RACSAM DOI 10.1007/s13398-016-0325-z ORIGINAL PAPER
( p, q)-Variant of Szász–Beta operators Ali Aral1 · Vijay Gupta2
Received: 1 February 2016 / Accepted: 23 August 2016 © Springer-Verlag Italia 2016
Abstract In the present paper, we introduce certain ( p, q) analogue of the Szász–Beta operators using ( p, q)-variant of Beta function of second kind. We present direct theorem in weighted spaces in terms of suitable weighted modulus of smoothness and Grüss-type inequality for mentioned operator. A Voronovskaya type theorem is also given. Keywords (p, q)-Beta function · (p, q)-Szász–Beta operators · Direct estimates · Modulus of continuity Mathematics Subject Classification 33B15 · 41A25
1 Introduction The ( p, q)-calculus which have many applications in areas of science and engineering was introduced in order to generalize the q-series by Gasper and Rahman [11]. The ( p, q)-series is derived as corresponding extensions of q-identities (for example [7,16]). The aim of this paper is to construct a Durrmeyer type operator as an application of ( p, q)-Beta integral, which was very recently defined by the authors in [5], using especially the ( p, q) differential and integration calculi. Very recently the extension of quantum-calculus is considered, in this way by Acar [2] who proposed ( p, q)-Szász-Mirakyan operators. Starting from this linear approximation process of discrete type, we construct and investigate an integral version of it. The
B
Ali Aral
[email protected] Vijay Gupta
[email protected]
1
Department of Mathematics, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey
2
Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110078, India
A. Aral, V. Gupta
new operators have Szász and Beta basis functions in summation and under the ( p, q)integral sign, respectively. We indicate sufficient conditions which ensure weighted uniform convergence of the mentioned sequence of the operators to the identity operator on the unbounded interval R+ = [0, ∞). We mentioned some previous results in this direction. Recently Gupta [12] introduced ( p, q)-Szász-Baskakov operators and established some direct results. Kantorovich variant of ( p, q) -Baskakov operators was defined in [13] Also authors investigated two Durrmeyer-type modifications of the ( p, q)-Szász–Mirakjan operators and ( p, q)-Baskakov operators, respectively in [5] and [6] . We can also mention other papers as Bernstein operators [17], Bernstein–Stancu operators [18], Bleimann–Butzer–Hahn operators [21] and Bernstein–Schurer operators [20]. Besides this, we also refer to some recent related work on this topic: e.g., [1,3,9,19]. At first we define the new operators and obtain some preliminary results of ( p, q)extension of them such that new operators the first two of the three Korovkin test functions are reproduced. In Sect. 3 we show that our operator is an approximation process on R+ and the errors of approximation are obtained using different modulus of smoothness. Section 4 deals with a Grüss type inequality on weighted spaces of continuous functions defined on a R+ with the help of weighted modulus of continuity. Finally a Voronovskaya type theorem in weighted spaces is highlighted. Some basic notations of ( p, q)-calculus are mentioned below: The ( p, q)-numbers are defined as [n] p,q =
pn − q n . p−q
Obviously, it may be seen that [n] p,q = p n−1 [n]q/ p . The ( p, q)-factorial is defined by [n] p,q ! =
n
[k] p,q , n ≥ 1, [0] p,q ! = 1.
k=1
The ( p, q)-binomial coefficient is given by [n] p,q ! n = , 0 ≤ k ≤ n. k p,q [n − k] p,q ![k] p,q ! Definition 1 The ( p, q)-power basis is defined below and it also has a link with q-power basis as (x ⊕ a)np,q = (x + a)( px + qa)( p 2 x + q 2 a) · · · ( p n−1 x + q n−1 a). (x a)np,q = (x − a)( px − qa)( p 2 x − q 2 a) · · · ( p n−1 x − q n−1 a). Definition 2 The ( p, q)-derivative of the function f is defined as D p,q f (x) =
f ( px) − f (q x) , x = 0 ( p − q)x
and D p,q f (0) = f (0), provided that f is differentiable at 0. Note also that for p = 1, the ( p, q)-derivative reduces to the q-derivative. The ( p, q)-derivative fulfils the following product rules D p,q ( f (x)g(x)) = f ( px)D p,q g(x) + g(q x)D p,q f (x) D p,q ( f (x)g(x)) = g( px)D p,q f (x) + f (q x)D p,q g(x).
( p, q)-Variant of Szász–Beta operators
Definition 3 Let n is a nonnegative integer, we define the ( p, q)-Gamma function as p,q (n + 1) =
( p q)np,q ( p − q)n
= [n] p,q !, 0 < q < p.
Proposition 1 The formula of ( p, q)-integration by part is given by b b f ( px)D p,q g(x)d p,q x = f (b)g(b) − f (a)q(a) − g(q x)D p,q f (x)d p,q x a
(1)
a
Let m, n ∈ N, we define ( p, q)-Beta function of second kind as ∞ x m−1 d p,q x. B p,q (m, n) = (1 ⊕ px)m+n 0 p,q An investigation of this equality appears in [5] . Theorem 1 Let m, n ∈ N. We have the following relation between ( p, q)-Beta and ( p, q)Gamma function: B p,q (m, n) = q [2−m(m−1)]/2 p −m(m+1)/2
p,q (m) p,q (n) . p,q (m + n)
Based on ( p, q)-calculus, very recently Acar [2] defined for x ∈ [0, ∞), 0 < q < p ≤ 1 the ( p, q) analogue of Szász operators as Sn, p,q ( f ; x) =
n
p,q
sn,k (x) f
k=0
where p,q
sn,k (x) =
[k] p,q q k−2 [n] p,q
,
(2)
q k(k−1)/2 1 ([n] p,q x)k . E p,q ([n] p,q x) [k] p,q !
Remark 1 [2] It can easily be verified by simple computation that Sn, p,q (1, x) = 1, Sn, p,q (t, x) = q x, Sn, p,q (t 2 , x) = pq x 2 + Sn, p,q (t 3 , x) = p 3 x 3 + Sn, p,q (t 4 , x) =
q2x . [n] p,q
( p 2 q + 2 pq 2 ) 2 q3 x + x [n] p,q [n]2p,q
p6 4 p3 pq( p 2 + 3 pq + 3q 2 ) 2 q4 x + ( p 2 + 2q + 3q 2 )x 3 + x + x 2 2 q q[n] p,q [n] p,q [n]3p,q
2 ( p, q)-Szász–Beta operators and moments In the year 2006 Gupta and Noor [15] proposed Szász–Beta operators and obtained some direct results in simultaneous approximation. Four years later Gupta and Aral [14] extended the studies and they proposed the q analogue of Szász–Beta operators. As ( p, q) analogue is further extension of q calculus, here we propose the ( p, q) variant of Szász–Beta operators as:
A. Aral, V. Gupta
Definition 4 Using ( p, q)-Beta function of second kind, we propose below for x ∈ [0, ∞), 0 < q < p ≤ 1 the ( p, q) analogue of Szász–Beta operators ∞ ∞ t k−1 1 p,q p,q Dn ( f ; x) = sn,k (x) f ( p k+1 qt)d p,q t B p,q (k, n + 1) 0 (1 ⊕ pt)k+n+1 p,q k=1 +
f (0) E p,q ([n] p,q x)
(3)
p,q
where sn,k (x) is as defined in (2). p,q
In the particular case p = 1; Dn , n ∈ N, turn into q -Szász–Beta operators, see [4]. We emphasize that our new operator has two advantages compared to the ( p, q)-Szász operators given in [2]: the operators of integral type generalization of the ( p, q)-Szász operators reproduce affine functions such that he ( p, q)-Sz ász operators operators do not enjoy this property and they reproduce only the constants ones. Also, using the following relations between ( p; q)-calculus and q-calculus [n] p,q = p n−1 [n]q/ p and replacing q with q/ p in q-Szász operators defined in [4], we can obtain ( p, q)-Szász operators. However, the new operators can not be obtained with same method from q-Szász– Beta operators defined in [4]. These operators are also different from those considered very recently by Gupta [12] as these reproduce linear functions. Lemma 1 For x ∈ [0, ∞), 0 < q < p ≤ 1, we have p,q
Dn (1; x) = 1, p,q
Dn (t; x) = x, [2] p,q q x p[n] p,q x 2 p,q Dn (t 2 ; x) = + , p[n − 1] p,q [n − 1] p,q p,q
Dn (t 3 ; x) =
p 3 [n]2p,q
x3 q 6 [n − 1] p,q [n − 2] p,q ( p[2] p,q + p 2 )[n] p,q ( p 2 q + 2 pq 2 )[n] p,q + + 6 x2 q 6 p 2 [n − 1] p,q [n − 2] p,q q [n − 1] p,q [n − 2] p,q [2] p,q ( p[2] p,q + p 2 ) x, + + 5 3 q 5 p 3 [n − 1] p,q [n − 2] p,q q p [n − 1] p,q [n − 2] p,q 6 [n]3p,q p3 p 4 p,q Dn (t 4 ; x) = 10 x + ( p 2 + 2q + 3q 2 )x 3 q [n − 1] p,q [n − 2] p,q [n − 3] p,q q 2 q[n] p,q pq( p 2 + 3 pq + 3q 2 ) 2 q4 + x + x [n]2p,q [n]3p,q
(( p[2] p,q + p 2 ) + 1)[n]2p,q ( p 2 q + 2 pq 2 ) 2 q3 3 3 p x + x + x + 3 8 p q [n − 1] p,q [n − 2] p,q [n − 3] p,q [n] p,q [n]2p,q ( p 3 + ( p[2] p,q + p 2 ))[n] p,q qx + 6 5 px 2 + [n] p,q p q [n − 1] p,q [n − 2] p,q [n − 3] p,q [2] p,q [3] p,q x. + 6 3 p q [n − 1] p,q [n − 2] p,q [n − 3] p,q
( p, q)-Variant of Szász–Beta operators
Proof By 3, we have p,q
Dn (1; x) = =
∞ k=1 ∞
p,q
sn,k (x)
1 B p,q (k, n + 1)
p,q
sn,k (x) +
k=0
∞
t k−1 (1 ⊕
0
pt)k+n+1 p,q
d p,q t +
1 E p,q ([n] p,q x)
1 E p,q ([n] p,q x) ∞
= =
q k(k−1)/2 1 1 ([n] p,q x)k + E p,q ([n] p,q x) [k] p,q ! E p,q ([n] p,q x) 1 E p,q ([n] p,q x)
k=1 ∞ k=0
q (k−1)k/2 ([n] p,q x)k = 1 [k] p,q !
Next p,q
Dn (t; x) = =
∞ k=1 ∞
p,q
sn,k (x) p,q
sn,k (x)
k=1
=
∞
∞
p k+1 q
0
tk (1 ⊕ pt)k+n+1 p,q
d p,q t
1 p k+1 q B p,q (k + 1, n) B p,q (k, n + 1)
sn,k (x)q −k+1 p,q
k=1
=
1 B p,q (k, n + 1)
[k] p,q [n] p,q
1 Sn, p,q (t; x) = x. q
Using similar consideration, the equality [k + 1] p,q = q k + p[k] p,q and Remark 1, we have p,q
Dn (t 2 ; x) = =
∞ k=1 ∞
p,q
sn,k (x) p,q
sn,k (x)
k=1
=
∞
1 B p,q (k, n + 1)
∞
p 2k+2 q 2
0
t k+1 (1 ⊕ pt)k+n+1 p,q
d p,q t
1 p 2k+2 q 2 B p,q (k + 2, n − 1) B p,q (k, n + 1)
sn,k (x) p −1 q −2k+3 p,q
k=1
=
[k + 1] p,q [k] p,q [n] p,q [n − 1] p,q
∞ 1 p,q sn,k (x)q −2k+3 (q k + p[k] p,q )[k] p,q p[n] p,q [n − 1] p,q k=1
=
1 p[n] p,q [n − 1] p,q
∞
sn,k (x)[q −k+3 [k] p,q + pq −2k+3 [k]2p,q )] p,q
k=1
[n] p,q q = Sn, p,q (t; x) + Sn, p,q (t 2 ; x) p[n − 1] p,q q[n − 1] p,q =
p[n] p,q x 2 [2] p,q q x + . p[n − 1] p,q [n − 1] p,q
A. Aral, V. Gupta
Next using [k + 1] p,q = q k + p[k] p,q , [k + 2] p,q = q k+1 + pq k + p 2 [k] p,q and Remark 1, we have p,q
Dn (t 3 ; x) = =
∞ k=1 ∞
p,q
sn,k (x) p,q
sn,k (x)
k=1
=
∞
1 B p,q (k, n + 1)
∞
p 3k+3 q 3
0
t k+2 (1 ⊕ pt)k+n+1 p,q
d p,q t
1 p 3k+3 q 3 B p,q (k + 3, n − 2) B p,q (k, n + 1)
sn,k (x) p −3 q −3k−3 p,q
k=1
=
[k + 2] p,q [k + 1] p,q [k] p,q [n] p,q [n − 1] p,q [n − 2] p,q
∞ 1 p,q sn,k (x) p 3 [n] p,q [n − 1] p,q [n − 2] p,q k=1
[q −k−3 [2] p,q [k] p,q + q −2k−3 ( p[2] p,q + p 2 )[k]2p,q + p 3 q −3k−3 [k]3p,q )] 3 3 p [n] p,q 1 Sn, p,q (t 3 ; x) = 3 p [n] p,q [n − 1] p,q [n − 2] p,q q9 ( p[2] p,q + p 2 )[n]2p,q [2] p,q [n] p,q 2 + Sn, p,q (t ; x) + Sn, p,q (t; x) q7 q5 =
p 3 [n]2p,q q 9 [n − 1] p,q [n − 2] p,q +
x3 +
[ p 3 + q 3 + 2 pq( p + q)][n] p,q 2 x q 8 p[n − 1] p,q [n − 2] p,q
( p 2 + pq + q)[2] p,q x. q 6 p 3 [n − 1] p,q [n − 2] p,q
Finally we can write p,q
Dn (t 4 ; x) = =
∞ k=1 ∞
p,q
sn,k (x) p,q
sn,k (x)
k=1
=
∞ k=1
=
1 B p,q (k, n + 1)
∞ 0
p 4k+4 q 4
t k+3 (1 ⊕ pt)k+n+1 p,q
d p,q t
1 p 4k+4 q 4 B p,q (k + 4, n − 3) B p,q (k, n + 1)
sn,k (x) p −6 q −4k−2 p,q
[k + 3] p,q [k + 2] p,q [k + 1] p,q [k] p,q [n] p,q [n − 1] p,q [n − 2] p,q [n − 3] p,q
∞ 1 p,q sn,k (x)q −4k−2 p 6 [n] p,q [n − 1] p,q [n − 2] p,q [n − 3] p,q k=1
( p 3 [k]2p,q + q k ( p[2] p,q + p 2 )[k] p,q + q 2k [2] p,q ) ( p 3 [k]2p,q + q k [k] p,q [3] p,q ) ∞
=
p,q 1 sn,k (x) p 6 [n] p,q [n − 1] p,q [n − 2] p,q [n − 3] p,q k=1 [k]4p,q p 6 q −10 [n]4p,q 4k−8 4 + q −8 p 3 (( p[2] p,q + p 2 ) + 1)[n]3p,q q [n] p,q
( p, q)-Variant of Szász–Beta operators
[k]3p,q q 3k−6 [n]3p,q
( p 3 + ( p[2] p,q + p 2 ))[n]2p,q q −6
+ [2] p,q [3] p,q [n] p,q q −4 =
[k] p,q k−2 q [n] p,q
[n]3p,q q 10 [n − 1] p,q [n − 2] p,q [n − 3] p,q +
[k]2p,q 2k−4 q [n]2p,q
Sn, p,q (t 4 ; x)
q −8 (( p[2] p,q + p 2 ) + 1)[n]2p,q p 3 [n − 1] p,q [n − 2] p,q [n − 3] p,q
Sn, p,q (t 3 ; x)
( p 3 + ( p[2] p,q + p 2 ))[n] p,q Sn, p,q (t 2 ; x) − 1] p,q [n − 2] p,q [n − 3] p,q [2] p,q [3] p,q + 6 4 Sn, p,q (t; x) p q [n − 1] p,q [n − 2] p,q [n − 3] p,q 6 [n]3p,q p 4 p3 = 10 x + ( p 2 + 2q + 3q 2 )x 3 2 q [n − 1] p,q [n − 2] p,q [n − 3] p,q q q[n] p,q pq( p 2 + 3 pq + 3q 2 ) 2 q4 + x + x [n]2p,q [n]3p,q +
+
p 6 q 6 [n
(( p[2] p,q + p 2 ) + 1)[n]2p,q
p 3 q 8 [n − 1] p,q [n − 2] p,q [n − 3] p,q
( p 2 q + 2 pq 2 ) 2 q3 3 3 × p x + x + x [n] p,q [n]2p,q ( p 3 + ( p[2] p,q + p 2 ))[n] p,q qx 2 + 6 5 px + [n] p,q p q [n − 1] p,q [n − 2] p,q [n − 3] p,q [2] p,q [3] p,q + 6 3 x. p q [n − 1] p,q [n − 2] p,q [n − 3] p,q
3 Weighted approximation Examining Lemma 1 and on the basis of weighted Korovkin’s theorem given in [10] we p,q observe that for any fixed p, q ∈ (0, 1) , Dn does not form an approximation process in weighted spaces. In order to gain this property, for each n ∈ N the constants p and q should be replaced by the sequences pn and qn . If for n sufficiently large pn → 1, qn → 1 and qnn → 1 and pnn → 1, then, then Lemma 1 guarantees mentioned weighted Korovkin theorem is satisfied, more exactly we can state: Let Hx 2 [0, ∞) be the set of all functions f defined on [0, ∞) satisfying the condition | f (x)| ≤ M f (1 + x 2 ), where M f is a constant depending only on f. By C x 2 [0, ∞), we denote the subspace of all continuous functions belonging to Hx 2 [0, ∞) . Also, let f (x) C x∗2 [0, ∞) be the subspace of all functions f ∈ C x 2 [0, ∞) , for which lim|x|→∞ 1+x 2 is | f (x)| ∗ finite. The norm on C x 2 [0, ∞) is f x 2 = supx∈[0, ∞) . 1 + x2
A. Aral, V. Gupta
Theorem 2 Let p = pn and q = qn satisfies 0 < qn < pn ≤ 1 and for n sufficiently large pn → 1, qn → 1 and qnn → 1 and pnn → 1. For each f ∈ C x∗2 [0, ∞) , we have
p ,q lim Dn n n ( f, x) − f x 2 = 0. n→∞
Proof Using the Theorem in [10] we see that it is sufficient to verify the following three conditions
p ,q
lim Dn n n t ν , x − x ν x 2 = 0, ν = 0, 1, 2. (4) n→∞
p ,q Dn n n
p ,q
Since (1, x) = 1 and Dn n n (t, x) = x, the first and second conditions of (4) is fulfilled for ν = 0 and ν = 1. Since [n] p,q = q n−1 + p[n − 1] p,q , we can write for n > 1
pn ,qn 2 p [n] pn ,qn x2
Dn (t , x) − x 2 x 2 ≤ −1 sup 2 [n − 1] pn ,qn x∈[0, ∞) 1 + x + ≤ which implies that
[2] pn ,qn qn pn [n − 1] pn ,qn
x 1 + x2 x∈[0, ∞) sup
[2] pn ,qn qn pn qnn−1 + pn2 − 1 + [n − 1] pn ,qn p[n − 1] pn ,qn
p ,q
lim Dn n n (t 2 , x) − x 2 x 2 = 0.
n→∞
Thus the proof is completed.
First we point out that our statements in previous sections are formulated for f ∈ C x∗2 [0, ∞) .In order to analyze the error of approximation of the operators on some weighted spaces, we consider the functions, satisfying certain polynomial growth at infinity, that is | f (t)| ≤ M(1 + t)m , for some M > 0 and m > 0. In this case we use the weight ρ (x) = (1 + x)−m , x ∈ I = [0, ∞) The polynomial weighted space associated to this weight is defined by Cρ (I ) = { f ∈ C (I ) : f ρ < ∞} where f ρ = sup ρ (x) | f (x)|
(5)
x∈I
Recently the case of weighted approximation by a broad class of linear positive operators, satisfying some conditions was considered in [8]. For a ∈ N0 , b > 0, c ≥ 0 we denote ϕ (x) = (1 + ax) (bx + c). For λ ∈ [0, 1] , f ∈ Cρ (I ) we consider the K -functional
∞ K 1,ϕ λ ( f, t)ρ = inf{ f − gρ + tϕ λ g ρ , g ∈ W1,λ (ϕ)},
∞ (ϕ) consist of all functions g ∈ C (I ) such that ϕ λ g < ∞. where W1,λ ρ ρ Also this K -functional is associated with the modulus of smoothness in following norm
C1 ωϕ2 λ ( f, t)ρ ≤ K 1,ϕ λ ( f, t)ρ ≤ C2 ωϕ2 λ ( f, t)ρ ,
( p, q)-Variant of Szász–Beta operators
where for f ∈ Cρ (I ) ωϕ2 λ ( f, t)ρ = sup
sup ρ (x) hϕ(x) f (x)
h∈(0,t] x∈I (ϕ,h)
(6)
and I (ϕ, h) = {x > 0 : hϕ (x) ≤ x} . The key for our new estimate is the following general result in [8] is the Theorem 3, which we cite here as: √ Theorem A Fix a ∈ N and set ϕ (x) = x (1 + ax). Let L n : Cρ (I ) → C(I ) be a sequence of positive linear operators satisfying the following conditions: (i) L n (e0 ) = e0 . (ii) There exists a constant C1 and a sequence {αn } such that L n ((t − x)2 ; x) ≤ C1 αn ϕ 2 (x) . (iii) There exists a constant C2 = C2 (m) such that for each n ∈ N, L n ((1 + t)m ; x) ≤ C2 (1 + x)m , x ≥ 0. (iv) There exists a constant C3 = C3 (m) such that for every m ∈ N,
(t − x)2 ρ (x) L n ; x ≤ C3 αn ϕ 2 (x) , x ≥ 0. ρ (t) Then there√exists a constant C such that for any f ∈ Cρ (I ) , x ≥ 0, n ∈ N such that αn ≤ 1/2 2 + a, f − L n f ρ ≤ Cωϕ2 ( f,
√
αn )ρ ,
where ωϕ2 ( f, t)ρ is the modulus in (6), with λ = 1. Lemma 2 For x ∈ [0, ∞), 0 < q < p ≤ 1, we have p,q
Dn ((t − x)2 ; x) ≤
2x (x + 1) p[n − 1] p,q
Proof Using the equality [n] p,q = q n−1 + p [n − 1] p,q we have
p 2 [n] p,q [2] p,q q x p,q 2 Dn ((t − x) ; x) = + − 1 x2 p[n − 1] p,q p [n − 1] p,q [2] p,q q x p 2 q n−1 + x 2 + ( p 3 − p)x 2 p[n − 1] p,q p[n − 1] p,q 2x 1 ≤ + x2 p[n − 1] p,q p[n − 1] p,q 2x (x + 1) ≤ . p[n − 1] p,q
=
p,q
Lemma 3 Let n ∈ N and m = 2. Then Dn defined by (3) is an operator from Cρ (R) into Cρ (R) . Moreover for any f ∈ Cρ (R) we have
A. Aral, V. Gupta
p,q Dn
and ρ (x)
p,q Dn
3 f ρ ≤ 4 + p[n − 1] p,q
(t − x)2 ;x ρ (t)
≤ C3
f ρ
2x (x + 1) , x ≥ 0. p[n − 1] p,q
Proof From (3) and Lemma 1, for f ∈ Cρ (R) we have p,q
Dn
p,q
( f (t) ; ·) ρ ≤ f ρ Dn
(1/ρ (t) ; ·) ρ p,q p,q p,q = f ρ sup ρ (x) Dn (1; x) + 2Dn (t; x) + Dn (t 2 ; x) x≥0
≤ ≤ ≤ ≤
p [n] p,q x 2 [2] p,q q x f ρ sup (1 + x) + 1 + 2x + p[n − 1] p,q [n − 1] p,q x≥0 p [n] p,q [2] p,q q f ρ 3 + + p[n − 1] p,q [n − 1] p,q [2] p,q q pq n−1 f ρ 3 + + + p2 p[n − 1] p,q [n − 1] p,q 3 f ρ 4 + . p[n − 1] p,q
−2
Lemma 1 implies that
2 p,q (t − x) p,q Dn ; x = Dn ((t − x)2 (1 + t)2 ; x) ρ (t) p,q
≤ 2Dn ((t − x)2 (1 + t 2 ); x) p,q
≤ 2Dn ((t − x)2 (1 + (t − x + x)2 ); x) p,q
≤ 4Dn ((t − x)2 (1 + (t − x)2 + x 2 ); x) p,q
p,q
≤ 4(1 + x 2 )Dn ((t − x)2 ; x) + 4Dn ((t − x)4 ; x). Next, p,q
Dn ((t − x)4 ; x) =
x p 6 q 6 [n − 1] p,q [n − 2] p,q [n − 3] p,q ( p 6 + p 3 q + [2] p,q ( p 4 q + pq 2 ) + p 5 q + p 2 q 2 + q 3 [2] p,q [3] p,q ) +
p 3 q 9 [n
x 2 [n] p,q − 1] p,q [n − 2] p,q [n − 3] p,q
( p 6 + 3 p 5 q + [2] p,q p 3 q 2 + 4 p 4 q 2 + 2 pq 2 + 2[2] p,q p 3 q 2 +2 p 3 q 3 + [2] p,q q 4 + pq 4 + p 2 q 4 ) + +
pq 11 [n
p 3 q 9 [n
x 2 4[2] p,q ( p 4 − pq 4 − p 2 q 3 ) − 2] p,q [n − 3] p,q
x 3 [n] p,q ( p6 + 2 p2 q + 3 p4 q 2 − 1] p,q [n − 2] p,q [n − 3] p,q
( p, q)-Variant of Szász–Beta operators
+ pq 3 + [2] p,q p 2 q 3 + p 3 q 3 ) x 3 6q[2] p,q x 3 4[n] p,q 3 3 2 4 3 6 ( p q + 2 p q + 2 pq + q ) + pq 11 [n − 1] p,q [n − 2] p,q p [n − 1] p,q
p 3 [n]3p,q 4 p 3 [n]3p,q 6 p[n] p,q 4 . +x −3 + + 12 − q [n − 1] p,q [n − 2] p,q [n − 3] p,q [n − 1] p,q [n − 1] p,q [n − 2] p,q
−
If we consider [n] p,q = p [n − 1] p,q + q n−1 [n] p,q = p 2 [n − 2] p,q + q n−2 [n] p,q = p 3 [n − 3] p,q + q n−3 we have p,q
Dn ((t − x)4 ; x) ≤
Cx p 6 q 11 [n − 1] p,q [n − 2] p,q [n − 3] p,q + + +
p 6 q 11 [n
Cx2 − 1] p,q [n − 2] p,q
p 6 q 11 [n
Cx2 Cx3 + 6 11 p q [n − 1] p,q [n − 2] p,q − 2] p,q [n − 3] p,q
Cx3 − 1] p,q
p 6 q 11 [n
1 p 12 p6 +x 4 −3 + 6 p 2 + 12 − 4 9 + O q q [n − 1] p,q 1 p 12 p6 , ≤ C x 3 (x + 1) −3 + 6 p 2 + 12 − 4 9 + O q q p [n − 1] p,q (7) where C is a positive constant. Thus
2 (1 + x 2 ) p,q 4 p,q (t − x) p,q ;x ≤ 4 ρ (x) Dn Dn ((t − x)2 ; x) + Dn ((t − x)4 ; x) 2 ρ (t) (1 + x) (1 + x)2 p,q
≤ 4Dn ((t − x)2 ; x) + ≤C
4 (1 + x)2
p,q
Dn ((t − x)4 ; x)
2x (x + 1) + C x (x + 1) p[n − 1] p,q
1 p 12 p6 −3 + 6 p + 12 − 4 9 + O q q p [n − 1] p,q 1 p 12 p6 2 . ≤ C x (x + 1) −3 + 6 p + 12 − 4 9 + O q q p [n − 1] p,q 2
A. Aral, V. Gupta
Theorem 3 Set ρ (x) = (1 + x)2 . For any f ∈ Cρ [0, ∞), x ≥ 0, n ∈ N, one has p,q ρ (x) f (x) − Dn ( f ; x)
12 6 1 p p 2 ≤ Cω . f, x (x + 1) −3 + 6 p 2 + 12 − 4 9 + O q q p [n − 1] p,q ρ
√ For n > 2 3 one has p,q
f (x) − Dn
where ϕ (x) =
( f ; x) ρ ≤ ωϕ2 √
1 p 12 p6 , f, −3 + 6 p 2 + 12 − 4 9 + O q q p [n − 1] p,q ρ
x (1 + x) and ωϕ2 ( f, t)ρ is modulus defined in 6 with λ = 1.
Proof For proof all the conditions in Theorem 2 and 3 for m = 2. √ A are verified in Lemmas From Lemma 2, we can see that ϕ (x) = x (x + 1) and αn = 1/ p[n − 1] p,q .
4 Grüss type inequality Consider two functions f, g ∈ Cρ [0, ∞) and define the positive bilinear functional p,q
Dn ( f, g; x) = Dn
p,q
( f g; x) − Dn
p,q
( f ; x) Dn
(g; x) .
To measure the rate of convergence of this positive bilinear functional on weighted spaces we use the modulus of smoothness defined by (6) and Theorem 3. Theorem 4 For any f ∈ Cρ [0, ∞), x ≥ 0, n ∈ N, one has Dn ( f, g)ρ 2 ≤
C ( f ). C (g),
where
1/2
12 6 1 p p C ( f ) = ωϕ2 f 2 , −3 + 6 p 2 + 12 − 4 9 + O q q p [n − 1] p,q 3 f ρ + 5+ p[n − 1] p,q 1/2
12 6 1 p p × ωϕ2 f, −3 + 6 p 2 + 12 − 4 9 + O . q q p [n − 1] p,q
Proof From Cauchy–Schwarz inequality, we have |Dn ( f, g; x)| ≤ Dn ( f, f ; x) Dn (g, g; x). We proceed as follows: p,q
p,q
Dn ( f, f ; x) = Dn ( f 2 ; x) − f 2 (x) + f 2 (x) − (Dn ( f ; x))2 p,q
p,q
p,q
= Dn ( f 2 ; x) − f 2 (x) + [ f (x) − Dn ( f ; x)][ f (x) + Dn ( f ; x)].
( p, q)-Variant of Szász–Beta operators
By (5) and Lemma 3 we have p,q
p,q
p,q
|Dn ( f, f ; x)| ≤ |Dn ( f 2 ; x) − f 2 (x)| + | f (x) − Dn ( f ; x)|ρ (x) ( f ρ + Dn ( f ) ρ ) 3 p,q p,q f ρ f (x) − Dn ( f ; x) . ≤ Dn ( f 2 ; x) − f 2 (x) + 5 + p[n − 1] p,q
It implies that
p,q f (x) − Dnp,q ( f ; x) |Dn ( f, f ; x)| |Dn ( f 2 ; x) − f 2 (x) | 3 f ρ ≤ + 5+ ρ 2 (x) ρ (x) p[n − 1] p,q ρ (x)
Using Theorem 3 we have
p,q
Dn ( f, f )ρ 2 ≤ Dn ( f 2 ) − f 2 ρ + 5 +
3 p[n − 1] p,q
p,q f ρ Dn ( f ) − f ρ
1/2
1 p 12 p6 ≤ f , −3 + 6 p + 12 − 4 9 + O q q p [n − 1] p,q 3 f ρ + 5+ p[n − 1] p,q 1/2
1 p 12 p6 2 2 × ωϕ f, −3 + 6 p + 12 − 4 9 + O . q q p [n − 1] p,q ωϕ2
2
2
This proves the theorem.
5 Voronovskaya Asymptotic formula Now we give the following Voronovskaya type theorem. In order the obtain following pointwise theorem we must assume that lim [n] pn ,qn ( pn − 1) = α
n→∞
and
lim [n] pn ,qn
n→∞
−3 + 6 pn2
pn12 p6 + 12 − 4 9n qn qn
= γ.
Theorem 5 Let f ∈ C(R+ ). If x ∈ R+ , f is two times differentiable in x and f is continuous in x, p = pn and q = qn satisfies 0 < qn < pn ≤ 1 and for n sufficiently large pn → 1, qn → 1 and qnn → 1 and pnn → 1. then the following holds true p ,q lim [n] pn ,qn Dn n n ( f, x) − f (x) = x (1 + αx) f (x) n→∞
Proof By the Taylor’s formula there exist η lying between x and y such that
f (x) f (t) = f (x) + f (x) (t − x) + (t − x)2 + h (t, x) (t − x)2 , 2
where h (y, x) :=
f (η) − f (x) 2
A. Aral, V. Gupta p ,qn
and h is a continuous function which vanishes at 0. Applying the operator Dn n equality, we get p ,qn
Dn n
to above
f (x) pn ,qn ((x − t)2 ; x) Dn 2 p ,q +Dn n n (h (t, x) (x − t)2 , x) p ,qn
( f, x) − f (x) = f (x) Dn n
(x − t; x) +
also we can write that p ,q p ,q [n] pn ,qn Dn n n ( f, x) − f (x) = f (x) [n] pn ,qn Dn n n ((x − t) ; x) f (x) p ,q [n] pn ,qn Dn n n ((x − t)2 ; x) 2 p ,q + [n] pn ,qn Dn n n (h (t, x) (x − t)2 , x). +
(8)
Using the equality [n] pn ,qn = q n−1 + p[n − 1] pn ,qn , we have p ,qn
lim [n] pn ,qn Dn n
n→∞
[n] pn ,qn [2] pn ,qn qn x pn [n − 1] pn ,qn p [n] pn ,qn + [n] p,q − 1 x2 [n − 1] pn ,qn pn qnn−1 = 2x + lim [n] pn ,qn + pn2 − 1 n→∞ [n − 1] pn ,qn
((x − t)2 ; x) = lim
n→∞
= 2x + 2αx 2 . In order to estimate the last term in (8), for every ε > 0 choose δ > 0 such that |h (t, x) | < ε for |t − x| < δ. Therefore if |t − x| < δ then h (t, x) (t − x)2 < ε (t − x)2 while if |t − x| ≥ δ, then since |h (t, x)| ≤ M we have h (t, x) (t − x)2 ≤ δM2 (t − x)4 . So we can write p ,qn
Dn n
p ,qn
(h (t, x) (t − x)2 , x) < ε Dn n
((t − x)2 , x) +
M pn ,qn Dn ((t − x)4 , x) δ2
From (7) we know that p ,qn
Dn n
1 pn12 p6 , ((t − x)4 , x) = x 3 (x + 1) −3 + 6 pn2 + 12 − 4 9n + O qn qn pn [n − 1] pn ,qn
and we conclude
p ,qn
lim n Dn n
n→∞
This proves the theorem.
(h (t, x) (t − x)4 , x) = 0.
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