The rate of pointwise approximation of positive linear operators based on q-integer
This paper is concerned with positive linear operators based on a q-integer. The rate of covergence of these operators are established. For these operators, we give Voronovskaya-type theorems and apply them to q Bernstein polynomials and q-Stancu operators.
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| author | Gupta, Vijay Nowak, G. Гупта, В. Новак, Г. |
| author_facet | Gupta, Vijay Nowak, G. Гупта, В. Новак, Г. |
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| description | This paper is concerned with positive linear operators based on a q-integer. The rate of covergence of these operators are established. For these operators, we give Voronovskaya-type theorems and apply them to q Bernstein polynomials and q-Stancu operators. |
| first_indexed | 2026-03-24T02:29:01Z |
| format | Article |
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UDC 517.9
G. Nowak (The Great Poland Univ. Soc. and Econ. in Sroda Wielkopolska, Poland),
Vijay Gupta (School Appl. Sci., Netaji Subhas Inst. Technology, New Delhi, India)
THE RATE OF POINTWISE APPROXIMATION
OF POSITIVE LINEAR OPERATORS BASED ON q-INTEGER
ШВИДКIСТЬ ПОТОЧКОВОГО НАБЛИЖЕННЯ ДОДАТНИХ
ЛIНIЙНИХ ОПЕРАТОРIВ, ЩО БАЗУЮТЬСЯ НА q-ЦIЛОМУ
This paper is concerned with positive linear operators based on a q-integer. The rate of covergence of these
operators are established. For these operators, we give Voronovskaya-type theorems and apply them to q-
Bernstein polynomials and q-Stancu operators.
Розглянуто додатнi лiнiйнi оператори, що базуються на q-цiлому. Встановлено швидкiсть збiжностi цих
операторiв. Теореми типу Вороновської наведено для цих операторiв та застосовано до q-полiномiв
Бернштейна та q-операторiв Станку.
1. Introduction. First formulate in what we know about q-calculus, which was initiatd by
Euler in the eighteenth century. Many remarkable results were obtained in the nineteenth
century. In 1910, Jackson [9] introduced the notion of the definite q-integral. He also was
the first to develop q-calculus in a systematic way. In the second half of the twentieth
century there was a significant increase of activity in the area of the q-calculus due to
applications of the q-calculus in mathematics and physics.
We now present definitions and facts from the q-calculus necessary for understanding
of this paper. We follow the terminology and notations from the recent book [10] (see
also [11]).
Definition 1.1. For an arbitrary function f(x), the q-differential is defined by
(dqf)(t) := f(qt)− f(t). In particular dqt = (q−1)t. The q-derivative of a function f
is defined by
Dqf(t) :=
(dqf)(t)
dqt
=
f(qt)− f(t)
(q − 1)t
, t 6= 0,
Dqf(0) = lim
t→0
Dqf(t)
and high q-derivatives are
D0
qf := f, Dn
q f := Dq(D
n−1
q f), n = 1, 2, 3, . . . .
Clearly, if f(x) is differentiable, then limq→1(Dqf)(x) = df(x)/dx.
Definition 1.2. Suppose a < b and 0 < b. In q-analysis, q-integral is defined as
b∫
0
f(t)dqt := (1− q)b
∞∑
j=0
qjf(qjb),
and
b∫
a
f(t)dqt :=
b∫
0
f(t)dqt−
a∫
0
f(t)dqt.
c© G. NOWAK, VIJAY GUPTA, 2011
350 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
THE RATE OF POINTWISE APPROXIMATION OF POSITIVE LINEAR OPERATORS . . . 351
Notice that limq→1
∫ b
a
f(t)dqt =
∫ b
a
f(t)dt.
For any fixed real number q > 0 and for nonnegative integer r, the q-integers of the
number [r]q is defined by
[r]q = (1− qr)/(1− q), for q 6= 1, [r]q = r, for q = 1.
Also [0]q = 0.
The q-factorial [r]q!, for r ∈ N0 = {0, 1, 2, . . .} is defined in the following
[r]q! = [1]q[2]q . . . [r]q, r = 1, 2, . . . , [0]q! = 1.
For the integers n, k, n ≥ k ≥ 0, the q-binomial or the Gaussian coefficients is defined
by (see [10, p. 12]) [
n
k
]
q
=
[n]q!
[k]q![n− k]q!
.
For f ∈ C[0; 1], δ > 0, we define the modulus of continuity ω(f ; δ) and the second
q-modulus of smoothness ωq,2(f ; δ) as follows:
ω(f ; δ) := sup
|h|≤δ
{
max
x∈[0;1−|h|]
|f(x+ h)− f(x)|
}
,
ωq,2(f ; δ) := sup
|h|≤δ
{
max
x∈[0;1−[2]q|h|]
|f(x+ [2]qh)− [2]qf(x+ h) + qf(x)|
}
.
It is clear that, if f ∈ C[0; 1], then
ωq,2(f ; δ)→ 0 for δ → 0
and
lim
q→1
ωq,2(f ; δ) = ω2(f ; δ),
where ω2(f ; δ) is a second modulus of smoothness. In the note, we obtain the estimates
for the rate of convergence for q-Bernstein – Stancu polynomials for 0 < q < 1, α ≥
≥ 0 in terms of ω(f ; ·) and ωq,2(f ; ·). Results are also new theorems for q-Bernstein
polynomials. In theorems and proofs we are using the q-differential and the q-integral.
In tis paper we present a few approximation theorems concerning with positive
operators based on q-integer. Typical examples these operators are: q-Bernstein operators
introduced by Phillips [15], generalized q-Bernstein operators introduced by Nowak
[13] and other. In third section we present these theorems for generalized q-Bernstein
operators. Now we are defining these operators.
For f ∈ C[0; 1], q > 0, α ≥ 0 and each positive integer n we introduce (see [13])
the following generalized q-Bernstein operators:
Bq,αn (f ;x) =
n∑
k=0
pq,αn,k(x)f
(
[k]q
[n]q
)
, (1.1)
where
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
352 G. NOWAK, VIJAY GUPTA
pq,αn,k(x) =
[
n
k
]
q
∏k−1
i=0
(x+ α[i]q)
∏n−1−k
s=0
(1− qsx+ α[s]q)∏n−1
i=0
(1 + α[i]q)
. (1.2)
Note, that an empty product in (1.2) denotes 1. In the case, where α = 0, Bq,αn (f ;x)
reduces to the well-known q-Bernstein polynomials introduced by Phillips [15] in 1997
Bn,q(f ;x) =
n∑
k=0
[
n
k
]
q
xk
n−k−1∏
i=0
(1− qix)f
(
[k]q
[n]q
)
. (1.3)
In the case, where q = 1, Bq,αn (f ;x) reduces to Bernstein – Stancu polynomials, introduced
by Stancu [26] in 1968
Sn(f ;x) =
n∑
k=0
(
n
k
)∏k−1
i=0
(x+ αi)
∏n−k−1
s=0
(1− x+ sα)∏n−1
i=0
(1 + iα)
f
(
k
n
)
.
When q = 1 and α = 0 we obtain the classical Bernstein polynomial defined by
Bn(f ;x) =
n∑
k=0
(
n
k
)
xk(1− x)n−kf
(
k
n
)
.
Basic facts on Bernstein polynomials, their generalizations and applications, can be found
e.g. in [12, 22 – 24]. In recent years, the q-Bernstein polynomials have attracted much
interest, and a great number of interesting results related to the Bn,q(f) polynomials
have been obtained (see [6, 8, 15 – 21, 27 – 30]). Some approximation properties of the
Stancu operators are presented in [3 – 5, 26].
Throughout, the symbolsK, K1, K2, . . .will mean some positive absolute constants,
not necessarily the same at each occurence.
2. Main result. Let (Ln)n≥1 be a sequence of positive linear operators on C[0, 1].
We introduce the standard notation for r ∈ N0 = {0, 1, . . .}, n ∈ N,
µr,n(x) = Lnf(x), where f(x) = (t− x)r,
ei := ei(x) = xi.
Theorem 2.1. Let (Ln)n≥1 be a sequence of linear positive operators, satisfying
µ0,n(x) = 1 and µ1,n(x) = 0. Suppose that f ∈ C[0, 1]. Then there exist q ∈ (0; 1)
such that for all q ∈ (q; 1) and all n ∈ N∣∣Ln(f ;x)− f(x)
∣∣ ≤ 2δnω(Dqf ; δn),
where δn = δn(x) =
√
µ2,n(x).
Proof. By the q-mean value theorem [25], there exists q ∈ (0; 1) such that ∀q ∈
∈ (q; 1) ∃ξ ∈ (t;x):
f(t)− f(x) = (t− x)Dqf(ξ).
As the operators Ln are linear and positive and on the fact that µ0,n(x) = 1, µ1,n(x) = 0
it follows immediately the equality
Ln(f ;x)− f(x) = Ln((e1 − x)Dqf(ξt,x);x) =
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
THE RATE OF POINTWISE APPROXIMATION OF POSITIVE LINEAR OPERATORS . . . 353
= Ln((e1 − x)(Dqf(ξt,x)−Dqf(x));x),
where ξt,x ∈ (u, v), u = min{x; t}, v = max{x; t}. Using the property of modulus of
continuity (see [12]), we get∣∣Dqf(ξt,x)−Dqf(x)
∣∣ ≤ ω(Dqf ; |ξt,x − x|) ≤
≤ (1 + δ−1n |ξt,x − x|)ω(Dqf ; δn) ≤ (1 + δ−1n |t− x|)ω(Dqf ; δn),
where δn > 0. Consequently,
|Ln(f ;x)− f(x)| ≤ ω(Dqf ; δn)Ln(|e1 − x|(1 + δ−1n |e1 − x|)).
The known Schwarz inequality and δn = δn(x) =
√
µ2,n(x) lead to theorem.
Theorem 2.2. Let f ∈ C[0; 1] and Ln(f ;x) be as in above theorem. Then there
exist q ∈ (0; 1) such that for all q ∈ (q; 1) the following inequality holds:∣∣Ln(f ;x)− f(x)
∣∣ ≤ K√µ2,n(x)‖D2
qf‖, (2.1)
for all n ∈ N, where K = K(n, x, q) =
1
[2]q
(√
µ2,n(x) + (1− q)
)
.
Proof. Using the q-Taylor formula [2] we write
f(t) = f(x) +Dqf(x)(t− x) +Rf,q,x(t),
where
Rf,q,x(t) =
t∫
x
(t− qv)D2
qf(v)dqv.
It is shown in [25] that there exist q ∈ (0; 1) such that for all q ∈ (q; 1), ξt,x ∈ (u, v),
u = min{x; t}, v = max{x; t}, which satisfies
Rf,q,x(t) =
D2
qf(ξt,x)
[2]q!
(t− x)(t− qx) =
D2
qf(ξt,x)
[2]q!
((t− x)2 + (1− q)(t− x)).
Schwarz’s inequality yields inequality (2.1) immediately.
Theorem 2.3. Let (Ln)n≥1 be a sequence of linear positive operators satisfying
µ0,n(x) = 1, µ1,n(x) = 0 and µ2,n(x) 6= 0. Suppose that f ∈ C[0, 1]. Then there exist
q ∈ (0; 1) such that for all q ∈ (q; 1) and all n ∈ N∣∣∣∣∣Ln(f ;x)− f(x)−
D2
qf(x)
[2]!
µ2,n(x)
∣∣∣∣∣ ≤ K(n, x, q)µ2,n(x)ω(D2
qf ; δn), (2.2)
where
δn = δn(q, x) = max
{
sup
x
(√
µ4,n(x)
µ2,n(x)
)
;
1
[n]q
}
and
K(n, x, q) =
1
[2]q
(
3 +
x
[n]q
√
µ2,n(x)
)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
354 G. NOWAK, VIJAY GUPTA
Proof. By the q-Taylor formula [25], there exists q ∈ (0; 1) such that ∀q ∈ (q; 1)
∃ξt,x ∈ (t;x), such that
f(t) = f(x) +Dqf(x)(t− x) +
D2
qf(x)
[2]q!
(t− x)(t− qx) + rf,q,x(t),
where
rf,q,x(t) =
D2
qf(ξt,x)−D2
qf(x)
[2]q!
(t− x)(t− qx).
The property µ0,n(x) = 1 and µ1,n(x) = 0, give us
Ln(f ;x)− f(x)−
D2
qf(x)
[2]!
µ2,n(x) = Ln(rf,q,x;x).
Using the property of modulus of continuity (see [12]), we get∣∣D2
qf(ξt)−D2
qf(x)
∣∣ ≤ (1 + δ−1|t− x|)ω(D2
qf ; δ),
where δ > 0. By te inequality 1/(1− q) ≤ 1/[n]q and simple calculation, we have
|Ln(rf,q,x;x)| ≤
≤
ω(D2
qf ; δ)
[2]q
µ2,n(x)
{
1 +
x
[n]q
√
µ2,n(x)
+ δ−1
(√
µ4,n(x)
µ2,n(x)
+
x
[n]q
)}
.
Choosing δn = max
{
supx
(√
µ4,n(x)
µ2,n(x)
)
;
1
[n]q
}
we get estimate (2.2) immediately.
Theorem 2.4. Let (Ln)n≥1 be a sequence of linear positive operators satisfying
µ0,n(x) = 0 and µ1,n(x) = 0. Suppose that f ∈ C[0, 1]. Then there exist q ∈ (0; 1)
such that for all q ∈ (q; 1) the following inequality holds:
‖Lnf − f‖ ≤
6
q
ωq,2(f ; δn)
for all n ∈ N, where δn = δn(q, x) =
√
µ2,n(x) max
{√
µ2,n(x); 1− q
}
.
Proof. For 0 < h ≤ 1
[2]q
min{x; 1− x} we define
gq,h(x) =
1
qh2
h/2∫
−h/2
h/2∫
−h/2
{
[2]qf(x+ t1 + t2)− f(x+ [2]qt1 + [2]qt2)
}
dqt1dqt2.
It is easy to check that
b∫
a
Dqf(t)dqt = f(b)− f(a).
Consequently∣∣D2
q(gq,h(x))
∣∣ =
1
qh2
∣∣∣(f(x+ [2]qh)− [2]qf(x+ h) + qf(x))+
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
THE RATE OF POINTWISE APPROXIMATION OF POSITIVE LINEAR OPERATORS . . . 355
+(f(x− [2]qh)− [2]qf(x− h) + qf(x))
∣∣∣ ≤ 2
qδ2
ωq,2(f ; δ).
By definition of q-integral
∣∣∣∣∫ a
−a
h(t)dqt
∣∣∣∣ ≤ 2a supu∈〈−a;a〉 |h(u)|. Therefore
∣∣f(x)− gq,h(x)
∣∣ =
=
1
qh2
∣∣∣∣∣∣∣
h/2∫
−h/2
h/2∫
−h/2
(f(x+ [2]q(t1 + t2))− [2]qf(x+ t1 + t2) + qf(x))dqt1dqt2
∣∣∣∣∣∣∣ ≤
≤ 1
q
ωq,2(f ; δ).
Using these inequalities and (2.1) we have
‖Lnf − f‖ ≤ ‖Ln(f − gq,h)‖+ ‖Lngq,h − gq,h‖+ ‖f − gq,h‖ ≤
≤ 2‖f − gq,h‖+ (µ2,n(x) + (1− q)
√
µ2,n(x))‖D2
qgq,h‖.
Choosing δn =
√
µ2,n(x) max{
√
µ2,n(x); 1− q} we get the desired estimate.
3. q-Stancu polynomial. The generalized q-Bernstein polynomial defined by (1.1)
can be simply expressed in terms of q-differences. For any functions f we define
∆0
qfj = fj
for j = 0, 1, . . . , n and, recursively,
∆k+1
q fj = ∆k
qfj+1 − qk∆k
qfj ,
for k = 0, 1, . . . , n− j − 1, where fj = f([j]/[n]). It is easily established by induction
that q-differences satisfy the relation
∆k
qfj =
k∑
i=0
(−1)iqi(i−1)/2
[
k
i
]
q
fj+k−i. (3.1)
Lemma 3.1 [13]. Let 0 < q < 1, α ≥ 0. The generalized q-Bernstein polynomial
may be expressed in the form
Bq,αn (f ;x) =
n∑
k=0
[
n
k
]
q
∆k
qf0
k−1∏
i=0
x+ α[i]q
1 + α[i]q
, (3.2)
for all n ∈ N and x ∈ [0; 1].
Let 0 < q < 1, α ≥ 0. From (3.1) and (3.2) by a simple calculation [13], we have
Bq,αn (1;x) = 1, (3.3)
Bq,αn (t;x) = x, (3.4)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
356 G. NOWAK, VIJAY GUPTA
Bq,αn (t2;x) =
1
1 + α
(
x(x+ α) +
x(1− x)
[n]q
)
, (3.5)
µq,α1,n(x) = 0, (3.6)
µq,α2,n(x) =
x(1− x)
1 + α
(
α+
1
[n]q
)
, (3.7)
for all n ∈ N and x ∈ [0; 1].
Lemma 3.2. Let 0 < q < 1, α ≥ 0. Then
Bq,αn (t3;x) =
1∏2
i=0
(1 + [i]qα)
2∑
k=0
1
[n]kq
W k(q, α, x), (3.8)
Bq,αn (t4;x) =
1∏3
i=0
(1 + [i]qα)
3∑
k=0
1
[n]kq
Wk(q, α, x), (3.9)
where
W 0(q, α, x) = x(x+ α)(x+ [2]qα),
W 1(q, α, x) = x(1− x)(x+ α)(2 + q),
W 2(q, α, x) = x(1− x)(1− [2]qx),
W0(q, α, x) = x(x+ α)(x+ [2]qα)(x+ [3]qα),
W1(q, α, x) = x(1− x)(x+ α)(x+ [2]qα)(q2 + 2q + 3),
W2(q, α, x) = x(1− x)(x+ α)×
×
{
(q2 + 3q + 3)x(x+ α)− [2]2q(x+ α− [2]qx(1 + [3]qα)
}
,
W3(q, α, x) = x(1− x){[2]qx([3]qx− q − 2) + 1− qα}
for all n ∈ N and x ∈ [0; 1].
Proof. The q-difference of the monomial xk of order greater than k are zero.
Consequently, we see from (3.2) that for all n ≥ k, Bq,αn (xk;x) is a polynomial of
degree k.
We deduce from (3.2) that
Bq,αn (t4;x) =
4∑
k=0
[
n
k
]
q
∆k
qf0
k−1∏
i=0
x+ α[i]q
1 + α[i]q
.
From (3.1), for f(x) = x4, we have
∆0
qf0 = f0 = 0, ∆1
qf0 = f1 − f0 = 1/[n]4q,
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THE RATE OF POINTWISE APPROXIMATION OF POSITIVE LINEAR OPERATORS . . . 357
∆2
qf0 = f2 − (1 + q)f1 + qf0 =
[2]qq
[n]4q
(q2 + 3q + 3),
∆3
qf0 = f3 − [3]qf2 + q[3]qf1 − q3f0 =
[2]q[3]qq
3
[n]4q
(q2 + 2q + 3),
∆4
qf0 = f4 − [4]qf3 + q(1 + q2)[3]qf2 − q3[4]qf1 + q6f0 =
[2]q[3]q[4]qq
6
[n]4q
.
Thanks to obvious equality qr[n− r]q = [n]q − [r]q, for r = 1, 2, 3, we have
Bq,αn (t4;x) =
x
[n]3q
+
[n]q − 1
[n]3q
(q2 + 3q + 3)
x(x+ α)
1 + α
+
+
([n]q − 1)([n]q − [2]2)
[n]3q
(q2 + 2q + 3)
x(x+ α)(x+ [2]qα)
(1 + α)(1 + [2]qα)
+
+
([n]q − 1)([n]q − [2]q)([n]q − [3]q)
[n]3q
3∏
i=0
x+ [i]qα
1 + [i]qα
.
By simple calulation we obtain (3.9). On the same way we can prove (3.8).
Lemma 3.3. Let q ∈ (0; 1), α ≥ 0. Then for every n ∈ N and x ∈ [0; 1]
µq,α4,n(x) ≤ Kx(1− x)
1 + α
1
[n]q
(
1
[n]q
+ α
)
.
Proof. In view of (3.3) – (3.9)
µq,α4,n(x) = Bq,αn (t4;x)− 4xBq,αn (t3;x) + 6x2Bq,αn (t2;x)− 3x4 =
=
3∑
i=0
1
[n]iq
Vi(q, α, x),
where
V0(q, α, x) =
x(x+ α)(x+ [2]qα)(x+ [3]qα)
(1 + α)(1 + [2]qα)(1 + [3]qα)
−
−4x2(x+ α)(x+ [2]qα)
(1 + α)(1 + [2]qα)
+
6x3(x+ α)
1 + α
− 3x4,
V1(q, α, x) =
x(1− x)(x+ α)(x+ [2]qα)
(1 + α)(1 + [2]qα)(1 + [3]qα)
(q2 + 2q + 3)−
−4x2(1− x)(x+ α)
(1 + α)(1 + [2]qα)
(2 + q) +
6x3(1− x)
1 + α
,
V2(q, α, x) =
x(1− x)(x+ α)
(1 + α)(1 + [2]qα)(1 + [3]qα)
×
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
358 G. NOWAK, VIJAY GUPTA
×
{
(q2 + 3q + 3)(x+ α)x− [2]2q(x+ α)− [2]qx(1 + [3]qα)
}
−
−4x2(1− x)(1− [2]qx)
(1 + α)(1 + [2]qα)
,
V3(q, α, x) =
x(1− x)
(1 + α)(1 + [2]qα)(1 + [3]qα)
{
[2]qx([3]qx− q − 2) + 1− qα
}
.
For 0 ≤ x ≤ 1, 0 < q < 1, α > 0, we have α/(1 + [2]qα) < 1, 1− q < 1/[n]q. Thanks
to that it is easy to see that
V0(q, α, x) =
x(1− x)α(1− q)2
(1 + α)(1 + [2]qα)(1 + [3]qα)
≤ x(1− x)
1 + α
1
[n]2q
,
|V1(q, α, x)| ≤ K1
x(1− x)
1 + α
(
1
[n]2q
+ α
)
,
|V2(q, α, x)| ≤ K2
x(1− x)
1 + α
,
|V3(q, α, x)| ≤ K3
x(1− x)
1 + α
.
Collecting the results we get estimate (3.6) immediately.
In next theorems we assume that (qn) and (αn) denotes a sequence such that 0 <
< qn ≤ 1, αn ≥ 0 and Bq,αn (f ;x) is defined by (1.1) with q = qn and α = αn.
Theorem 3.1. Suppose that f ∈ C[0, 1]. Then there exist q ∈ (0; 1) such that for
all q ∈ (q; 1)
|Bqn,αn
n (f ;x)− f(x)| ≤ 2
√
x(1− x)
1 + αn
δnω(Dqf ; δn),
where δn = (1/[n]qn + αn)1/2 and n ∈ N.
Theorem 3.2. Let f : [0, 1] → R. Then there exist q ∈ (0; 1) such that for all
q ∈ (q; 1) the following inequality holds:
∣∣Bqn,αn
n (f ;x)− f(x)
∣∣ ≤ x
√
(1− x)(1− qnx)
[2]q(1 + αn)
(
αn +
1
[n]qn
)
‖D2
qf‖
for all n ∈ N.
Theorem 3.3. At assumptions from the previous theorem we have∣∣∣∣∣Bqn,αn
n (f ;x)− f(x)−
D2
qf(x)
[2]q(1 + αn)
x(1− x)
(
αn +
1
[n]qn
)∣∣∣∣∣ ≤
≤ Kx
√
1− x
(
αn +
1
[n]qn
)
ω
(
D2
qf ; δn
)
,
where δn = max
{
1
[n]
1/2
qn
;
1
[n]q
}
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
THE RATE OF POINTWISE APPROXIMATION OF POSITIVE LINEAR OPERATORS . . . 359
It is easy to check that limq→1(D2
qf)(x) = df ′′(x)/dx. Therefore, we have the
following corollary.
Corollary 3.1. If f ∈ C(2)[0; 1] and qn → 1, an → 0 as n→∞, then
lim
n→∞
(
1
[n]qn
+ αn
)−1
(1 + αn){Bqn,αn
n (f ;x)− f(x)} =
f ′′(x)
2
x(1− x)
uniformly on [0; 1].
It will be noted that, for α = 0, thit is corollary obtain by Videnskii [27] for
q-Bernstein polynomial defined by (1.3).
Remark 3.1. For the function f(t) = t2 takes place the exact equality(
1
[n]qn
+ αn
)−1
(1 + αn)
{
Bqn,αn
n (t2;x)− x2
}
=
(x2)′′
2
x(1− x).
without passing to the limit.
Theorem 3.4. If f is continuous on [0, 1], then there exist q ∈ (0; 1) such that for
all qn ∈ (q; 1) the following inequality holds:
‖Bqn,αn
n f − f‖ ≤ 9
4q
ωq,2(f ; δn),
where δn = max
{
(αn + 1/[n]qn)1/2; 1− q
}
.
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360 G. NOWAK, VIJAY GUPTA
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Received 20.02.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
|
| id | umjimathkievua-article-2721 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:01Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/42/6015eeee1e99df7f59ba1f41e4c81b42.pdf |
| spelling | umjimathkievua-article-27212020-03-18T19:34:39Z The rate of pointwise approximation of positive linear operators based on q-integer Швидкiсть поточкового наближення додатних лiнiйних операторiв, що базуються на q-цiлому Gupta, Vijay Nowak, G. Гупта, В. Новак, Г. This paper is concerned with positive linear operators based on a q-integer. The rate of covergence of these operators are established. For these operators, we give Voronovskaya-type theorems and apply them to q Bernstein polynomials and q-Stancu operators. Розглянуто додатнi лiнiйнi оператори, що базуються на q-цiлому. Встановлено швидкiсть збiжностi цих операторiв. Теореми типу Вороновської наведено для цих операторiв та застосовано до q-полiномiв Бернштейна та q-операторiв Станку. Institute of Mathematics, NAS of Ukraine 2011-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2721 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 3 (2011); 350-360 Український математичний журнал; Том 63 № 3 (2011); 350-360 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2721/2198 https://umj.imath.kiev.ua/index.php/umj/article/view/2721/2199 Copyright (c) 2011 Gupta Vijay; Nowak G. |
| spellingShingle | Gupta, Vijay Nowak, G. Гупта, В. Новак, Г. The rate of pointwise approximation of positive linear operators based on q-integer |
| title | The rate of pointwise approximation of positive linear operators based on q-integer |
| title_alt | Швидкiсть поточкового наближення додатних лiнiйних операторiв, що базуються на q-цiлому |
| title_full | The rate of pointwise approximation of positive linear operators based on q-integer |
| title_fullStr | The rate of pointwise approximation of positive linear operators based on q-integer |
| title_full_unstemmed | The rate of pointwise approximation of positive linear operators based on q-integer |
| title_short | The rate of pointwise approximation of positive linear operators based on q-integer |
| title_sort | rate of pointwise approximation of positive linear operators based on q-integer |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2721 |
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