((t − x)4 + (s − y)4 ; x, y) !1/2 !1/2 K n,α ((t − x)4 ; x) + K n,α ((s − y)4 ; y) ≤ K n,n,α,α (ε 2 (t, s; x, y); x, y) .
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From Theorem 3, K n,n,α,α (ε 2 (t, s; x, y); x, y) → 0 as n → ∞ uniformly in I 2 and using Lemma 4 lim n 2 K n,n,α,α ((t − x)4 ; x) = 3x 2 (x − 1)2 , uniformly in I 2 , and n→∞
lim n 2 K n,n,α,α ((s − y)4 ; y) = 3y 2 (y − 1)2 , uniformly in I 2 , we get n→∞ 4 4 lim n K n,n,α,α ε(t, s; x, y) (t − x) + (s − y) ; x, y = 0, n→∞
uniformly in (x, y) ∈ I 2 . Using Lemma 4, we get lim n K n,α ((t − x); x) = −x +
n→∞
1 1+α
and lim n K n,α ((s − y); y) = −y + n→∞
1 1+α
lim n K n,α ((t − x)2 ; x) = x(1 − x) and lim n K n,α ((s − y)2 ; y) = y(1 − y), n→∞ n→∞
uniformly in (x, y) ∈ I 2 . Using Lemma 4, we obtain lim n K n,α ((t − x); x)K n,α ((s − y); y) = 0,
n→∞
uniformly in (x, y) ∈ I 2 . Collecting the above estimates and the Eq. (3.3), we obtain the required result. Next, we consider the degree of approximation for the bivariate operators (3.1) by using of the Lipschitz class. For 0 < ρ1 ≤ 1 and 0 < ρ2 ≤ 1, we discuss the Lipschitz class Li p M (ρ1 , ρ2 ) for the bivariate extension as follows: | f (t, s) − f (x, y)| ≤ M|t − x|ρ1 |s − y|ρ2 . Theorem 5 Let f ∈ Li p M (ρ1 , ρ2 ), then we have ρ /2
ρ /2
||K n 1 ,n 2 ,α1 ,α2 ( f ) − f || ≤ M n 11 ,α1 n 22 ,α2 , where n 1 ,α1 = ||K n 1 ,α1 ((t − ·)2 ; ·)||, n 2 ,α2 = ||K n 2 ,α2 ((s − ·)2 ; ·)||. Proof By our hypothesis, we can write |K n 1 ,n 2 ,α1 ,α2 ( f ; x, y) − f (x, y)| ≤ K n 1 ,n 2 ,α1 ,α2 (| f (t, s) − f (x, y)|; x, y)
≤ M K n 1 ,n 2 ,α1 ,α2 (|t − x|ρ1 |s − y|ρ2 ; x, y)
= M K n 1 ,α1 (|t − x|ρ1 ; x)K n 2 ,α2 (|s − y|ρ2 ; y). 2 2 , and w2 = ρ22 , u 2 = 2−ρ , Now, using the Hölder’s inequality with w1 = ρ21 , u 1 = 2−ρ 1 2 we obtain ρ1 2−ρ1 |K n 1 ,n 2 ,α1 ,α2 ( f ; x, y) − f (x, y)| ≤ M K n 1 ,α1 ((t − x)2 ; x) 2 K n 1 ,α1 (e0 ; x) 2 ρ2 2−ρ2 × K n 2 ,α2 ((s − y)2 ; y) 2 K n 2 ,α2 (e0 ; y) 2 ρ /2
ρ /2
≤ M n 11 ,α1 n 22 ,α2 , from which the desired result is immediate. Theorem 6 Let f ∈ C 1 (I 2 ). Then, we have
" " ||K n 1 ,n 2 ,α1 ,α2 ( f ) − f || ≤ || f x || n 1 ,α1 + || f y || n 2 ,α2 ,
where n 1 ,α1 and n 2 ,α2 are defined as in Theorem 5.
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Proof By our hypothesis, we may write
t
f (t, s) − f (x, y) = x
f w (w, s)dw +
s
f u (x, u)du.
y
Now, operating K n 1 ,n 2 ,α1 ,α2 (·; x, y) on both sides, we get
Kn
1 ,n 2 ,α1 ,α2 ( f ; x, y) − f (x, y) ≤ K n 1 ,n 2 ,α1 ,α2
t x
f w (w, s)dw; x, y
+ K n 1 ,n 2 ,α1 ,α2
s
f u (x, u)du; x, y .
y
Since
t x
f w (w, s)dw ≤ || f x || |t − x| and
s y
f u (x, u)du
≤ || f y || |s − y|,
we have
K n ,n ,α ,α ( f ; x, y) − f (x, y) ≤ || f x ||K n ,α (|t − x|; x) + || f y ||K n ,α (|s − y|; y). 1 2 1 2 1 1 2 2 Now, using the Cauchy–Schwarz inequality and Lemma 4, we have
Kn
1 ,n 2 ,α1 ,α2
1 1 ( f ; x, y) − f (x, y) ≤ || f x || K n 1 ,α1 ((t − x)2 ; x) 2 K n 1 ,α1 (e0 ; x) 2 1 1 + || f y || K n 2 ,α2 ((s − y)2 ; y) 2 K n 2 ,α2 (e0 ; y) 2 " " ≤ || f x || n 1 ,α1 + || f y || n 2 ,α2 , ∀(x, y) ∈ I 2 .
Let C 2 (I 2 ) be the space of all functions f ∈ C(I 2 ) such that to C(I 2 ). The norm on the space C 2 (I 2 ) is defined as || f ||C 2 (I 2 )
# 2 # ∂i f # = || f ||C(I 2 ) + # ∂xi
∂i f ∂xi
# i #∂ f +# # ∂ yi C(I 2 )
# # # #
i=1
, ∂∂ yfi for i = 1, 2 belong i
# # # #
C(I 2 )
.
The Peetre’s K -functional of the function f ∈ C(I 2 ) is defined by K( f ; δ) =
inf
{|| f − g||C(I 2 ) + δ||g||C 2 (I 2 ) , δ > 0}.
g∈C 2 (I 2 )
Also, K( f ; δ) ≤ Cω2 ( f ;
√
δ)
√ holds for all δ > 0. The constant C is independent of δ and f and ω2 ( f ; δ) is the second order modulus of continuity. The usual modulus of continuity of f ∈ C(I 2 ) will be denoted by ω ( f ; δ) .
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Theorem 7 For the function f ∈ C(I 2 ), we have
K n ,n ,α ,α ( f ; x, y) − f (x, y) ≤ 4K f ; δn ,n ,α ,α (x, y) 1 2 1 2 1 2 1 2 ⎞ ⎛ 2 2 1 1 1 1 ⎠ +ω⎝ f; −x + −y + + (n 1 + 1) 1 + α1 (n 2 + 1) 1 + α2 " ≤ Cω2 f ; δn 1 ,n 2 ,α1 ,α2 (x, y) ⎞ ⎛ 2 2 1 1 1 1 ⎠ −x + −y + + +ω⎝ f; (n 1 + 1) 1 + α1 (n 2 + 1) 1 + α2 where δn 1 ,n 2 ,α1 ,α2 (x, y) = α12 (1+α1 )2 (1+2α1 )
1 2 n 1 +1 δn 1 ,α1
(x) +
and δn22 ,α2 = n 2 y(1 − y) +
1 2 n 2 +1 δn 2 ,α2 2 α2 . (1+α2 )2 (1+2α2 )
(y) , δn21 ,α1 = n 1 x(1 − x) +
Proof We consider the auxiliary operators as follows: n1 x 1 n2 y K n∗1 ,n 2 ,α1 ,α2 ( f ; x, y) = K n 1 ,n 2 ,α1 ,α2 ( f ; x, y) − f + , n 1 + 1 (1 + α1 )(n 1 + 1) n 2 + 1 1 + + f (x, y) . (1 + α2 )(n 2 + 1) Let g ∈ C 2 (I 2 ) and t, s ∈ I. By using the Taylor’s expansion, we can write t ∂ 2 g(u, y) ∂g(x, y) ∂g(x, y) (t − u) du + g(t, s) − g(x, y) = (t − x) + (s − y) 2 ∂x ∂u ∂y x s ∂ 2 g(x, v) (s − v) dv. (3.4) + ∂v 2 y Applying the operator K n∗1 ,n 2 ,α1 ,α2 on the above equation and then, from Lemma 4, K n∗1 ,n 2 ,α1 ,α2 ((t − x) ; x, y) = 0 and K n∗1 ,n 2 ,α1 ,α2 ((s − y) ; x, y) = 0, we get K n∗1 ,n 2 ,α1 ,α2 (g; x, y) − g(x, y) t ∂ 2 g(u, y) ∗ (t − u) du; x, y = K n 1 ,n 2 ,α1 ,α2 ∂u 2 x s ∂ 2 g(x, v) ∗ + K n 1 ,n 2 ,α1 ,α2 (s − v) dv; x, y ∂v 2 y t ∂ 2 g(u, y) = K n 1 ,n 2 ,α1 ,α2 (t − u) du; x, y ∂u 2 x n 1 x 2 1 n 1 +1 + (1+α1 )(n 1 +1) n1 x 1 ∂ g(u, y) du; x, y − + −u n 1 + 1 (1 + α1 )(n 1 + 1) ∂u 2 x s ∂ 2 g(x, v) + K n 1 ,n 2 ,α1 ,α2 (s − v) dv; x, y ∂v 2 y n 2 y 2 1 n 2 +1 + (1+α2 )(n 2 +1) n2 y ∂ g(x, v) 1 dv; x, y . − + −v n 2 + 1 (1 + α2 )(n 2 + 1) ∂v 2 y
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Hence, applying Remark 1, we obtain
∗
K
n 1 ,n 2 ,α1 ,α2
(g; x, y) − g(x, y)
2 1 1 gC 2 ( I 2 ) ≤ K n 1 ,n 2 ,α1 ,α2 (t − x) ; x, y + −x + (n 1 + 1) 1 + α1 2 1 1 2 gC 2 ( I 2 ) −y + + K n 1 ,n 2 ,α1 ,α2 (s − y) ; x, y + (n 2 + 1) 1 + α2 $ 2 2 2 2 < δ δ (x) + (y) gC 2 ( I 2 ) , n 1 + 1 n 1 ,α1 n 2 + 1 n 2 ,α2
where δn21 ,α1 = n 1 x(1 − x) +
2
α12 (1+α1 )2 (1+2α1 )
and δn22 ,α2 = n 2 y(1 − y) +
α22 . (1+α2 )2 (1+2α2 )
( f ; x, y) ≤ K n 1 ,n 2 ,α1 ,α2 ( f ; x, y)
n1 x 1 n2 y 1
+ | f (x, y)| +
f + , + n 1 + 1 (1 + α1 )(n 1 + 1) n 2 + 1 (1 + α2 )(n 2 + 1)
≤ 3 f.
∗
K
n 1 ,n 2 ,α1 ,α2
Hence, for all f ∈ C I 2 and g ∈ C 2 I 2 , we get
( f ; x, y) − f (x, y)
∗ n1 x 1 n2 y
≤ K n 1 ,n 2 ,α1 ,α2 ( f ; x, y) − f (x, y) + f + , n 1 + 1 (1 + α1 )(n 1 + 1) n 2 + 1
1 − f (x, y)
+ (1 + α2 )(n 2 + 1)
∗
≤ K n 1 ,n 2 ,α1 ,α2 ( f − g; x, y) + | f (x, y) − g (x, y)|
n1 x 1 n2 y + K n∗1 ,n 2 ,α1 ,α2 (g; x, y) − g (x, y) +
f + , n 1 + 1 (1 + α1 )(n 1 + 1) n 2 + 1
1 + − f (x, y)
(1 + α2 )(n 2 + 1) $ 2 2 ≤ 4 f − g + δn21 ,α1 (x) + δn22 ,α2 (y) gC 2 ( I 2 ) n1 + 1 n2 + 1 ⎞ ⎛ 2 2 1 1 1 1 ⎠. −x + −y + +ω⎝ f; + (n 1 + 1) 1 + α1 (n 2 + 1) 1 + α2
Kn
1 ,n 2 ,α1 ,α2
Taking the infimum on the right hand side over all g ∈ C 2 I 2 , we get
Kn
( f ; x, y) − f (x, y) ≤ 4K f ; δn 1 ,n 2 ,α1 ,α2 (x, y) ⎛ ⎞ 2 2 1 1 1 1 ⎠ +ω⎝ f; + −x + −y + (n 1 + 1) 1 + α1 (n 2 + 1) 1 + α2 1 ,n 2 ,α1 ,α2
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" ≤ Cω2 f ; δn 1 ,n 2 ,α1 ,α2 (x, y) ⎛ ⎞ 2 2 1 1 1 1 ⎠. +ω⎝ f; + −x + −y + (n 1 + 1) 1 + α1 (n 2 + 1) 1 + α2 This completes the proof of the theorem.
Acknowledgements The authors wishes to thank the referee for her/his suggestions which definitely improved the final form of this paper.
References 1. Acar, T., Gupta, V., Aral, A.: Rate of convergence for generalized Szász operators. Bull. Math. Sci. 1(1), 99–113 (2011) 2. Acu, A.M., Muraru, C.V.: Approximation properties of bivariate extension of q-Bernstein–Schurer– Kantorovich operators. Results Math. 67, 265–279 (2015) 3. Agrawal, P.N., Ispir, N., Kajla, A.: Approximation properties of Lupa¸s–Kantorovich operators based on Polya distribution. Rend. Circ. Mat. Palermo 65, 185–208 (2016) 4. Angeloni, L., Vinti, G.: Approximation in variation by homothetic operators in multidimensional setting. Differ. Integral Equ. 26(5–6), 655–674 (2013) 5. Bˇarbosu, D., Muraru, C.V.: Approximating B-continuous functions using GBS operators of Bernstein– Schurer–Stancu type based on q-integers. Appl. Math. Comput. 259, 80–87 (2015) 6. Bai, G.-D., Shaw, S.-Y.: Rate of approximation for functions with derivatives of bounded variation. Anal. Math. 28(3), 171–196 (2002) 7. Bojanic, R., Cheng, F.: Rate of convergence of Bernstein polynomials of functions with derivative of bounded variation. J. Math. Anal. Appl. 141, 136–151 (1989) 8. Coroianu, L., Gal, S.: L p -approximation by truncated max-product sampling operators of Kantorovichtype based on Fejer kernel. J. Integral Equ. Appl. 29, 349–364 (2017) 9. Cluni, F., Costarelli, D., Minotti, A.M., Vinti, G.: Applications of sampling Kantorovich operators to thermographic images for seismic engineering. J. Comput. Anal. Appl. 19(4), 602–617 (2015) 10. Costarelli, D., Vinti, G.: Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Integral Equ. Appl. 26(4), 455–481 (2014) 11. Costarelli, D., Vinti, G.: Convergence for a family of neural network operators in Orlicz spaces. Math. Nachr. 290(2–3), 226–235 (2017) 12. Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450, 1083–1103 (2017) 13. Do˘gru, O., Gupta, V.: Korovkin-type approximation propeties of bivariate q-Meyer–König and Zeller operators. Calcolo 43(1), 51–63 (2006) 14. Gonska, H., P˘alt˘anea, R.: Simultaneous approximation by a class of Bernstein–Durrmeyer operators preserving linear functions. Czech. Math. J. 60(135), 783–799 (2010) 15. Gupta, V., Vasishtha, V., Gupta, M.K.: Rate of convergence of summation-integral type operators with derivatives of bounded variation. J. Inequal. Pure Appl. Math. 4(2), 1–18 (2003). (Article 34) 16. Ispir, N.: Rate of convergence of generalized rational type Baskakov operators. Math. Comput. Model. 46, 625–631 (2007) 17. Ispir, N., Aral, A., Do˘gru, O.: On Kantorovich process of a sequence of the generalized linear positive operators. Numer. Funct. Anal. Optim. 29(5–6), 574–589 (2008) 18. Karsli, H.: Rate of convergence of new Gamma type operators for functions with derivatives of bounded variation. Math. Comput. Model. 45, 617–624 (2007) 19. Özarslan, M.A., Aktuˇglu, H.: Local approximation properties for certain King type operators. Filomat 27(1), 173–181 (2013) 20. Özarslan, M.A., Duman, O.: Smoothness properties of modified Bernstein–Kantorovich operators. Numer. Funct. Anal. Optim. 37, 92–105 (2016) 21. Özarslan, M.A., Duman, O., Kannaˇglu, C.: Rates of convergence of certain King-type operators for functions with derivative of bounded variation. Math. Comput. Model. 52, 334–345 (2010) 22. Paltanea, R.: A generalization of Kantorovich operators and a shape-preserving property of Bernstein operators. Bull. Transilv. Univ. Brasov Ser. III 5(54), 65–68 (2012)
123
Author's personal copy Modified Bernstein–Kantorovich operators for functions… 23. Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stand. 45, 239–245 (1950) 24. Vinti, G., Zampogni, L.: Approximation results for a general class of Kantorovich type operators. Adv. Nonlinear Stud. 14(4), 991–1011 (2014) 25. Wafi, A., Khatoon, S.: Approximation by generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces. Thai. J. Math. 2, 53–66 (2004) 26. Wafi, A., Khatoon, S.: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cent. Eur. J. Math. 6(2), 325–334 (2008)
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