Bernstein operators of second kind and blending systems

Daniela Inoan$^\ast $, Fadel Nasaireh$^\ast $ Ioan Raşa$^\ast $

October 27, 2016.

$^\ast $Technical University of Cluj-Napoca, Department of Mathematics, Memorandumului str. no. 28, Cluj-Napoca, Romania, e-mail: {daniela.inoan, Fadel.Nasaireh,
Ioan.Rasa}@math.utcluj.ro.

We consider the fundamental polynomials associated with the Bernstein operators of second kind. They form a blending system for which we study some shape preserving properties. Modified operators are introduced; they have better interpolation properties. The corresponding blending system is also studied.

MSC. 47A58

Keywords. Blending system, total positivity, shape preserving properties

1 Introduction

A system $(f_0, f_1, \dots , f_m)$ of nonnegative, continuous functions on an interval $[a,b]$ is said to be a blending system if

\[ \sum _{i=1}^m f_i(x)=1, \ \textrm{for all} \ x \in [a,b]. \]

If $P_0, P_1, \dots , P_m \in \mathbb {R}^s$ are some given points, using a blending system one can define the curve

\[ \gamma (t)=\sum _{i=1}^m f_i(t)P_i, \quad t\in [a,b]. \]

The points $P_0, P_1, \dots , P_m$ are called control points of the curve $\gamma $ with respect to the blending system $(f_0, f_1, \dots , f_m)$.

Blending systems are important instruments in Computer Aided Geometric Design (see [ 1 ] , [ 2 ] and the references therein). Of particular interest are the blending systems for which some shape properties of the curve $\gamma $ are inherited from the properties of the control polygon $P_0P_1\dots P_m $. Bernstein and B-spline bases are well-known systems of totally positive blending functions that preserve monotonicity and convexity. In [ 1 ] , [ 3 ] were given general results that connect shape preservation with the total positivity of the system of functions.

We will study some properties of two particular blending systems derived from an operator introduced by P. Soardi in [ 9 ] .

Let $n \geq 1$, $n \in \mathbb {N} $ and $m=[n/2]$. The positive linear Soardi operator $\beta _n $ is defined for a function $f \in C[0,1] $ and $x \in [0,1]$ by

\[ \beta _nf(x)= \sum _{k=0}^m f\left( \tfrac {n-2m+2k}{n}\right) w_{n,k}(x), \]

where $w_{n,k}$ are the fundamental polynomials

\begin{align*} & w_{n,k}(x)=\\ & =\tfrac {n+1-2m+2k}{(n+1)2^{n+1}x}\tbinom {n+1}{m-k}\Big((1-x)^{m-k}(1+x)^{n+1-m+k}\! -\! (1-x)^{n+1-m+k}(1+x)^{m-k} \Big). \end{align*}

Monotonicity or convexity preserving properties and a Voronovskaja-type formula for this operator can be found in [ 8 ] . In [ 7 ] there were given some inequalities for generalized convex functions that involve the operator $\beta _n$. A recursive de Casteljau type algorithm for this operator was described in [ 5 ] .

Theorem 1

The system $(w_{n,0}(x), w_{n,1}(x), \dots , w_{n,m}(x))$, $x \in [0,1]$ is a blending system.

Proof â–¼

It follows directly from the fact that $\beta _n1=1$, which is proved in [ 9 ] .

2 Total positivity

We recall (see [ 6 ] , [ 4 ] for instance) that a system of functions $(u_0, \dots ,u_m)$ defined on an interval $I$ is totally positive (TP) if for any $t_0, \dots , t_p \in I$ with $t_0 {\lt}t_1 {\lt} \dots {\lt} t_p$ the corresponding collocation matrix

\begin{equation} \label{eq:total-posit} M\left( \begin{array}{c} u_0, \dots , u_m \\ t_0, \dots , t_p \end{array} \right)= \left( u_j(t_i)\right)_{i=0,\dots ,p; j=0,\dots ,m} \end{equation}

has only nonnegative minors.

The system is totally positive of order $r$ ($TP_r$), $1\leq r \leq m+1$ if for any collocation matrix (1), all the $k \times k$ minors, $k \in \{ 1, \dots , r\} $ are nonnegative.

A system of functions $(u_0, \dots ,u_m)$ is a Chebyshev system if all its square collocation matrices $M\left( \begin{array}{c} u_0, \dots , u_m \\ t_0, \dots , t_m \end{array} \right)$ have positive determinant. If the determinant is nonnegative, the system is weak Chebyshev.

Lemma 2

Let $0{\lt}a_0{\lt}a_1 {\lt} \dots {\lt} a_m$, $m \in \mathbb {N}$. The system

\[ (\sinh a_0t, \sinh a_1t, \dots , \sinh a_mt) \]

is totally positive on $[0,\infty )$.

Proof â–¼

Let $0\leq t_0{\lt}t_1 {\lt} \dots {\lt}t_m$. We study the minors of the corresponding collocation matrix.

For $s \in \{ 0, \dots ,m\} $ denote

\[ M(s,t)=\sinh a_st=\sum _{i=0}^\infty \tfrac {a_s^{2i+1}t^{2i+1}}{(2i+1)!},\quad K(s,i)=\tfrac {a_s^{2i+1}}{(2i+1)!}, \quad E(i,t)=t^{2i+1}. \]

Let $\{ s_1, \dots ,s_k\} \subset \{ 0,\dots ,m\} $ be row indices and $\{ s_1^\prime , \dots ,s_k^\prime \} \subset \{ 0,\dots ,m\} $ column indices in a minor of the collocation matrix. Then, using the basic composition formula (see for instance [ 6 ] ) we have:

\begin{align*} & \left| \begin{array}{ccc} \displaystyle \sinh a_{s_1}t_{s_1^\prime } & \dots & \sinh a_{s_k}t_{s_1^\prime } \\ \dots & \dots & \dots \\ \sinh a_{s_1}t_{s_k^\prime } & \dots & \sinh a_{s_k}t_{s_k^\prime } \end{array} \right| =\\ & = M \left( \begin{array}{c} s_1, \dots ,s_k \\ t_{s_1^\prime }, \dots , t_{s_k^\prime } \end{array} \right) \\ & = \sum _{0 \leq i_1 {\lt} \dots {\lt} i_k {\lt} \infty } K \left( \begin{array}{c} s_1, \dots ,s_k \\ i_1, \dots , i_k \end{array} \right) E \left( \begin{array}{c} i_1, \dots , i_k \\ t_{s_1^\prime }, \dots , t_{s_k^\prime } \end{array} \right)\\ & = \sum _{0 \leq i_1 {\lt} \dots {\lt} i_k {\lt} \infty } \! \! \! \tfrac {1}{(2i_1+1)!\dots (2i_k+1)!} \left| \begin{array}{ccc} \displaystyle a_{s_1}^{2i_1+1} & \dots & a_{s_k}^{2i_1+1} \\ \dots & \dots & \dots \\ a_{s_1}^{2i_k+1} & \dots & a_{s_k}^{2i_k+1} \end{array} \right| \cdot \left| \begin{array}{ccc} \displaystyle t_{s_1^\prime }^{2i_1+1} & \dots & t_{s_k^\prime }^{2i_1+1} \\ \dots & \dots & \dots \\ t_{s_1^\prime }^{2i_k+1} & \dots & t_{s_k^\prime }^{2i_k+1} \end{array} \right|\\ & \geq 0. \end{align*}

The next property was mentioned in [ 8 ] ; we give here a more detailed proof.

Theorem 3

The system $(w_{n,0}, \dots , w_{n,m})$ is totally positive on $[0,1]$.

Proof â–¼

We have

\begin{equation*} \begin{split} & w_{n,k}(x)=\\ & =\tfrac {n+1-2m+2k}{(n+1)2^{n+1}}\tbinom {n+1}{m-k} \tfrac {1}{x}(1-x^2)^{\frac{n+1}{2}} \left[ \left( \tfrac {1+x}{1-x} \right)^{\frac{n+1}{2}-m+k} - \left( \tfrac {1+x}{1-x} \right)^{-\frac{n+1}{2}+m-k} \right] \end{split}\end{equation*}

so with the exception of a strictly positive factor, the system is

\[ \left( ( e^t )^{\frac{n+1}{2}-m} - (e^t )^{-\frac{n+1}{2}+m}, \dots , ( e^t )^{\frac{n+1}{2}} - ( e^t)^{-\frac{n+1}{2}} \right) \]

where we used the notation $\frac{1+x}{1-x}=e^t$. We get $t=\log \frac{1+x}{1-x}$, an increasing function on $[0,\infty )$ and the system

\[ ({\mathcal{S}}) \quad (\sinh a_0t, \sinh a_1t, \dots , \sinh a_mt), \]

with $a_j=\frac{n+1}{2}-m+j$.

From Lemma 2 we have that the system $({\mathcal{S}})$ is totally positive and according to [ 4 ] , page 161, this implies the total positivity of $(w_{n,0}, \dots , w_{n,m})$.

Theorem 4

For each $j=0,1,\dots , m$,

\[ ({S_j}) \quad (w_{n,j}(x), w_{n,j+1}(x), \dots , w_{n,m}(x)) \]

is a Chebyshev system on $[0,1]$.

Proof â–¼

Let $x \in (0,1)$. As in the previous proof, we see that instead of $(S_j)$ is enough to study the system $(\sinh a_jt, \sinh a_{j+1}t, \dots , \sinh a_mt)$, with $0{\lt}a_j{\lt} \dots {\lt} a_m$ and $t \in (0,\infty )$. For any $0{\lt}t_j{\lt}t_{j+1}{\lt} \dots {\lt} t_m$, the determinant $\Delta =\left| \begin{array}{ccc} \displaystyle \sinh a_{j}t_{j} & \dots & \sinh a_{m}t_{j} \\ \dots & \dots & \dots \\ \sinh a_{j}t_{m} & \dots & \sinh a_{m}t_{m} \end{array} \right|$ can be written like in the proof of Lemma 2 and we obtain that it is strictly positive.

Let $x\in (0,1]$ and consider $0{\lt} x_j {\lt} \dots {\lt} x_m \leq 1$. The situation differs from the previous case only if $x_m=1$. Then

\begin{equation*} \begin{split} & \left| \begin{array}{cccc} w_{n,j}(x_j) & w_{n,j+1}(x_j) & \dots & w_{n,m}(x_j) \\ \dots & \dots & \dots & \dots \\ w_{n,j}(1) & w_{n,j+1}(1) & \dots & w_{n,m}(1) \end{array} \right| = \\ & = \left| \begin{array}{cccc} w_{n,j}(x_j) & w_{n,j+1}(x_j) & \dots & w_{n,m}(x_j) \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & 1 \end{array} \right| \\ & = \left| \begin{array}{cccc} w_{n,j}(x_j) & w_{n,j+1}(x_j) & \dots & w_{n,m-1}(x_j) \\ \dots & \dots & \dots & \dots \\ w_{n,j}(x_{m-1}) & w_{n,j+1}(x_{m-1}) & \dots & w_{n,m-1}(x_{m-1}) \end{array} \right| {\gt}0, \end{split}\end{equation*}

like in the previous case.

Let $x \in [0,1)$ and consider $0 \leq x_j {\lt} \dots {\lt} x_m {\lt} 1$. The situation differs from the first case only if $x_1=0$. Then, instead of $\Delta $ we will have, with $0=t_j{\lt} t_{j+1} {\lt} \dots {\lt}t_m$,

\begin{equation*} \begin{split} & \left| \begin{array}{ccc} \tfrac {\sinh a_jt_j}{t_j} & \dots & \tfrac {\sinh a_mt_j}{t_j} \\ \sinh a_jt_{j+1} & \dots & \sinh a_mt_{j+1} \\ \dots & \dots & \dots \\ \sinh a_jt_{m} & \dots & \sinh a_mt_{m} \end{array} \right| =\! \! \sum _{0 \leq i_j {\lt} \dots {\lt} i_m {\lt} \infty } \! \! \! \tfrac {1}{(2i_j+1)!}\dots \tfrac {1}{(2i_m+1)!} \Delta _1 \cdot \Delta _2 {\gt}0, \end{split}\end{equation*}

since we have $\displaystyle \Delta _1= \left| \begin{array}{ccc} a_j^{2i_j+1} & \dots & a_m^{2i_j+1} \\ \dots & \dots & \dots \\ a_j^{2i_m+1} & \dots & a_m^{2i_m+1} \end{array} \right| {\gt}0$,
$\Delta _2=\left| \begin{array}{cccc} t_j^{2i_j} & t_{j+1}^{2i_j+1} & \dots & t_{m}^{2i_j+1} \\ \dots & \dots & \dots & \dots \\ t_j^{2i_m} & t_{j+1}^{2i_m+1} & \dots & t_{m}^{2i_m+1} \end{array} \right|=0$ for $i_j{\gt}0$ and $\Delta _2{\gt}0$ for $i_j=0$.

The case $x \in [0,1]$, with $0=x_j{\lt}x_{j+1} {\lt} \dots {\lt} x_m=1$ can be treated as a combination of the previous three cases.

3 Shape preserving properties

A blending system $(f_0, \dots ,f_m)$ is said to be monotonicity preserving [ 1 ] if $\alpha _0 \leq \alpha _1 \leq \dots \leq \alpha _m$ implies that $\sum _{i=0}^m \alpha _i f_i(t)$ is an increasing function.

The system is said to be strictly monotonicity preserving if $\alpha _0 {\lt} \alpha _1 {\lt} \dots {\lt} \alpha _m$ implies that $\sum _{i=0}^m \alpha _i f_i(t)$ is strictly increasing.

In the case of a monotonicity preserving blending system, if the projections of the control points $P_0, \dots ,P_m$ onto a line are ordered, then the projections of the points of the curve $\gamma $ on the same line are also ordered.

Lemma 5

[ 1 , Corollary 2.4 ] A totally positive blending system is monotonicity preserving.

This allows us to state, following directly from Theorem 3:

Theorem 6

Soardi’s blending system $(w_{n,0}, \dots , w_{n,m})$ is monotonicity preserving.

Lemma 7

[ 1 , Proposition 2.1 ] Let $(f_0, \dots ,f_m)$ be a blending system on $[a,b]$ and let

\[ g_i=\sum _{j=i}^m f_j, \quad i=0,\dots , m. \]

Then:

a) $(f_0, \dots ,f_m)$ is monotonicity preserving if and only if $g_1, \dots , g_m$ are increasing functions.

b) $(f_0, \dots ,f_m)$ is strictly monotonicity preserving if and only if $g_1, \dots , g_m$ are increasing functions and $\sum _{i=1}^m g_i$ is strictly increasing.

Theorem 8

Soardi’s blending system $(w_{n,0}, \dots , w_{n,m})$ is strictly monotonicity preserving.

Proof â–¼

According to Theorem 6 and Lemma 7 we have to prove only that the function $\sum _{i=1}^m g_i=\sum _{i=1}^m iw_{n,i}$ is strictly increasing.

It was proved in [ 8 , Theorem 2.1 ] , that if a function $f \in C[0,1]$ is strictly increasing then $\beta _n f$ is also strictly increasing.

Let $f \in C[0,1]$ be such that $f\left( \frac{n-2m+2k}{n} \right)=k$, for $k=0,\dots ,m$ and such that it is strictly increasing. Then also

\[ \sum _{i=1}^m iw_{n,i}=\sum _{i=1}^m f\left( \tfrac {n-2m+2i}{n} \right)w_{n,i}=\beta _n f \]

is strictly increasing.

As a consequence of the property of monotonicity preserving, the blending system $(w_{n,0},\dots ,w_{n,m})$ is also length diminishing (see [ 1 ] ), that is the length of the curve $\sum _{i=0}^m P_iw_{n,i}$ is smaller then the length of the control polygon $P_0P_1\dots P_m$.

The fact that Soardi’s blending system is totally positive and a Chebyshev system implies also some other properties: geometrically (strictly) convexity preserving (see [ 1 , Theorems 3.9 and 3.10 ] ) or geometrically (strictly) k-convexity preserving (see [ 1 , Proposition 5.7 and 5.8 ] ).

4 A modified Soardi operator

Starting from Soardi’s operator we define $\Gamma _n :C[0,1]\to C[0,1]$ by

\[ \Gamma _nf=\tfrac {e_1}{\beta _ne_1}\beta _nf+\left( 1- \tfrac {e_1}{\beta _ne_1} \right) f(0). \]

This is a positive linear operator. It easy to prove, by direct computation, that $\Gamma _ne_0=e_0$, $\Gamma _ne_1=e_1$ and $\Gamma _nf(0)=f(0)$, $\Gamma _nf(1)=f(1)$ for every $f \in C[0,1]$. To obtain a blending system we consider two situations: $n$ being even or odd.

For $n=2m$, we have

\[ \Gamma _n f(x)= \sum _{k=0}^m f\left( \tfrac {n-2m+2k}{n}\right) \gamma _{n,k}(x), \]

where

\begin{align*} \gamma _{n,0}(x)=& \tfrac {x}{\beta _ne_1(x)}\left( w_{n,0}(x)-1 \right) +1, \\ \gamma _{n,k}(x)=& \tfrac {x}{\beta _ne_1(x)} w_{n,k}(x), \quad \textrm{for} \ k=1,2, \dots ,m. \end{align*}

For $n=2m+1$, we have

\[ \Gamma _n f(x)= f(0)\theta _{n,0}(x)+\sum _{k=0}^m f\left( \tfrac {n-2m+2k}{n}\right) \theta _{n,k+1}(x), \]

where

\begin{align*} \theta _{n,0}(x)=& -\tfrac {x}{\beta _ne_1(x)}+1 , \\ \theta _{n,k+1}(x)=& \tfrac {x}{\beta _ne_1(x)} w_{n,k}(x), \quad \textrm{for} \ k=0,1, \dots ,m. \end{align*}

Theorem 9

For $n=2m$ the system $(\gamma _{n,0}(x), \gamma _{n,1}(x), \dots , \gamma _{n,m}(x))$, $x \in [0,1]$ is a blending system.

For $n=2m+1$ the system $(\theta _{n,0}(x), \theta _{n,1}(x), \dots , \theta _{n,m+1}(x))$, $x \in [0,1]$ is a blending system.

Proof â–¼

It follows directly from $\Gamma _ne_0=e_0$.

Lemma 10

For any $n \in \mathbb {N}$, the function $g:[0,1] \to \mathbb {R}$, $g(x)=\frac{x}{\beta _n e_1(x)}$ is strictly increasing.

Proof â–¼

Denote $h(x)=\beta _n e_1(x) -x \beta _n^\prime e_1(x)$. We have $ h^\prime (x)=-x \beta _n^{\prime \prime } e_1(x) \leq 0 $, since $\beta _n e_1(x)$ is a convex function (see [ 8 , Theorem 2.1 ] ). Thus $h$ is decreasing on $[0,1]$. By direct computation we get $h(1)=\frac{1}{n}{\gt}0$, which implies $h(x){\gt}0$ for any $x\in [0,1]$. Finally the conclusion follows from $g^\prime (x)=\frac{h(x)}{(\beta _n e_1(x))^2}{\gt}0$.

Theorem 11

a) For any even number $n=2m$, the blending system $(\gamma _{n,0}(x), \gamma _{n,1}(x), \dots , \gamma _{n,m}(x))$ is strictly monotonicity preserving.

b) For any odd number $n=2m+1$, the blending system

\[ (\theta _{n,0}(x), \theta _{n,1}(x), \dots , \theta _{n,m+1}(x)) \]

is strictly monotonicity preserving.

Proof â–¼

We will use Lemma 7. For $n=2m$ and $i=1, \dots , m$ we have

\[ g_i=\sum _{j=i}^m \gamma _{n,j}(x)=\tfrac {x}{\beta _n e_1(x)} \sum _{j=i}^m w_{n,j}(x) \]

By the previous lemma, the function $\frac{x}{\beta _n e_1(x)}$ is strictly increasing. Also the function $\sum _{j=i}^m w_{n,j}(x)$ is strictly increasing since $(w_{n,0}, \dots , w_{n,m})$ is monotonicity preserving. So $g_i$ is increasing and Lemma 7 gives the monotonicity preserving property of $(\gamma _{n,0}(x), \gamma _{n,1}(x), \dots , \gamma _{n,m}(x))$.

Moreover, the function

\[ \sum _{i=1}^m g_i= \tfrac {x}{\beta _n e_1(x)} \sum _{i=1}^m iw_{n,i}(x) \]

is strictly increasing, so the blending system is strictly monotonicity preserving.

The case $n=2m+1$ can be treated in the same way.

Bibliography

1: J.M. Carnicer, M. García-Esnaola, J.M. Peña, Bases with Convexity Preserving Properties, Advanced Topics in Multivariate Approximation, F.Fontanella, K. Jetter and P.J. Laurent (eds.), World Scientific Publishing 1996, pp. 17–31.
2: J.M. Carnicer, M. García-Esnaola, J.M. Peña, Convexity of rational curves and total positivity, J. Comput. Appl. Math., 71 (2) (1996), pp. 365–382. $\includegraphics[scale=0.1]{ext-link.png}$
3: M.S. Floater, Total Positivity and Convexity Preservation, J. Approx. Theory, 96 (1) (1999), pp. 46–66. $\includegraphics[scale=0.1]{ext-link.png}$
4: Tim N.T. Goodman, Total Positivity and the Shape of Curves, Total Positivity and its Applications, M. Gasca and Ch. Micchelli (eds.), Kluver Academic Publishers, 1996, pp. 157–186.
5: D. Inoan, I. Raşa, A recursive algorithm for Bernstein operators of second kind, Numer. Algor., 64 (4) (2013), pp .699–706. $\includegraphics[scale=0.1]{ext-link.png}$
6: S. Karlin, Total Positivity, Stanford University Press, California, 1968.
7: I. Raşa, Classes of convex functions associated with Bernstein operators of second kind, Math. Ineq. Appl., 9(4) (2006), pp. 599–605.
8: I. Raşa, On Soardi’s Bernstein operators of second kind, Rev. Anal. Numér. Theór. Approx., 29 (2000), no. 2 , pp. 191–199. $\includegraphics[scale=0.1]{ext-link.png}$
9: P. Soardi, Bernstein polynomials and random walks on hypergroups, Probability measures on groups $X$, Oberwolfach (1990), Plenum, New York 1991, pp. 387–393.