(p,q)-Beta Functions and Applications in Approximation

0.25 MB·

by  Gradimir V. Milovanovic

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7 (p,q)-Beta Functions and Applications
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in Approximation
]
A
C
Gradimir V. Milovanovi´c, Vijay Gupta and Neha Malik
.
h
t
a
m Abstract. In the present paper, we consider (p,q)-analogue of theBeta
operatorsand usingit,weproposetheintegralmodification ofthegen-
[
eralizedBernsteinpolynomials.Weestimatesomedirectresultsonlocal

2 and global approximation. Also, we illustrate some graphs for the con-
v vergence of (p,q)-Bernstein-Durrmeyeroperators for different values of
7 theparameters p and q using Mathematicapackage.
0
Mathematics Subject Classification (2010). Primary 33B15; Secondary
3
41A25.
6
0 Keywords.(p,q)-Betafunction,(p,q)-Gammafunction,Bernstein poly-
. nomial, Durrmeyervariant,direct results.
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0
6
1
:
v
1. Introduction
i
X
In the last decade, the generalizations of several operators to quantum vari-
r
a ant have been introduced and their approximation behavior have been dis-
cussed [see for instance [1], [5], [8], [12], [10] etc]. The further generalization
ofquantumcalculusisthepost-quantumcalculus,denotedby(p,q)-calculus.
Recently, some researchers started working in this direction (cf. [2], [6], [13],
[16]). Some basic definitions and theorems, which are mentioned below may
be found in these papers and references therein.
pn qn
[n]p,q :=pn−1+pn−2q+pn−3q2+···+pqn−2+qn−1 = p−q ·
−
n
The (p,q)-factorial is given by [n] ! = [k] , n 1, [0] ! = 1. The
p,q p,q ≥ p,q
k=1
(p,q)-binomial coefficient satisfies Q
n [n] !
p,q
= , 0 k n.
hkip,q [n−k]p,q![k]p,q! ≤ ≤
ThispaperwassupportedbytheSerbianMinistryofEducation,ScienceandTechnological
Development(No.#OI174015).
2 G.V. Milovanovi´c,Vijay Gupta and Neha Malik
The (p,q)-power basis is defined as
(x a)n = (x a)(px qa)(p2x q2a) (pn 1x qn 1a).
⊖ p,q − − − ··· − − −
The (p,q)-derivative of the function f is defined as
f(px) f(qx)
D f(x)= − , x=0.
p,q
(p q)x 6
−
Asaspecialcasewhenp=1,the(p,q)-derivativereducestotheq-derivative.
The (p,q)-derivative fulfils the following product rules
D (f(x)g(x)) = f(px)D g(x)+g(qx)D f(x),
p,q p,q p,q
D (f(x)g(x)) = g(px)D f(x)+f(qx)D g(x).
p,q p,q p,q
Obviously D (x a)0 =0 and for n 1, we have
p,q ⊖ p,q ≥
Dp,q(x⊖a)np,q = [n]p,q(px⊖a)pn,−q1,
D (a x)n = [n] (a qx)n 1.
p,q ⊖ p,q − p,q ⊖ p,−q
Letf be anarbitraryfunctionanda R. The (p,q)-integraloff(x) on
∈
[0,a] (see [15]) is defined as
a
∞ pk pk p
f(x)d x=(q p)a f a if <1
p,q − qk+1 qk+1 q
Z0 Xk=0 (cid:18) (cid:19) (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
a
∞ qk qk q
f(x)d x=(p q)a f a if <1.
p,q − pk+1 pk+1 p
Z0 Xk=0 (cid:18) (cid:19) (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
The formula of (p,q)-integration by part is given by (cid:12) (cid:12)
b
f(px)D g(x)d x = f(b)g(b) f(a)g(a)
p,q p,q
−
Z
a
b
g(qx)D f(x)d x (1.1)
p,q p,q
−
Z
a
Very recently Gupta and Aral [7] proposed the (p,q) analogue of usual
Durrmeyer operators by considering some other form of (p,q) Beta func-
tions, which is not commutative. In the present article, we define different
(p,q)-variantof Beta function of first kind and find an identity relationwith
(p,q)-Gamma functions. It is observed that (p,q)-Beta functions may sat-
isfy the commutative property, by multiplying the appropriate factor while
choosing (p,q) Beta function. As far as the approximation is concerned, or-
der is important in post-quantum calculus. We propose a generalization of
Durrmeyer type operators and establish some direct results.
(p,q)-Beta Functions and Applications 3
2. (p,q)-Gamma and (p,q)-Beta Functions
Definition 2.1 ([14]). Let n is a nonnegative integer, we define the (p,q)-
Gamma function as
(p q)n
Γ (n+1)= ⊖ p,q =[n] !, 0<q <p.
p,q (p q)n p,q
−
Definition 2.2. Let m,n N, we define (p,q)-Beta integral as
∈
1
Bp,q(m,n)= xm−1(1⊖qx)pn,−q1dp,qx. (2.1)
Z
0
Theorem2.3. The (p,q)-Gamma and (p,q)-Beta functions fulfil thefollowing
fundamental relation
Γ (m)Γ (n)
B (m,n)=p(n 1)(2m+n 2)/2 p,q p,q , (2.2)
p,q − −
Γ (m+n)
p,q
where m,n N.
∈
Proof. For any m,n N since
∈
1
Bp,q(m,n)= xm−1(1⊖qx)pn,−q1dp,qx,
Z
0
using (1.1) for f(x) = (x/p)m−1 and g(x) = (1 x)n /[n] with the
− ⊖ p,q p,q
equality Dp,q(1⊖x)n =−[n]p,q(1⊖qx)n−1 we have
1
[m 1]
B (m,n) = − p,q xm 2(1 qx)n d x
p,q pm 1[n] − ⊖ p,q p,q
− p,q Z
0
[m 1]
= − p,q B (m 1,n+1). (2.3)
pm 1[n] p,q −
− p,q
Also we can write for positive integer n
1
B (m,n+1) = xm 1(1 qx)n d x
p,q − ⊖ p,q p,q
Z
0
1
= xm−1(1⊖qx)pn,−q1 pn−1−qnx dp,qx
Z
0 (cid:0) (cid:1)
1 1
= pn−1 xm−1(1⊖qx)pn,−q1dp,qx−qn xm(1⊖qx)pn,−q1dp,qx
Z Z
0 0
= pn 1B (m,n) qnB (m+1,n).
− p,q p,q
−
4 G.V. Milovanovi´c,Vijay Gupta and Neha Malik
Using (2.3), we have
[m]
B (m,n+1)=pn 1B (m,n) qn p,q B (m,n+1),
p,q − p,q − pm[n] p,q
p,q
which implies that
pn qn
Bp,q(m,n+1)=pn+m−1pn+m−qn+mBp,q(m,n).
−
Further, by definition of (p,q) integration
1
1
B (m,1) = xm 1d x=
p,q − p,q
[m] ·
Z p,q
0
We immediately have
B (m,n) = pn+m−2 pn−1−qn−1 B (m,n−1)
p,q pn+m−1−qn+m−1 p,q
= pn+m−2 pn−1−qn−1 pn+m−3 pn−2−qn−2 B (m,n−2)
pn+m−1−qn+m−1 pn+m−2−qn+m−2 p,q
= pn+m−2 pn−1−qn−1 pn+m−3 pn−2−qn−2 ···
pn+m−1−qn+m−1 pn+m−2−qn+m−2
p−q
×pm B (m,1)
pm+1−qm+1 p,q
= pn+m−2 pn−1−qn−1 pn+m−3 pn−2−qn−2 ···
pn+m−1−qn+m−1 pn+m−2−qn+m−2
p−q 1
×pm
pm+1−qm+1[m]
p,q
(p⊖q)n−1
= pm+(m+1)+···+(m+n−2) p,q (p−q)
(pm⊖qm)n
p,q
(p⊖q)n−1
= p(n−1)(2m+n−2)/2 p,q (p−q). (2.4)
(pm⊖qm)n
p,q
Following [14], we have (a b)n+m =(a b)n (apn bqn)m thus (2.4)
⊖ p,q ⊖ p,q ⊖ p,q
leads to
Bp,q(m,n) = p(n−1)(22m+n−2)(p(mp⊖⊖qq)mnp,−)qn1 (p−q)
p,q
= p(n−1)(22m+n−2)((pp−⊖qq))nnp,−−q11 · ((pp⊖−qq))mpm,q−−11 · (p(p⊖−q)qm)m−−11(p(mp−⊖qq)mn−)n1 (p−q)
p,q p,q
= p(n−1)(22m+n−2)((pp−⊖qq))nnp,−−q11 · ((pp⊖−qq))mpm,q−−11 · ((pp−⊖qq))mm++nn−−11
p,q
= p(n−1)(22m+n−2)Γp,q(m)Γp,q(n)·
Γ (m+n)
p,q
This completes the proof of the theorem. (cid:3)
(p,q)-Beta Functions and Applications 5
Remark 2.4. Thefollowingobservationshavebeenmadefor(p,q)Betafunc-
tions:
For m,n N, we have
• ∈
B (m,n+1)=pn 1B (m,n) qnB (m+1,n).
p,q − p,q p,q
−
The (p,q)-Beta integral defined by (2.1) is not commutative. In order
•
to make commutative, we may consider the following form
1
Bp,q(m,n)= pm(m−1)/2xm−1(1⊖qx)pn,−q1dp,qx.
Z
0
e
For this form, (p,q)-Gamma and (p,q)-Beta functions fulfill the follow-
ing fundamental relation
Γ (m)Γ (n)
B (m,n)=p(2mn+m2+n2 3m 3n+2)/2 p,q p,q , (2.5)
p,q − −
Γ (m+n)
p,q
whereem,n N.Obviouslyforform(2.5),wegetB (m,n)=B (n,m).
p,q p,q
∈
e e
3. (p,q) Bernstein-Type Operators and Moments
For n N and k 0, we have the following identity, which can be easily
∈ ≥
verified using the principle of mathematical induction:
n
n
pk(k 1)/2xk(1 x)n k =pn(n 1)/2. (3.1)
k p,q − ⊖ p,−q −
Xk=0h i
Using the above identity, we consider the (p,q)-analogueof Bernstein opera-
tors for x [0,1] and 0<q <p 1 as
∈ ≤
n pn k[k]
B (f,x)= bp,q(1,x)f − p,q , (3.2)
n,p,q n,k [n]
k=0 (cid:18) p,q (cid:19)
X
where the (p,q)-Bernstein basis is defined as
n
bp,q(1,x)= p[k(k 1) n(n 1)]/2xk(1 x)n k.
n,k k p,q − − − ⊖ p,−q
h i
Remark 3.1. Other form of the (p,q)-analogue of Bernstein polynomials has
been recently considered by Mursaleen et al. [13].
Remark 3.2. Using the identity (3.1) and the following recurrence
relation (for m 1, Up,q (x)=B (e ,x)=B (tm,x)):
≥ n,m n,p,q m n,p,q
[n] Up,q (px)=pnx(1 px)D [Up,q (x)]+[n] pxUp,q (px),
p,q n,m+1 − p,q n,m p,q n,m
the (p,q)-Bernstein polynomial satisfy
pn 1x(1 x)
B (e ,x)=1, B (e ,x)=x, B (e ,x)=x2+ − − ,
n,p,q 0 n,p,q 1 n,p,q 2
[n]
p,q
where e =ti, i=0,1,2.
i
6 G.V. Milovanovi´c,Vijay Gupta and Neha Malik
Recently, Gupta and Wang (see [9]) discussed the q-variant of certain
Bernstein-Durrmeyer type operators. We now extend these studies and pro-
pose the following(p,q)-Bernstein-Durrmeyeroperatorsbasedon(p,q)-Beta
function.
For x [0,1] and 0 < q < p 1, the (p,q)-analogue of Bernstein-
∈ ≤
Durrmeyer operators is defined as
n
Dp,q(f,x) = [n+1] p (n k+1)(n+k)/2bp,q(1,x)
n p,q − − n,k
k=1
X
1
bp,q (t)f(t)d t+bp,q(1,x)f(0), (3.3)
× n,k 1 p,q n,0
−
Z
0
where bp,q(1,x) is defined by (3.2) and
n,k
n
bpn,,qk(t)= k p,qtk(1⊖qt)pn,−qk.
h i
It maybe remarkedhere that for p=1, these operatorswill notreduce
to the q-Durrmeyer operators; but for p = q = 1, these will reduce to the
Durrmeyer operators.
Lemma3.3. Let e =tm, m N 0 , then for x [0,1] and 0<q <p 1,
m
∈ ∪{ } ∈ ≤
we have
p[n] x
Dp,q(e ,x)=1, Dp,q(e ,x)= p,q ,
n 0 n 1 [n+2]
p,q
(p+q)pn+1[n] x ([n] pn 1)p2q[n] x2
Dp,q(e ,x)= p,q + p,q− − p,q
n 2 [n+2] [n+3] [n+2] [n+3] ·
p,q p,q p,q p,q
Proof. Using (2.2) and (2.1) and Remark 3.2, we have
n
Dp,q(e ,x) = [n+1] p [(n+1 k)(n+k)/2]bp,q(1,x)
n 0 p,q − − n,k
k=1
X
1
n
× k 1 tk−1(1⊖qt)np,+q1−kdp,qt+bpn,,q0(1,x)
Z0 (cid:20) − (cid:21)p,q
n
n
= [n+1] p [(n+1 k)(n+k)/2]bp,q(1,x)
p,q − − n,k k 1
k=1 (cid:20) − (cid:21)
X
B (k,n k+2)+bp,q(1,x)
× p,q − n,0
n
[n] ![k 1] !
= [n+1]p,q p−[(n+1−k)(n+k)/2]bpn,,qk(1,x) [np+,q1 −k] p,q!
p,q
k=1 −
X
[k 1] ![n k+1] !
× p[(n+1−k)(n+k)/2] − p[,nq+1−] ! p,q +bpn,,q0(1,x)
p,q
= B (1,x)=1.
n,p,q
(p,q)-Beta Functions and Applications 7
Next, applying Remark 3.2, we have
n
Dnp,q(e1,x) = [n+1]p,q p−[(n+1−k)(n+k)/2]bpn,,qk(1,x)
k=1
X
1
n
tk(1 qt)n+1 kd t
× k 1 ⊖ p,q − p,q
Z (cid:20) − (cid:21)p,q
0
n
= [n+1] p [(n+1 k)(n+k)/2]bp,q(1,x)
p,q − − n,k
k=1
X
n
B (k+1,n k+2)
p,q
× k 1 −
(cid:20) − (cid:21)p,q
n
n
= [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x)
p,q − − n,k k 1
k=1 (cid:20) − (cid:21)p,q
X
[k] ![n k+1] !
p(n+1−k)(n+k+2)/2 p,q − p,q
× [n+2] !
p,q
n
[k]
= pn k+1bp,q(1,x) p,q
− n,k [n+2]
p,q
k=1
X
p[n] n pn k[k] p[n] x
= p,q bp,q(1,x) − p,q = p,q
[n+2] n,k [n] [n+2] ·
p,q p,q p,q
k=1
X
Further, using the identity [k+1] =pk+q[k] and by Remark 3.2,
p,q p,q
we get
n
Dp,q(e ,x) = [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x)
n 2 p,q − − n,k
k=1
X
1
n
× k 1 tk+1(1⊖qt)np,+q1−kdp,qt
Z0 (cid:20) − (cid:21)p,q
n
= [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x)
p,q − − n,k
k=1
X
n
B (k+2,n k+2)
p,q
× k 1 −
(cid:20) − (cid:21)p,q
n
n
= [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x)
p,q − − n,k k 1
k=1 (cid:20) − (cid:21)p,q
X
[k+1] ![n k+1] !
p(n+1−k)(n+k+4)/2 p,q − p,q
× [n+3] !
p,q
n
[k] [k+1]
= p2(n k+1) bp,q(1,x) p,q p,q
− n,k [n+2] [n+3]
p,q p,q
k=1
X
8 G.V. Milovanovi´c,Vijay Gupta and Neha Malik
i.e.,
Dp,q(e ,x) = n p2(n−k+1) bp,q(1,x) [k]p,q(pk+q[k]p,q)
n 2 n,k [n+2] [n+3]
Xk=1 p,q p,q
pn+2[n] n pn−k[k]
= p,q bp,q(1,x) p,q
[n+2] [n+3] n,k [n]
p,q p,q Xk=1 p,q
p2q[n]2 n pn−k[k] 2
+ p,q bp,q(1,x) p,q
[n+2] [n+3] n,k (cid:18) [n] (cid:19)
p,q p,q Xk=1 p,q
pn+2[n] x p2q[n]2 pn−1x(1−x)
= p,q + p,q x2+
[n+2] [n+3] [n+2] [n+3] (cid:18) [n] (cid:19)
p,q p,q p,q p,q p,q
pn+2[n] x p2q[n]2 x2 pn+1q[n] x(1−x)
= p,q + p,q + p,q
[n+2] [n+3] [n+2] [n+3] [n+2] [n+3]
p,q p,q p,q p,q p,q p,q
(p+q)pn+1[n] x ([n] −pn−1)p2q[n] x2
= p,q + p,q p,q ·
[n+2] [n+3] [n+2] [n+3]
p,q p,q p,q p,q
(cid:3)
Remark 3.4. Using above lemma, we can obtain the following central mo-
ments:
(p[n] [n+2] )x
1◦ Dnp,q((t−x),x)= p,[qn−+2] p,q
p,q
(p+q)pn+1[n] x
2◦ Dnp,q (t−x)2,x = [n+2] [n+3p],q
p,q p,q
(cid:0) (cid:1)
[([n] pn 1)p2q[n] 2p[n] [n+3] +[n+2] [n+3] ]x2
p,q − p,q p,q p,q p,q p,q
+ − − .
[n+2] [n+3]
p,q p,q
Lemma 3.5. Let n be a given natural number, then
6 1
Dp,q((t x)2,x) ϕ2(x)+ ,
n − ≤ [n+2] [n+2]
p,q (cid:18) p,q(cid:19)
where ϕ2(x)=x(1 x), x [0,1].
− ∈
Proof. In view of Lemma 3.3, we obtain
(p+q)pn+1[n] x
Dp,q((t x)2,x)= p,q
n − [n+2] [n+3]
p,q p,q
[([n] pn 1)p2q[n] 2p[n] [n+3] +[n+2] [n+3] ]x2
p,q − p,q p,q p,q p,q p,q
+ − − .
[n+2] [n+3]
p,q p,q
By direct computations, using the definition of the (p,q)-numbers, we
get
(p+q)pn+1[n]p,q =pn+1(p+q)(pn−1+pn−2q+pn−3q2+ +pqn−2+qn−1)>0
···
for every q (q ,1).
0
∈
Furthermore, the expression
(p+q)pn+1[n] +([n] pn 1)p2q[n] 2p[n] [n+3] +[n+2] [n+3]
p,q p,q − p,q p,q p,q p,q p,q
− −
(p,q)-Beta Functions and Applications 9
is equal to
(p+q)pn+1[n] +p2q[n]2 pn+1q[n]
p,q p,q − p,q
2p[n] (pn+2+qpn+1+q2pn+q3[n] )
p,q p,q
−
+(pn+1+qpn+q2[n] )(pn+2+qpn+1+q2pn+q3[n] ) 6.
p,q p,q
≤
In conclusion, for x [0,1], we have
∈
6
Dp,q((t x)2,x) δ2(x), (3.4)
n − ≤ [n+2] n
p,q
which was to be proved. (cid:3)
4. Local and Global Estimates
Inthissection,weestimatesomedirectresults,viz.,localandglobalapprox-
imation in terms of modulus of continuity.
Our first main result is a local theorem.
Forthis,wedenoteW2 = g C[0,1]:g C[0,1] ,forδ >0,K functional
′′
∈ ∈ −
is defined as
(cid:8) (cid:9)
K2(f,δ)=inf f g +η g′′ :g W2 ,
k − k k k ∈
where norm- . denotes the u(cid:8)niform norm on C[0,1]. F(cid:9)ollowing the well
kk
known inequality due to DeVore and Lorentz [3], there exists an absolute
constant C >0 such that
K (f,δ) Cω (f,√δ), (4.1)
2 2
≤
where the second order modulus of smoothness for f C[0,1] is defined as
∈
ω (f,√δ)= sup sup f(x+h) f(x).
2
| − |
0<h √δx,x+h [0,1]
≤ ∈
The usual modulus of continuity for f C[0,1] is defined as
∈
ω(f,δ)= sup sup f(x+h) f(x).
| − |
0<h δx,x+h [0,1]
≤ ∈
Theorem 4.1. Let n > 3 be a natural number and let 0 < q < p 1, q =
0
≤
q (n) (0,p) be defined as in Lemma 3.5. Then, there exists an absolute
0
∈
constant C >0 such that
2x
|Dnp,q(f,x)−f(x)|≤Cω2 f,[n+2]p−,1q/2δn(x) +ω f,[n+2] ,
(cid:16) (cid:17) (cid:18) p,q(cid:19)
where f C[0,1], δ2(x) =ϕ2(x)+ 1 , ϕ2(x)=x(1 x), x [0,1] and
∈ n [n+2]p,q − ∈
q (q ,1).
0
∈
Proof. For f C[0,1] we define
∈
p[n] x
Dp,q(f,x) = Dp,q(f,x)+f(x) f p,q .
n n − [n+2]
(cid:18) p,q(cid:19)
e
10 G.V. Milovanovi´c,Vijay Gupta and Neha Malik
Then, by Lemma 3.3, we immediately observe that
D˜p,q(1,x)=Dp,q(1,x)=1
n n
and
p[n] x
D˜p,q(t,x)=Dp,q(t,x)+x p,q =x.
n n − [n+2]
p,q
By applying Taylor’s formula
t
g(t)=g(x)+(t x)g (x)+ (t u)g (u)du,
′ ′′
− −
Z
x
we get
t
Dp,q(g,x) = g(x)+Dp,q (t u)g (u)du,x
n n  − ′′ 
Z
x
e e  t 
= g(x)+Dnp,q (t−u)g′′(u)du,x
Z
x
 
p[n]p,qx
[n+2]p,q
p[n] x
p,q
u g′′(u)du.
− [n+2] −
Z (cid:18) p,q (cid:19)
x
Thus, Dp,q(g,x) g(x)
| n − |
p[n]p,qx
e t [n+2]p,q
p[n] x
Dp,q t u g (u) du ,x + p,q u g (u)du
≤ n (cid:12)(cid:12)Zx | − |·| ′′ | (cid:12)(cid:12)  (cid:12)(cid:12) Zx (cid:12)(cid:12) [n+2]p,q − (cid:12)(cid:12)| ′′ | (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)  (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) p[n](cid:12) x (cid:12) 2 (cid:12) (cid:12) (cid:12)
≤ Dnp,q((t−x)2,x)kg′′k+ [n+p2,]q −x kg′′k. (4.2)
(cid:18) p,q (cid:19)
Also, we have
p[n] x 2
Dp,q((t x)2,x)+ p,q x
n − [n+2] −
(cid:18) p,q (cid:19)
2
6 1 (p[n] [n+2] )x
ϕ2(x)+ + p,q− p,q
≤ [n+2] [n+2] [n+2]
p,q (cid:18) p,q(cid:19) (cid:18) p,q (cid:19)
10 1
ϕ2(x)+ . (4.3)
≤ [n+2] [n+2]
p,q (cid:18) p,q(cid:19)
Hence, by (4.2) and with the condition n>3 and x [0,1], we have
∈
10
|Dnp,q(g,x)−g(x)|≤ [n+2] δn2(x) kg′′k. (4.4)
p,q
e