Approximation by (p,q)-Baskakov-Beta operators

0.39 MB·

by  Neha Malik

APPROXIMATION BY (p,q)-BASKAKOV-BETA OPERATORS
Neha Malik and Vijay Gupta
Department of Mathematics, Netaji Subhas Institute of Technology
Sector 3 Dwarka, New Delhi-110078, India
neha.malik [email protected]
6 [email protected]
1
0
2

r
p
Abstract. In the present paper, we consider (p,q)-analogue of the Baskakov-Beta
A
operators and using it, we estimate some direct results on approximation. Also, we

9 represent the convergence of these operators graphically using MATLAB.
1
Key Words. (p,q)-Beta function, (p,q)-Gamma function.

] AMS Subject Classification. 33B15, 41A25.
A
C
1. Introduction
.
h
at Approximation theory has been an engaging field of research with abstract approx-
m imation to the core (cf. [15]). Varied operators with their approximation properties,
[ mainly the quantitative one, have been discussed and studied by many researchers. It has

2 been seen that the generalizations of several well known operators to quantum-calculus
v (q-calculus) were introduced in the last three decades and their approximation behavior
8
were also discussed (see [3], [10], [11] and [12]). Further generalization of quantum variant
0
3 is the post-quantum calculus, denoted by (p,q)-calculus. Very recently, some researchers
6
studied in this direction (see [4], [9] and [17]). Few basic definitions and notations men-
0
. tioned below may be found in these papers and references therein.
2
The (p,q)-numbers are given by
0
6
1 [n] := pn−1 +pn−2q +pn−3q2 +···+pqn−2 +qn−1
p,q
:
v (cid:26) pn−qn , if p (cid:54)= q (cid:54)= 1;
i = p−q
X n , if p = q = 1.
r
a n
The (p,q)-factorial is given by [n] ! = (cid:81) [r] , n (cid:62) 1, [0] ! = 1. The (p,q)-binomial
p,q p,q p,q
r=1
coefficient satisfies
(cid:20) n (cid:21) [n] !
= p,q , 0 (cid:54) r (cid:54) n.
r [n−r] ![r] !
p,q p,q p,q
Let n be a non-negative integer, the (p,q)-Gamma function is defined as
(p(cid:9)q)n
p,q
Γ (n+1) = = [n] !, 0 < q < p,
p,q (p−q)n p,q
1
where (p(cid:9)q)n = (p−q)(p2 −q2)(p3 −q3)···(pn −qn).
p,q
The (p,q)-integral for 0 < q < p (cid:54) 1 (generalized Jackson integral) is defined as
a
(cid:90) (cid:88)∞ qi (cid:18)aqi (cid:19)
f(x)d x = (p−q)a f , x ∈ [0,a]. (1)
p,q pi+1 pi+1
i=0
0
By simple computation, we get
a
(cid:90) an+1
xn d x = ·
p,q
[n+1]
p,q
0
Also, the integral (1) includes the nodes x = x (p,q) = aqi , i = 0,1,..., geometrically
i i pi+1
distributed in (0,+∞), not only in (0,a), as in the case p = 1 (standard Jackson’s q-
integral). Moreover, one may observe that only a finite number of nodes in (1) are outside
(0,a), i.e., those x for which qi > pi+1. Thus, the above definition of (p,q)-integral may
i
be well utilized to define the (p,q)-extensions of well-known results.
For m,n ∈ N, the (p,q)-Beta function of second kind considered in [2] is given by
∞
(cid:90) tm−1
B (m,n) = d t,
p,q (1⊕pt)m+n p,q
p,q
0
where the (p,q)-power basis is given by
(1⊕pt)m+n = (1+pt)(p+pqt)(p2 +pq2t)···(pm+n−1 +pqm+n−1t).
p,q
Using the (p,q)-integration by parts:
b b
(cid:90) (cid:90)
f(px) D g(x) d x = f(b)g(b)−f(a)g(a)− g(qx) D f(x) d x,
p,q p,q p,q p,q
a a
it was shown in [2] that the following relation is satisfied by the (p,q)-analogues of Beta
and Gamma functions:
q Γ (m)Γ (n)
p,q p,q
B (m,n) = ·
p,q (pm+1 qm−1)m/2 Γ (m+n)
p,q
As a special case, if p = q = 1, B(m,n) = Γ(m) Γ(n)/Γ(m + n). It may be observed
that in (p,q)-setting, order is important, which is the reason why (p,q)-variant of Beta
function does not satisfy commutativity property, i.e., B (m,n) (cid:54)= B (n,m).
p,q p,q
For n ∈ N, x ∈ [0,∞) and 0 < q < p (cid:54) 1, the (p,q)-analogue of Baskakov operators
can be defined as
(cid:88)∞ (cid:18)pn−1[k] (cid:19)
B (f,x) = bp,q(x)f p,q ,
n,p,q n,k qk−1[n]
p,q
k=0
where (p,q)-Baskakov basis function is given by
(cid:20) n+k −1 (cid:21) xk
bp,q(x) = pk+n(n−1)/2qk(k−1)/2 ·
n,k k (1⊕x)n+k
p,q p,q
Gupta[9]consideredthisformof(p,q)-BaskakovoperatorswhilestudyingitsKantorovich
variant. This form was also considered by T. Acar et. al. in [5].
Remark 1. It has been observed in [9] that the (p,q)-Baskakov operators satisfy the fol-
lowing recurrence relation:
[n] Tp,q (qx) = qpn−1 x(1+px) D [Tp,q (x)]+[n] qxTp,q (qx),
p,q n,m+1 p,q n,m p,q n,m
∞ (cid:16) (cid:17)m
where Tp,q (x) := B (e ,x) = (cid:80) bp,q(x) pn−1[k]p,q .
n,m n,p,q m n,k qk−1[n]p,q
k=0
Then, we have
B (e ,x) = 1, B (e ,x) = x,
n,p,q 0 n,p,q 1
[n+1] x2 +pn−1qx
p,q
B (e ,x) = ,
n,p,q 2
q[n]
p,q
where e (t) = ti, i = 0,1,2. In case p = 1, we get the q-Baskakov operators [1], [11]. If
i
p = q = 1, then these operators reduce to the well known Baskakov operators.
2. Construction of Operators and Moments
In the year 1985, Sahai-Prasad [16] introduced the Durrmeyer variant of the well known
Baskakov operators. However, there were some technical problems in the main estimates
of [16], which were later improved by Sinha et. al. [19]. In this continuation, in 1994,
Gupta proposed yet another Durrmeyer type generalization of Baskakov operators by
taking the weights of Beta basis function. The operators discussed in [8] provide better
approximation in simultaneous approximation than the usual Baskakov-Durrmeyer oper-
ators, studied in [19]. This motivated us to study further in this direction and here, we
propose the (p,q)-variant of Baskakov-Beta operators.
For n ∈ N, x ∈ [0,∞) and 0 < q < p (cid:54) 1, the (p,q)-Baskakov-Beta operators are
defined by:
∞
(cid:88)∞ 1 (cid:90) tk
Dp,q(f,x) = bp,q(x) f(q2pn+kt) d t, (2)
n n,k B (k +1,n) (1⊕pt)n+k+1 p,q
k=0 p,q p,q
0
(cid:20) (cid:21)
n+k −1
where bp,q(x) = pk+n(n−1)/2 qk(k−1)/2 xk ·
n,k k (1⊕x)np,+qk
p,q
In the present article, we estimate the moments of these operators by using (p,q)-Beta
functions and establish some direct results in terms of modulus of continuity of first and
second order using K-functional. Finally, we provide weighted approximation estimate
alongwith the rate of convergence.
Lemma 1. The following equalities hold:
(1) Dp,q(1,x) = 1;
n
(2) Dp,q(t,x) = [n]p,qx+pn−2 q, for n > 1;
n [n−1]p,q
(3) Dp,q(t2,x) = x2[n]p,q(cid:16)[n]p,q+pqn(cid:17)+x[n]p,q{pn−3 q2+2pn−2q+pn−1}+ p2n−5q[2]p,q , for n > 2.
n q [n−1]p,q [n−2]p,q q [n−1]p,q [n−2]p,q [n−1]p,q [n−2]p,q
Proof. By Remark 1, we have
∞
(cid:88)∞ 1 (cid:90) tk
Dp,q(1,x) = bp,q(x) d t
n n,k B (k +1,n) (1⊕pt)n+k+1 p,q
k=0 p,q p,q
0
∞
(cid:88) 1
= bp,q(x) B (k +1,n)
n,k B (k +1,n) p,q
p,q
k=0
∞
(cid:88)
= bp,q(x)
n,k
k=0
= B (1,x) = 1.
n,p,q
Next, using [k +1] = qk +p[k] , we have
p,q p,q
∞
(cid:88)∞ 1 (cid:90) tk+1q2pn+k
Dp,q(t,x) = bp,q(x) d t
n n,k B (k +1,n) (1⊕pt)n+k+1 p,q
k=0 p,q p,q
0
∞
(cid:88) 1
= bp,q(x) q2pn+kB (k +2,n−1)
n,k B (k +1,n) p,q
p,q
k=0
∞
(cid:88) Γ (n+k +1)
= bp,q(x) p,q q2pn+k
n,k q[2−(k+1)k]/2p−(k+1)(k+2)/2Γ (k +1)Γ (n)
p,q p,q
k=0
Γ (k +2)Γ (n−1)
× q[2−(k+2)(k+1)]/2p−(k+2)(k+3)/2 p,q p,q
Γ (n+k +1)
p,q
∞
(cid:88) [k +1]
= bp,q(x) pn−2 q1−k p,q
n,k [n−1]
p,q
k=0
∞
1 (cid:88)
= bp,q(x) pn−2 q1−k (qk +p[k] )
[n−1] n,k p,q
p,q
k=0
pn−2 q (cid:88)∞ [n] (cid:88)∞ pn−1[k]
= bp,q(x)+ p,q bp,q(x) p,q·
[n−1] n,k [n−1] n,k qk−1[n]
p,q p,q p,q
k=0 k=0
Using Remark 1, we have
pn−2 q [n]
Dp,q(t,x) = B (1,x)+ p,q B (t,x)
n [n−1] n,p,q [n−1] n,p,q
p,q p,q
pn−2 q [n]
p,q
= ·1+ ·x
[n−1] [n−1]
p,q p,q
[n] x+pn−2 q
p,q
= ·
[n−1]
p,q
Further, using the identity [k +2] = qk+1 +pqk +p2[k] , we get
p,q p,q
∞
(cid:88)∞ 1 (cid:90) tk+2q4p2(n+k)
Dp,q(t2,x) = bp,q(x) d t
n n,k B (k +1,n) (1⊕pt)n+k+1 p,q
k=0 p,q p,q
0
∞
(cid:88) 1
= bp,q(x) q4p2(n+k) B (k +3,n−2)
n,k B (k +1,n) p,q
p,q
k=0
∞
(cid:88) Γ (n+k +1)
= bp,q(x) p,q q4p2(n+k)
n,k q[2−(k+1)k]/2 p−(k+1)(k+2)/2 Γ (k +1) Γ (n)
p,q p,q
k=0
Γ (k +3) Γ (n−2)
× q[2−(k+3)(k+2)]/2 p−(k+3)(k+4)/2 p,q p,q
Γ (n+k +1)
p,q
∞
(cid:88) [k +2] [k +1]
= bp,q(x) p2n−5 q1−2k p,q p,q
n,k [n−1] [n−2]
p,q p,q
k=0
∞
= 1 (cid:88)bp,q(x) p2n−5 q1−2k (cid:0)qk+1 +pqk +p2[k] (cid:1)
[n−1] [n−2] n,k p,q
p,q p,q
k=0
(cid:0) (cid:1)
× qk +p[k]
p,q
p2n−5q2 +p2n−4q (cid:88)∞
= bp,q(x)
[n−1] [n−2] n,k
p,q p,q
k=0
pn−3q[n] (cid:88)∞ pn−1[k]
+ p,q bp,q(x) p,q
[n−1] [n−2] n,k qk−1[n]
p,q p,q p,q
k=0
2pn−2[n] (cid:88)∞ pn−1[k]
+ p,q bp,q(x) p,q
[n−1] [n−2] n,k qk−1[n]
p,q p,q p,q
k=0
[n]2 (cid:88)∞ p2n−2[k]2
+ p,q bp,q(x) p,q·
q [n−1] [n−2] n,k q2k−2[n]2
p,q p,q k=0 p,q
Again, using Remark 1
p2n−5q2 +p2n−4q pn−3q[n]
Dp,q(t2,x) = B (1,x) + p,q B (t,x)
n [n−1] [n−2] n,p,q [n−1] [n−2] n,p,q
p,q p,q p,q p,q
2pn−2[n] [n]2
+ p,q B (t,x) + p,q B (t2,x)
n,p,q n,p,q
[n−1] [n−2] q [n−1] [n−2]
p,q p,q p,q p,q
p2n−5q2 +p2n−4q x[n] {pn−3q +2pn−2}
p,q
= +
[n−1] [n−2] [n−1] [n−2]
p,q p,q p,q p,q
(cid:26) pn−1x (cid:18) px(cid:19)(cid:27) [n]2
+ x2 + 1+ p,q
[n] q q [n−1] [n−2]
p,q p,q p,q
(cid:16) (cid:17)
x2[n] [n] + pn
p,q p,q q x[n]p,q{pn−3q2 +2pn−2q +pn−1}
= +
q [n−1] [n−2] q [n−1] [n−2]
p,q p,q p,q p,q
p2n−5q[2]
p,q
+ ·
[n−1] [n−2]
p,q p,q
(cid:3)
Remark 2. Let n > 2 and x ∈ [0,∞), then for 0 < q < p (cid:54) 1, we have the central
moments as follows:
µp,q(x) := Dp,q((t−x),x)
n,1 n
x([n] −[n−1] )+pn−2q
p,q p,q
=
[n−1]
p,q
and
µp,q(x) := Dp,q((t−x)2,x)
n,2 n
(cid:110) (cid:16) (cid:17) (cid:111)
x2 [n] [n] + pn +q[n−1] [n−2] −2q[n] [n−2]
p,q p,q q p,q p,q p,q p,q
=
q[n−1] [n−2]
p,q p,q
x{[n] (pn−3q2 +2pn−2q +pn−1)−2pn−2q2[n−2] } [2] p2n−5q
p,q p,q p,q
+ + ·
q[n−1] [n−2] [n−1] [n−2]
p,q p,q p,q p,q
3. Direct Estimations
In this section, we prove direct results using two different approaches, i.e., K-functional
and Steklov mean. We also represent the convergence of the (p,q)-Baskakov-Beta opera-
tors using the software MATLAB.
Wedenotethenorm||f|| = sup |f(x)|onC [0,∞),theclassofrealvaluedcontinuous
B
x∈[0,∞)
bounded functions. For f ∈ C [0,∞) and δ > 0, the m-th order modulus of continuity is
B
defined as
ω (f,δ) = sup sup |∆mf(x)|,
m h
0(cid:54)h(cid:54)δ x∈[0,∞)
where ∆ is the forward difference and ∆m = ∆ (cid:0)∆m−1(cid:1) for m (cid:62) 1. In case m = 1, we
h h h h
mean the usual modulus of continuity denoted by ω(f,δ).
The Peetre’s K-functional is defined as
(cid:8) (cid:9)
K (f,δ) = inf ||f −g||+δ||g(cid:48)(cid:48)|| : g ∈ C2[0,∞) ,
2 B
g∈C2[0,∞)
B
where
C2[0,∞) = {g ∈ C [0,∞) : g(cid:48),g(cid:48)(cid:48) ∈ C [0,∞)}.
B B B
Theorem 1. Let f ∈ C [0,∞) and 0 < q < p (cid:54) 1, then for every x ∈ [0,∞) and n > 2,
B
the following inequality holds:
(cid:18) (cid:113) (cid:19)
|Dp,q(f,x)−f(x)| (cid:54) ω(cid:0)f,|µp,q(x)|(cid:1)+C ω f, µp,q(x)+(cid:0)µp,q(x)(cid:1)2 ,
n n,1 2 n,2 n,1
where C is some positive constant.
Proof. Consider the following operator:
(cid:18)[n] x+pn−2 q(cid:19)
Dˇp,q(f,x) = Dp,q(f,x)−f p,q +f(x), x ∈ [0,∞) (3)
n n [n−1]
p,q
Let g ∈ C2[0,∞) and x, t ∈ [0,∞). By Taylor’s expansion, we have
B
t
(cid:90)
g(t) = g(x)+(t−x) g(cid:48)(x)+ (t−u) g(cid:48)(cid:48)(u) du,
x
Applying Dˇp,q, we get
n
 
t
(cid:90)
Dˇp,q(g,x)−g(x) = g(cid:48)(x)Dˇp,q((t−x),x)+Dˇp,q (t−u) g(cid:48)(cid:48)(u) du,x.
n n n
x
Hence,
|Dˇp,q(g,x)−g(x)|
n
[n]p,qx+pn−2 q
(cid:54) (cid:18)Dp,q(cid:12)(cid:12)(cid:12)(cid:90)t |t−u| |g(cid:48)(cid:48)(u)| du(cid:12)(cid:12)(cid:12), x(cid:19)+(cid:12)(cid:12)(cid:12) [n(cid:90)−1]p,q (cid:18)[n]p,qx+pn−2 q −u(cid:19)g(cid:48)(cid:48)(u) du(cid:12)(cid:12)(cid:12)
n (cid:12) (cid:12) (cid:12) [n−1] (cid:12)
p,q
x x
[n]p,qx+pn−2 q
(cid:54) Dp,q((t−x)2, x) (cid:107)g(cid:48)(cid:48)(cid:107)+(cid:12)(cid:12)(cid:12) [n(cid:90)−1]p,q (cid:18)[n]p,qx+pn−2 q −u(cid:19) g(cid:48)(cid:48)(u) du(cid:12)(cid:12)(cid:12)
n (cid:12) [n−1] (cid:12)
p,q
x
(cid:40) (cid:41)
(cid:18)x ([n] −[n−1] )+pn−2q(cid:19)2
(cid:54) µp,q(x)+ p,q p,q (cid:107)g(cid:48)(cid:48)(cid:107).
n,2 [n−1]
p,q
Now, using operators (2), we have, by Lemma 1,
∞
(cid:88)∞ 1 (cid:90) tk
|Dp,q(f,x)| (cid:54) bp,q(x) |f(q2pn+kt)| d t.
n n,k B (k +1,n) (1⊕pt)n+k+1 p,q
k=0 p,q p,q
0
Hence, by (3), |Dp,q(f,x)| (cid:54) 3(cid:107)f(cid:107).
n
Therefore
|Dp,q(f,x)−f(x)|
n
(cid:54) |Dˇp,q(f −g,x)−(f −g)(x)|+|Dˇp,q(g,x)−g(x)|+(cid:12)(cid:12)(cid:12)f (cid:18)[n]p,qx+pn−2 q(cid:19)−f(x)(cid:12)(cid:12)(cid:12)
n n (cid:12) [n−1] (cid:12)
p,q
(cid:40) (cid:41)
(cid:18)x ([n] −[n−1] )+pn−2q(cid:19)2
(cid:54) 4 (cid:107)f −g(cid:107)+ µp,q(x)+ p,q p,q (cid:107)g(cid:48)(cid:48)(cid:107)
n,2 [n−1]
p,q
(cid:18) |x ([n] −[n−1] )+pn−2q|(cid:19)
p,q p,q
+ω f, .
[n−1]
p,q
Lastly, taking infimum over all g ∈ C2[0,∞), and using the inequality K (f,δ) (cid:54)
√ B 2
Cω (f, δ), δ > 0 due to [6], we get the desired assertion. (cid:3)
2
Example 1. We show comparisons and some illustrative graphs for the convergence of
(p,q)-analogue of Baskakov-Beta operators Dp,q(f,x) for different values of the parameters
n
p and q, such that 0 < q < p (cid:54) 1.
For x ∈ [0,∞), p = 0.9 and q = 0.8, the convergence of the operators Dp,q(f,x) to the
n
function f, where f(x) = 18x2 −12x+2015, for different values of n is illustrated using
MATLAB.
Figure 1. D0.9,0.8(f,x) for x ∈ [0,∞), when f(x) = 18x2−12x+2015.
n
Example 2. For x ∈ [0,∞), p = 0.9 and q = 0.75, the convergence of the operators
Dp,q(f,x) to the function f, where f(x) = 25x2 − 2x + 7, for different values of n is
n
illustrated using MATLAB.
Figure 2. D0.9,0.75(f,x) for x ∈ [0,∞), when f(x) = 25x2−2x+7.
n
Let B [0,∞) be the space of all real valued functions on [0,∞) satisfying the condition
σ
|f(x)| (cid:54) C σ(x), where C > 0 and σ(x) is a weight function. Let C [0,∞) be the
f f σ
space of all continuous functions in B [0,∞) with the norm (cid:107)f(cid:107) = sup |f(x)| and
σ σ σ(x)
x∈[0,∞)
(cid:110) (cid:111)
C0[0,∞) = f ∈ C [0,∞) : lim |f(x)| < ∞ . Weconsiderσ(x) = (1+x2)inthefollowing
σ σ σ(x)
x→∞
two results. Also, we denote the modulus of continuity on f on the closed interval [0,κ],
κ > 0 by
ω (f,δ) = sup|t−x| (cid:54) δ sup |f(t)−f(x)|.
κ
x,t∈[0,κ]
It is easy to see that for f ∈ C [0,∞), the modulus of continuity ω (f,δ) tends to zero.
σ κ
Following is a theorem on the rate of convergence for the operators Dp,q(f,x).
n
Theorem 2. Let f ∈ C [0,∞), 0 < q < p (cid:54) 1 and ω (f,δ) be its modulus of continuity
σ κ+1
on the finite interval [0,κ+1] ⊂ [0,∞), where κ > 0. Then, for every n > 2,
(cid:16) (cid:113) (cid:17)
||Dp,q(f)−f|| (cid:54) L µp,q(x)+ω +2ω f, L µp,q(x) ,
n C[0,κ] n,2 κ+1 n,2
where L = 6 C (1+κ2)(1+κ+κ2).
f
Proof. For x ∈ [0,κ] and t > κ+1, since t−x > 1, therefore we have
|f(t)−f(x)| (cid:54) C (2+x2 +t2) (cid:54) (2+3x2 +2(t−x)2) (cid:54) 6 C (1+κ2)(t−x)2. (4)
f f
And for x ∈ [0,κ] and t (cid:54) κ+1, we have
(cid:18) (cid:19)
|t−x|
|f(t)−f(x)| (cid:54) ω (f,|t−x|) (cid:54) 1+ ω (f,δ), (5)
κ+1 κ+1
δ
with δ > 0. Using above two equations (4) and (5),
(cid:18) (cid:19)
|t−x|
|f(t)−f(x)| (cid:54) 6 C (1+κ2)(t−x)2 + 1+ ω (f,δ),
f κ+1
δ
for t (cid:62) 0.
Now,
|Dp,q(f,x)−f(x)| (cid:54) Dp,q(|f(t)−f(x)|,x)
n n
(cid:18) 1 (cid:19)1/2
(cid:54) 6 C (1+κ2)Dp,q((t−x)2,x)+ω (f,δ) 1+ Dp,q((t−x)2,x)
f n κ+1 δ n
Using Remark 2 and Schwarz’s inequality, for every 0 < q < p (cid:54) 1 and x ∈ [0,κ], we have
(cid:18) (cid:19)
1 (cid:113)
|Dp,q(f,x)−f(x)| (cid:54) 6 C (1+κ2) µp,q(x)+ω (f,δ) 1+ µp,q(x)
n f n,2 κ+1 δ n,2
(cid:18) (cid:19)
1(cid:113)
(cid:54) L µp,q(x)+ω (f,δ) 1+ µp,q(x) .
n,2 κ+1 δ n,2
(cid:113)
Taking δ = L µp,q(x), the conclusion holds. (cid:3)
n,2
Following is a direct estimate in weighted approximation.
Theorem 3. Let p = p , q = q satisfying 0 < q < p (cid:54) 1 and p → 1, q → 1, pn → a,
n n n n n n n
qn → b as n → ∞. Then, for f ∈ C0[0,∞), we have
n σ
lim (cid:107)Dp,q(f)−f(cid:107) = 0.
n σ
n→∞
Proof. Due to the well-known Bohman-Korovkin theorem in [7], it is sufficient to verify
the following equation holds:
(cid:0) (cid:1)
lim (cid:107)Dp,q tk,x −xk(cid:107) = 0, for k = 0, 1, 2.
n σ
n→∞
By Lemma 1, the result immediately follows for k = 0.
Again, by Lemma 1, we have:
(cid:12) (cid:12)
(cid:107)Dp,q(t,x)−x(cid:107) = sup (cid:12)(cid:12)[n]p,qx+pn−2 q −x(cid:12)(cid:12) 1
n σ (cid:12) [n−1] (cid:12) (1+x2)
x∈[0,∞) p,q
(cid:12)(cid:12)(cid:26)x([n]p,q −[n−1]p,q)+pn−2q(cid:27)(cid:12)(cid:12) 1
= sup (cid:12) (cid:12)
(cid:12) [n−1] (cid:12) (1+x2)
x∈[0,∞) p,q