PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of springeropenLink to Publisher's site
Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 167.
Published online 2017 July 17. doi:  10.1186/s13660-017-1443-7
PMCID: PMC5514192

Representation of (pq)-Bernstein polynomials in terms of (pq)-Jacobi polynomials

Abstract

A representation of (pq)-Bernstein polynomials in terms of (pq)-Jacobi polynomials is obtained.

Keywords: (pq)-Bernstein polynoimals; (pq)-Pearson difference equation; (pq)-orthogonal solutions; (pq)-difference operator

Introduction

Classical univariate Bernstein polynomials were introduced by Bernstein in a constructive proof for the Stone-Weierstrass approximation theorem [1], and they are defined as [2]

bin(x)=(ni)xi(1x)ni,i=0,1,,n.

They form a basis of polynomials and satisfy a number of important properties as non-negativity (bin(x)0 for 0 ≤ x ≤ 1), partition of unity (i=0nbin(x)=1) or symmetry (bin(x)=bnin(1x)).

For a given real-valued defined and bounded function f on the interval [0, 1], the nth Bernstein polynomial for f is

Bn(f)(x)=k=0nbkn(x)f(kn).

Then, for each point x of continuity of f, we have Bn(f)(x) → f(x) as n → ∞. Moreover, if f is continuous on [0, 1] then Bn(f) converges uniformly to f as n → ∞. Also, for each point x of differentiability of f, we have Bn(f)(x)f(x) as n → ∞ and if f is continuously differentiable on [0, 1] then Bn(f) converges to f uniformly as n → ∞.

Bernstein polynomials have been generalized in the framework of q-calculus. More precisely, Lupaş [3] initiated the application of q-calculus in area of the approximation theory, and introduced the q-Bernstein polynomials. Later on, Philips [4] proposed and studied other q-Bernstein polynomials. In both the classical case and in its q-analogs, expansions of Bernstein polynomials have been obtained in terms of appropriate orthogonal bases [5, 6].

Mursaleen et al. [7] recently introduced first the concept of (pq)-calculus in approximation theory and studied the (pq)-analog of Bernstein operators. The approximation properties for these operators based on Korovkin’s theorem and some direct theorems were considered [8]. Also, many well-known approximation operators have been introduced using these techniques, such as Bleimann-Butzer-Hahn operators [9] and Szász-Mirakyan operators [10]. Very recently Milovanović et al. [11] considered a (pq)-analog of the beta operators and using it proposed an integral modification of the generalized Bernstein polynomials. (pq)-analogs of classical orthogonal polynomials have been characterized in [12].

The main aim of this work is to obtain a representation of (pq)-Bernstein polynomials in terms of suitable (pq)-orthogonal polynomials, where the connection coefficients are proved to satisfy a three-term recurrence relation. For this purpose, we have divided the work in two sections. First, we present the basic definitions and notations. Later, in Section 3 we obtain the main results of this work relating (pq)-Bernstein polynomials and (pq)-Jacobi orthogonal polynomials.

Basic definitions and notations

Next, we summarize the basic definitions and results which can be found in [1318] and the references therein.

The (pq)-power is defined as

((a,b);(p,q))k=j=0k1(apjbqj)with ((a,b);(p,q))0=1.
1

The (pq)-hypergeometric series is defined as

Φsr((a1p,a1q),,(arp,arq)(b1p,b1q),,(bsp,bsq)|(p,q);z)=j=0((a1p,a1q),,(arp,arq);(p,q))j((b1p,b1q),,(bsp,bsq);(p,q))jzj((p,q);(p,q))j((1)j(q/p)j(j1)2)1+sr,
2

where

((a1p,a1q),,(arp,arq);(p,q))j=s=1r((asp,asq);(p,q))j,

and rs ∈ ℤ+ and a1pa1q, …, arparqb1pb1q, …, bspbsqz ∈ ℂ.

The (pq)-difference operator is defined as (see e.g. [14])

(Dp,qf)(x)=Lpf(x)Lqf(x)(pq)x,x0,
3

where the shift operator is defined by

ah(x) = h(ax), 
4

and (𝒟p,qf)(0) = f(0), provided that f is differentiable at 0.

The (pq)-Bernstein polynomials are defined as

bin(x;p,q)=pn(1n)/2[ni]p,qpi(i1)/2xi((1,x);(p,q))ni,
5

and can be expanded in the basis {xk}k≥0 as

bin(x;p,q)=k=in(1)kiq(ki)(ki1)/2p12((i1)i+k(k2n+1))[nk]p,q[ki]p,qxk.
6

From the definition of (pq)-Bernstein polynomials it is possible to derive the basic properties of (pq)-Bernstein polynomials.

  1. Partition of unity
    i=0nbin(x;p,q)=1.
  2. End-point properties
    bin(0;p,q)={1,i=0,0,otherwise,bin(1;p,q)={1,i=n,0,otherwise.

The (pq)-Jacobi polynomials are defined by

Pn(x;α,β;p,q)=2Φ1((pn,qn),(pα+β+n+1,qα+β+n+1)(pβ+1,qβ+1)|(p,q);xqαp),
7

and they satisfy the second order (pq)-difference equation

qx(qxp)p2(Dp,q2y)(x)+(x(pα+β+2qαβq2)pβ+2qβ+pqp2(pq))Lp((Dp,qy)(x))+[n]p,q(qpn2pα+β1qαβnpq)Lpqy(x)=0.
8

The (pq)-Jacobi polynomials satisfy the three-term recurrence relation

P0(x;α,β;p,q)=1,P1(x;α,β;p,q)=xB0(α,β;p,q),Pn+1(x;α,β;p,q)=(xBn(α,β;p,q))Pn(x;α,β;p,q)Cn(α,β;p,q)Pn1(x;α,β;p,q),

where

Bn(α,β;p,q)=pn+2qα+n+1(pq)2[α+β+2n]p,q[α+β+2n+2]p,q×((pβ+qβ)qα+β+2n+1(p+q)(pα+qα)pβ+nqβ+n+(pβ+qβ)pα+β+2n+1)
9

and

Cn(α,β;p,q)=pβ+2n+3q2α+β+2n+1[n]p,q[α+n]p,q[β+n]p,q[α+β+n]p,q[α+β+2n1]p,q([α+β+2n]p,q)2[α+β+2n+1]p,q.
10

Representation of (pq)-Bernstein polynomials in terms of (pq)-Jacobi polynomials

Lemma 3.1

The (pq)-Bernstein polynomials satisfy the following first order (pq)-difference equation:

(px1)x(Dp,qbin)(x;p,q)+(p1n[n]p,qx+pi[i]p,q)bin(px;p,q)=0.
11

Proof

The result can be obtained by equating the coefficients in xj.

If we introduce the first order (pq)-difference operator

Li,n = (px − 1)xDp,q + (−p1−n[n]p,qxpi[i]p,q)ℒp
12

then

Li,nbin(x;p,q)=0.

Lemma 3.2

The (pq)-Jacobi polynomials satisfy the following structure relation:

x(px1)Dp,q(Pn(p2x;α,β;p,q))=[n]p,qpn2Pn+1(p3x;α,β;p,q)+ϖ1(n)Pn(p3x;α,β;p,q)+ϖ2(n)Pn1(p3x;α,β;p,q),
13

where

ϖ1(n)=[n]p,q((p+q)qα+npβ+n+pα+β+2n+1+qα+β+2n+1)[α+β+n+1]p,q(pq)[α+β+2n]p,q[α+β+2n+2]p,q,ϖ2(n)=qα+npβ+2n+1[n]p,q[α+n]p,q[β+n]p,q[α+β+n]p,q[α+β+n+1]p,q[α+β+2n1]p,q([α+β+2n]p,q)2[α+β+2n+1]p,q.

Proof

The result follows from (7) by equating the coefficients in xj.

Theorem 3.1

The (pq)-Bernstein polynomials defined in (5) have the following representation in terms of (pq)-Jacobi polynomials defined in (7):

bin(x;p,q)=k=0nHk(i,n;α,β;p,q)Pk(p2x;α,β;p,q),
14

where the connection coefficients Hk(inαβpq) satisfy the following three-term recurrence relation:

Hk1(i,n;α,β;p,q)Λ1(k1,i,n;α,β;p,q)+Hk(i,n;α,β;p,q)Λ2(k,i,n;α,β;p,q)+Hk+1(i,n;α,β;p,q)Λ3(k+1,i,n;α,β;p,q)=0,
15

valid for 1 ≤ k ≤ n − 1 with initial conditions

Hn+1(inαβpq) = 0, 
16

Hn(i,n;α,β;p,q)=(1)n+1q12(1n)npn(n+3)/2+k(k+1)/2[ni]p,q,
17

and

{Λ1(k,i,n;α,β;p,q)=pk2[k]p,qpn2[n]p,q,Λ2(k,i,n;α,β;p,q)=pi[i]p,qp2n[n]p,qBk(α,β;p,q)+ϖ1(k),Λ3(k,i,n;α,β;p,q)=pn2[n]p,qCk(α,β;p,q)+ϖ2(k).
18

Proof

In order to obtain the result we shall apply the so-called Navima algorithm (see e.g. [19, 20] and the references therein) for solving connection problems. If we apply the first order linear operator Li,n defined in (12) to both sides of (14) we have

0=k=0nHk(i,n;α,β;p,q)Li,nPk(p2x;α,β;p,q)=k=0nHk(i,n;α,β;p,q)((px1)xDp,q(Pk(p2x;α,β;p,q))+(p1n[n]p,qx+pi[i]p,q)Pk(p3x;α,β;p,q)).

From the three-term recurrence relation for (pq)-Jacobi polynomials it yields

(p1n[n]p,qx+pi[i]p,q)Pk(p3x;α,β;p,q)=pn2[n]p,qPk+1(p3x;α,β;p,q)+p2ni(pn+2[i]p,q+pi[n]p,qBk(α,β;p,q))Pk(p3x;α,β;p,q)pn2[n]p,qCk(α,β;p,q)Pk1(p3x;α,β;p,q).

Therefore, by using the structure relation for (pq)-Jacobi polynomials (13) we have

(px1)xDp,q(Pk(p2x;α,β;p,q))+(p1n[n]p,qx+pi[i]p,q)Pk(p3x;α,β;p,q)=Λ1(k,i,n;α,β;p,q)Pk+1(p3x;α,β;p,q)+Λ2(k,i,n;α,β;p,q)Pk(p3x;α,β;p,q)+Λ3(k,i,n;α,β;p,q)Pk1(p3x;α,β;p,q),

where Λi(kinαβpq) are given in (18).

As a consequence,

0=k=0nHk(i,n;α,β;p,q)(Λ1(k,i,n;α,β;p,q)Pk+1(p3x;α,β;p,q)+Λ2(k,i,n;α,β;p,q)Pk(p3x;α,β;p,q)+Λ3(k,i,n;α,β;p,q)Pk1(p3x;α,β;p,q)).

By using the linear independence of {Pk(p3xαβpq)} we obtain the three-term recurrence relation (15) for the connection coefficients Hk(inαβpq), where the initial conditions are obtained by equating the highest power in xk.

Conclusions

In this work we have obtained a three-term recurrence relation for the coefficients in the expansion of (pq)-Bernstein polynomials in terms of (pq)-Jacobi polynomials. For our purposes some auxiliary results both for (pq)-Bernstein polynomials and (pq)-Jacobi polynomials have been derived.

Acknowledgements

The authors thank both reviewers for their comments. This work has been partially supported by the Ministerio de Ciencia e Innovación of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015-004 and R 2016/022. The first author thanks the hospitality of Departamento de Estatística, Análise Matemática e Optimización of Universidade de Santiago de Compostela, and Departamento de Matemática Aplicada II of Universidade de Vigo during her visit.

Footnotes

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, FS, IA, MMJ, and JJN contributed to each part of this study equally and read and approved the final version of the manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Bernstein S. Démonstration du théorème de Weierstrass fondé sur le calcul des probabilities. Commun. Soc. Math. Kharkov. 1912;13:1–2.
2. Lorentz GG. Bernstein Polynomials. Toronto: University of Toronto Press; 1953.
3. Lupaş A. Seminar on Numerical and Statistical Calculus (Cluj-Napoca, 1987) Cluj-Napoca: Univ. “Babeş-Bolyai”; 1987. A q-analogue of the Bernstein operator; pp. 85–92.
4. Phillips GM. Bernstein polynomials based on the q-integers. Ann. Numer. Math. 1997;4(1-4):511–518.
5. Area I, Godoy E, Woźny P, Lewanowicz S, Ronveaux A. Formulae relating little q-Jacobi, q-Hahn and q-Bernstein polynomials: application to q-Bézier curve evaluation. Integral Transforms Spec. Funct. 2004;15(5):375–385. doi: 10.1080/10652460410001727491. [Cross Ref]
6. Ronveaux A, Zarzo A, Area I, Godoy E. Bernstein bases and Hahn-Eberlein orthogonal polynomials. Integral Transforms Spec. Funct. 1998;7(1-2):87–96. doi: 10.1080/10652469808819188. [Cross Ref]
7. Mursaleen M, Ansari KJ, Khan A. On (pq)(p,q)-analogue of Bernstein operators. Appl. Math. Comput. 2015;266:874–882.
8. Kang SM, Rafiq A, Acu A-M, Ali F, Kwun YC. Some approximation properties of (pq)(p,q)-Bernstein operators. J. Inequal. Appl. 2016;2016(169)
9. Mursaleen M, Nasiruzzaman M, Khan A, Ansari KJ. Some approximation results on Bleimann-Butzer-Hahn operators defined by (pq)(p,q)-integers. Filomat. 2016;30(3):639–648. doi: 10.2298/FIL1603639M. [Cross Ref]
10. Acar T. (pq)(p,q)-generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. 2016;39(10):2685–2695. doi: 10.1002/mma.3721. [Cross Ref]
11. Milovanović GV, Gupta V, Malik N. (pq)(p,q)-Beta functions and applications in approximation. Bol. Soc. Mat. Mexicana. 2016
12. Masjed-Jamei M, Soleyman F, Area I, Nieto JJ. On (pq)(p,q)-classical orthogonal polynomials and their characterization theorems. Adv. Differ. Equ. 2017;2017 doi: 10.1186/s13662-017-1236-9. [Cross Ref]
13. Burban IM, Klimyk AU. PQP,Q-differentiation, PQP,Q-integration, and PQP,Q-hypergeometric functions related to quantum groups. Integral Transforms Spec. Funct. 1994;2(1):15–36. doi: 10.1080/10652469408819035. [Cross Ref]
14. Chakrabarti R, Jagannathan R. A (pq)(p,q)-oscillator realization of two-parameter quantum algebras. J. Phys. A. 1991;24(13):L711–L718. doi: 10.1088/0305-4470/24/13/002. [Cross Ref]
15. Gasper G, Rahman M. Basic Hypergeometric Series. 2. Cambridge: Cambridge University Press; 2004.
16. Kac V, Cheung P. Quantum Calculus. New York: Springer; 2002.
17. Koekoek R, Lesky PA, Swarttouw RF. Hypergeometric Orthogonal Polynomials and Their q-Analogues. Berlin: Springer; 2010.
18. Sadjang, PN: On the fundamental theorem of (pq)(p,q)-calculus and some (pq)(p,q)-Taylor formulas. Technical report (2013). arXiv:1309.3934v1
19. Godoy E, Ronveaux A, Zarzo A, Area I. Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case. J. Comput. Appl. Math. 1997;84(2):257–275. doi: 10.1016/S0377-0427(97)00137-4. [Cross Ref]
20. Area I, Godoy E, Ronveaux A, Zarzo A. Inversion problems in the q-Hahn tableau. J. Symb. Comput. 1999;28(6):767–776. doi: 10.1006/jsco.1999.0338. [Cross Ref]

Articles from Springer Open Choice are provided here courtesy of Springer