# Shape Preserving Properties forq-Bernstein-Stancu Operators

Authors: Yali Wang; Yinying Zhou
Journal: Journal of Mathematics
 (2014)

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## Abstract

We investigate shape preserving forq-Bernstein-Stancu polynomials
Bnq,α(f;x) introduced by Nowak in 2009. When α=0, Bnq,α(f;x) reduces to the well-known
q-Bernstein polynomials introduced by Phillips in 1997; when q=1, Bnq,α(f;x) reduces to
Bernstein-Stancu polynomials introduced by Stancu in 1968; when q=1, α=0, we obtain classical Bernstein polynomials. We prove that basic Bnq,α(f;x) basis is a normalized
totally positive basis on [0,1] and q-Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on [0,1].


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## Body

## 1. Introduction

Letq>0. For each nonnegative integer r, we define the q-integer [r]q as
(1)[r]q≡[r]≔{(1-qr)(1-q),q≠1,r,q=1,
we then define q-factorial [r]! as
(2)[r]q!≡[r]!≔[r][r-1]⋯[1],[0]!=1,
and we next define a q-binomial coefficient as
(3)[nr]q≡[nr]≔[n][n-1]⋯[n-r+1][r]!=[n]![r]![n-r]!,
for integers n≥r≥0 and as zero otherwise. Also, we use the q-Pochhammer symbol defined as for any c∈ℂ(4)(c;q)0≔1,(c;q)n≔∏k=0n-1‍(1-cqk),(n≥1),(c;q)∞≔∏k=0∞‍(1-cqk),(0<q<1).Forf∈ℂ[0,1], q>0, α≥0, and each positive integer n, we will investigate the following q-Bernstein-Stancu operator introduced by Nowak in 2009 [1]:
(5)Bnq,α(f;x)=∑k=0n‍Bn,kq,α(x)f([k][n]),
where
(6)Bn,kq,α(x)=[nk]∏i=0k-1(x+α[i])∏s=0n-k-1(1-qsx+α[s])∏i=0n-1(1+α[i]).Note that empty product in (6) denotes 1.In this case, whenα=0, Bnq,α(f;x) reduces to the well-known q-Bernstein polynomials introduced by Phillips [2] in 1997:
(7)Bn,q(f;x)=∑k=0n[nk]xk∏i=0n-k-1(1-qix)f([k][n]).
When q=1, Bnq,α(f;x) reduces to Bernstein-Stancu polynomials introduced by Stancu [3] in 1968:
(8)Sn(f;x)=∑k=0n(nr)∏i=0k-1(x+αi)∏s=0n-k-1(1-x+sα)∏i=0n-1(1+iα)×f(kn).
When q=1 and α=0, we obtain the classical Bernstein polynomials defined by
(9)Bn(f;x)=∑k=0n‍(nr)xk(1-x)n-kf(kn).Now, we review and state some general properties ofq-Bernstein-Stancu operators.It follows directly from the definition thatq-Bernstein-Stancu operators possess the endpoint interpolation property, that is,
(10)Bnq,α(f;0)=f(0),Bnq,α(f;1)=f(1),hhhhhhhh∀q>0andalln∈ℕ,
and leave invariant linear function:
(11)Bnq,α(at+b)=ax+b,∀q>0andalln∈ℕ.They are also degree reducing on polynomials; that is, if𝒫m is a polynomial of degree m, then Bnq,α(𝒫m) is a polynomials of degree ≤(m,n).Takinga=0, b=1 in (11), we conclude that
(12)∑k=0n‍Bn,kq,α(x)=1,∀n∈ℕ.In 2009, Nowak proved that theq-Bernstein-Stancu operators can be expressed in terms of q-differences [1]:
(13)Bnq,α(f;x)=∑k=0n[nk]△qkf0∏s=0k-1x+α[s]1+α[s],
where
(14)△qkf0=[k]![n]kqk(k-1)/2f[0;1[n];⋯;[k][n]].At the same time, he still showed that, for0<q<1, α≥0,
(15)Bnq,α(1;x)=1,Bnq,α(t;x)=x,Bnq,α(t2;x)=11+α(x(x+α)+x(1-x)[n]).For a real-valued functionf on an interval I, we define S-(f) to be the number of sign changes of f; that is,
(16)S-(f)=sup⁡S-(f(x0),…,f(xm)),
where the supremum is taken over all increasing sequence (x0,…,xm) in I, for all m. We say that Ln is variation-diminishing if
(17)S-(Ln(f))≤S-(f).
Similarly, for a matrix T, we say T is variation-diminishing if, for any vector V for which TV is defined, then S-1(TV)≤S-1(V).Let(Ln)n≥1 be a sequence of positive linear operators on ℂ[0,1]. We say that Ln is monotonicity preserving if Ln(f) is increasing (decreasing) for an increasing (decreasing) function f on [0,1]. We say that Ln is convexity preserving if Ln(f) is convex (concave) for a convex (concave) function f on [0,1].Letq∈(0,1), x∈[0,1] and let Bnq,α=(Bn,0q,α(x),Bn,1q,α(x),…,Bn,nq,α(x)) be the sequence of basic q-Bernstein-Stancu polynomials, and denote by Πn the sequence of all polynomials of degree at most n; then Bnq,α is a basis for Πn (see [1]). Hence, there exists a nonsingular transformation matrix Sn,q1,α1;q2,α2 from Bnq1,α1 to Bnq2,α2 such that
(18)Bn,0q2,α2(x)⋮Bn,nq2,α2(x)=Sn,q1,α1;q2,α2(Bn,0q1,α1⋮Bn,nq1,α1).A matrix is said to be totally positivity (TP) if all its minors are nonnegative. It is well known that totally positivity matrix is various-diminishing. We say that a sequenceϕ(x)=(ϕ0(x),…,ϕn(x)) of real-value function is TP on an interval I if, for any points x0<x1⋯<xn in I, the collocation matrix (ϕj(xi))i,j=0n is TP on I. If ϕ is TP on I and ∑i=0n‍ϕi(x)=1, x∈I, (so that its collocation matrix is stochastic), we say that ϕ is normalized totally positive system on I.Theorem 1.
Forα>0, q∈(0,1), q-Bernstein-Stancu basis Bnq,α is a normalized totally positivity basis on [0,1].Theorem 2.
Forα>0, q∈(0,1), q-Bernstein-Stancu operators Bnq,α(f;x) are variation-diminishing, monotonicity preserving, and convexity preserving.


## 2. Proof of Theorems1 and 2

Lemma 3 (see [4]).
A finite matrix is totally positive if and only if it is a product of 1-banded matrices with nonnegative elements, where a matrixA=(ai,j) is called 1-banded matrix if, for some l, ai,j≠0, implies l≤j-i≤l+1.Lemma 4 (see [5]).
Letϕ=(ϕ0(x),…,ϕn(x)) and ψ=(ψ0(x),…,ψn(x)) be the base of Πn and let S be the transformation matrix from ψ to ϕ; that is,
(19)ϕ0(x)⋮ϕn(x)=S(ψ0(x)⋮ψn(x)).
If S is a totally positive matrix and ψ is a totally positive system on [0,1], so is ϕ.Lemma 5 (see [6]).
If the sequenceϕ=(ϕ0(x),…,ϕn(x)) is totally positivity on [0,1], then, for any numbers a0,…,an,
(20)S-1(a0ϕ0(x)+⋯+anϕn(x))≤S-1(a0,…,an).Proof of Theorem1.
We recall that theq-Bernstein-Stancu operators Bnq,α:ℂ[0,1]→𝒫 are defined by
(21)Bnq,α(x)(f;x)=∑k=0n‍f([k][n])Bn,kq,α,
where
(22)Bn,kq,α(x)=[nk]∏j=0k-1(x+α[j])∏j=0n-k-1(1-qjx+α[j])∏j=0n-1(1+α[j]).
Thus
(23)Bn,kq,α(x)=[nk]∏j=0k-1(x+α[j])∏j=0n-k-1(1-qjx+α[j])∏j=0n-1(1+α[j])=[nk]∏j=0n-k-1(1+α[j])∏j=0n-1(1+α[j])×∏j=0k-1(x+α[j])∏j=0n-k-1(1-sjx)=[nk]∏j=n-kn-1(1+α[j])-1∏j=0k-1(x+rj)∏j=0n-k-1(1-sjx)=[nk]∏j=n-kn-1(1+α[j])-1pn,kq,α,
where rj=α[j], sj=(qj/(1+α[j])), and
(24)pn,kq,α=∏j=0k-1‍(x+rj)∏j=0n-k-1‍(1-sjx).
Clearly, from the definition we know that, for arbitrary positive numbersa0,…,an, if the sequence (ϕ0(x),…,ϕn(x)) is totally positive on [0,1], then so is the sequence (a0ϕ0,…,anϕn). We want to prove that Bnq,α=(Bn,0q,α(x),Bn,1q,α(x),…,Bn,nq,α(x)) on [0,1] is totally positive system, provided to prove that Pnq,α=(Pn,0q,α(x),…,Pn,nq,α(x)) is totally positivity system on [0,1]; we use Heping Wang’s methods (see [5]) to prove that Pnq,α=(Pn,0q,α(x),…,Pn,nq,α(x)) is totally positivity system on [0,1]. For 0≤i, k≤n and fixed q∈(0,1), we define
(25)iRkn(x)={xk∏j=0n-k-1‍(1-sjx),n-k≤i,xk(1-x)n-k-i∏j=0i-1‍(1-sjx),n-k>i,iPkn(x)={∏j=0k-1‍(x+rj)∏j=0n-k-1‍(1-sjx),k≤i,xk-i∏j=0i-1‍(x+rj)∏j=0n-k-1‍(1-sjx),k>i,
where sj, rj, j=0,…,n are given in (24). Clearly, for 0≤k≤n,
(26)nPkn(x)=Pn,kq,α(x),0Pkn=nRkn(x)=xk∏j=0n-k-1‍(1-sjx),0Rkn(x)=xk(1-x)n-k.
For 0≤i<n, it follows from the definition of iRkn(x) that, for k≥n-i,
(27)i+1Rkn(x)=iRkn(x),
and, for k<n-i,
(28)i+1Rkn(x)=xk(1-x)n-k-i-1∏j=0i‍(1-sjx)=xk(1-x)n-k-i(1-x)-1(1-sjx)∏j=0i-1‍(1-sjx)=xk(1-x)n-k-i(1-x)-1((1-x)+(1-sj)x)×∏j=0i-1‍(1-sjx)=xk(1-x)n-k-i∏j=0i-1‍(1-sjx)+xk+1(1-x)n-k-i-1(1-sj)∏j=0i-1‍(1-sjx)=iRkn(x)+(1-si)iRk+1n(x).
Similarly, from the definition of iPkn(x), we get that, for k≤i,
(29)i+1Pkn=iPkn(x),
and, for k>i,
(30)i+1Pkn(x)=xk-i-1∏j=0i‍(x+rj)∏j=0n-k-1‍(1-sjx)=xk-i-1(x+ri)∏j=0i-1‍(x+rj)∏j=0n-k-1‍(1-sjx)=xk-i-1(ri-risn-kx+(1+risn-k)x)×∏j=0i-1‍(x+rj)∏j=0n-k-1‍(1-sjx)=rixk-i-1∏j=0i-1‍(x+rj)×∏j=0n-k‍(1-sjx)+(1+risn-k)xk-i×∏j=0i-1‍(x+rj)∏j=0n-k-1‍(1-sjx)=riiPk-1n(x)+(1+risn-k)iPkn(x).
Hence, if we let(31)[i+1P0n(x)⋮i+1Pnn(x)]=S(i)[iP0n(x)⋮iPnn(x)],(32)[i+1R0n(x)⋮i+1Rnn(x)]=T(i)[iR0n(x)⋮iRnn(x)],
then
(33)S(i)=(1⋱1ri1+riSn-i-1⋱⋱ri1+riS0),T(i)=[11-Si⋱⋱11-SJ1⋱1].
From (26) and (31), we obtain
(34)Pn,0q,α⋮Pn,nq,α(x)=[Pn,0q,α(x)Pn,1q,α(x)⋮Pn,nq,α(x)]=[nP0n(x)nP1,nn(x)⋮nPn,nn(x)]=S(n-1)[n-1P0n(x)n-1P1,nn(x)⋮n-1Pn,nn(x)]=S(n-1)S(n-2)⋯S(1)S(0)[0P0n(x)0P1n(x)⋮0Pnn(x)]=S(n-1)S(n-2)⋯S(1)S(0)[nR0n(x)nR1n(x)⋮nRnn(x)]=S(n-1)S(n-2)⋯S(1)S(0)T(n-1)[(n-1)R0n(x)(n-1)R1n(x)⋮(n-1)Rnn(x)]=S(n-1)S(n-2)⋯S(1)S(0)T(n-1)T(n-2)[(n-2)R0n(x)(n-2)R1n(x)⋮(n-2)Rnn(x)]=S(n-1)S(n-2)⋯S(1)S(0)T(n-1)T(n-2)⋯T(1)T(0)×[0R0n(x)0R1n(x)⋮0Rnn(x)]=S(n-1)S(n-2)⋯S(1)S(0)T(n-1)T(n-2)⋯T(1)T(0)×0R0n(x)⋮0R0n(x).
Obviously,S(i), T(i), i=0,1,…,n-1 are 1-banded matrixes with nonnegative elements. Since the sequence of functions,
(35)((1-x)n,x(1-x)n-1,x2(1-x)n-2,…,xn-1(1-x),xn),
is totally positive on [0,1], by (26), (34) and Lemmas 3 and 4, we obtain that Pnq,α is a totally positive system on [0,1]. The proof of Theorem 1 is complete.Proof of Theorem2.
The proof of Theorem2 follows from Theorem 1. From Theorem 1, we know that q-Bernstein-Stancu basis
(36)Bnq,α=(Bn,0q,α(x),…,Bn,nq,α)
is totally positive for x∈[0,1]. By Lemma 5 we obtain that
(37)S-1(Bnq,α(f;x))=S-1(∑r=0n‍f([k]q[n]q)Bn,kq,α(x))≤S-1(f([0]q[n]q),f([1]q[n]q),…,f([n]q[n]q))≤S-1(f(x)),
which means that the q-Bernstein-Stancu operators Bnq,α are variation-diminishing. Since q-Bernstein-Stancu polynomials reproduce linear functions, we get for any function f and any linear polynomial P,
(38)S-1(Bnq,α(f)-p)=S-1(Bnq,α(f-p))≤S-1(f-p).
A standard reasoning based on (38) and endpoint interpolation property of Bnq,α yields that Bnq,α are monotonicity preserving and convexity preserving (see [7], pp. 287-288). Theorem 2 is proved.



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