Abstract

We investigate shape preserving for -Bernstein-Stancu polynomials introduced by Nowak in 2009. When , reduces to the well-known -Bernstein polynomials introduced by Phillips in 1997; when , reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; when , , we obtain classical Bernstein polynomials. We prove that basic basis is a normalized totally positive basis on and -Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on .

1. Introduction

Let . For each nonnegative integer , we define the -integer as we then define -factorial as and we next define a -binomial coefficient as for integers and as zero otherwise. Also, we use the -Pochhammer symbol defined as for any

For , , , and each positive integer , we will investigate the following -Bernstein-Stancu operator introduced by Nowak in 2009 [1]: where

Note that empty product in (6) denotes 1.

In this case, when , reduces to the well-known -Bernstein polynomials introduced by Phillips [2] in 1997: When , reduces to Bernstein-Stancu polynomials introduced by Stancu [3] in 1968: When and , we obtain the classical Bernstein polynomials defined by

Now, we review and state some general properties of -Bernstein-Stancu operators.

It follows directly from the definition that -Bernstein-Stancu operators possess the endpoint interpolation property, that is, and leave invariant linear function:

They are also degree reducing on polynomials; that is, if is a polynomial of degree , then is a polynomials of degree ≤.

Taking , in (11), we conclude that

In 2009, Nowak proved that the -Bernstein-Stancu operators can be expressed in terms of -differences [1]: where

At the same time, he still showed that, for , ,

For a real-valued function on an interval , we define to be the number of sign changes of ; that is, where the supremum is taken over all increasing sequence in , for all . We say that is variation-diminishing if Similarly, for a matrix , we say is variation-diminishing if, for any vector for which is defined, then .

Let be a sequence of positive linear operators on . We say that is monotonicity preserving if is increasing (decreasing) for an increasing (decreasing) function on . We say that is convexity preserving if is convex (concave) for a convex (concave) function on .

Let , and let be the sequence of basic -Bernstein-Stancu polynomials, and denote by the sequence of all polynomials of degree at most ; then is a basis for (see [1]). Hence, there exists a nonsingular transformation matrix from to such that

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 sequence of real-value function is TP on an interval if, for any points in , the collocation matrix is TP on . If is TP on and , , (so that its collocation matrix is stochastic), we say that is normalized totally positive system on .

Theorem 1. For , , -Bernstein-Stancu basis is a normalized totally positivity basis on .

Theorem 2. For , , -Bernstein-Stancu operators are variation-diminishing, monotonicity preserving, and convexity preserving.

2. Proof of Theorems 1 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 matrix is called 1-banded matrix if, for some , , implies .

Lemma 4 (see [5]). Let and be the base of and let be the transformation matrix from to ; that is, If is a totally positive matrix and is a totally positive system on , so is .

Lemma 5 (see [6]). If the sequence is totally positivity on , then, for any numbers ,

Proof of Theorem 1. We recall that the -Bernstein-Stancu operators are defined by where Thus where , , and
Clearly, from the definition we know that, for arbitrary positive numbers , if the sequence is totally positive on , then so is the sequence . We want to prove that on is totally positive system, provided to prove that is totally positivity system on ; we use Heping Wang’s methods (see [5]) to prove that is totally positivity system on . For , and fixed , we define where , , are given in (24). Clearly, for , For , it follows from the definition of that, for , and, for , Similarly, from the definition of , we get that, for , and, for ,
Hence, if we let then
From (26) and (31), we obtain
Obviously, , , are 1-banded matrixes with nonnegative elements. Since the sequence of functions, is totally positive on , by (26), (34) and Lemmas 3 and 4, we obtain that is a totally positive system on . The proof of Theorem 1 is complete.