APPROXIMATION OF CONVEX FUNCTIONS BY CONVEX SPLINES AND CONVEXITY PRESERVING CONTINUOUS LINEAR OPERATORS

Let $f$$f$ff$f$ be a real-valued function defined on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$. By $[f\left(x_0\right),\dots ,f\left(x_i\right)]$$\left[f\left(x_0\right),\dots ,f\left(x_i\right)\right]$[f(x_(0)),dots,f(x_(i))]\left[f\left(x_{0}\right), \ldots, f\left(x_{i}\right)\right]$[f\left(x_0\right),\dots ,f\left(x_i\right)]$ we mean the $l$$l$ll$l$-th divided difference of $f$$f$ff$f$ :

where $a\le x_0<\dots <x_l\le b$$a\le x_0<\dots <x_l\le b$a <= x_(0) < dots < x_(l) <= ba \leq x_{0}<\ldots<x_{l} \leq b$a\le x_0<\dots <x_l\le b$ and ${\omega }_l(x)=(x-x_0)\dots (x-x_l),l==0,1,2,\dots$${\omega }_l(x)=\left(x-x_0\right)\dots \left(x-x_l\right),l==0,1,2,\dots$omega_(l)(x)=(x-x_(0))dots(x-x_(l)),l==0,1,2,dots\omega_{l}(x)=\left(x-x_{0}\right) \ldots\left(x-x_{l}\right), l= =0,1,2, \ldots${\omega }_l(x)=(x-x_0)\dots (x-x_l),l==0,1,2,\dots$.

The function $f$$f$ff$f$ is a convex function of order $l$$l$ll$l$ if $[f\left(x_0\right),\dots ,f\left(x_l\right)]⩾0$$\left[f\left(x_0\right),\dots ,f\left(x_l\right)\right]⩾0$[f(x_(0)),dots,f(x_(l))] >= 0\left[f\left(x_{0}\right), \ldots, f\left(x_{l}\right)\right] \geqslant 0$[f\left(x_0\right),\dots ,f\left(x_l\right)]⩾0$ for every choice of $a\le x_0<\dots <x_l\le b$$a\le x_0<\dots <x_l\le b$a <= x_(0) < dots < x_(l) <= ba \leq x_{0}<\ldots<x_{l} \leq b$a\le x_0<\dots <x_l\le b$. The set of all continuous convex functions of order $l$$l$ll$l$ on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ will be denoted by $K_l[a,b]$$K_l[a,b]$K_(l)[a,b]K_{l}[a, b]$K_l[a,b]$. When convexity problems are discussed, it is customary to assume that $l⩾2$$l⩾2$l >= 2l \geqslant 2$l⩾2$. However, it will be convenient here to allow also the values $l=1$$l=1$l=1l=1$l=1$ and $l=0$$l=0$l=0l=0$l=0$. Consequently, $K_1[a,b]$$K_1[a,b]$K_(1)[a,b]K_{1}[a, b]$K_1[a,b]$ will be the set of all continuous, non-decreasing functions on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ and $K_0[a,b]$$K_0[a,b]$K_(0)[a,b]K_{0}[a, b]$K_0[a,b]$ will be the set of all continuous, nonnegative functions on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$.

Our main result is a generalization of the following geometrically obvious result:

A continuous, non-decreasing function $f$$f$ff$f$ on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ can be approximated uniformly on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ by functions of the form

where $n⩾2$$n⩾2$n >= 2n \geqslant 2$n⩾2$ and $c_k\in (a,b),k=1,\dots ,n-1$$c_k\in (a,b),k=1,\dots ,n-1$c_(k)in(a,b),k=1,dots,n-1c_{k} \in(a, b), k=1, \ldots, n-1$c_k\in (a,b),k=1,\dots ,n-1$, and

More precisely, the points $c_k,k=1,\dots ,n-1$$c_k,k=1,\dots ,n-1$c_(k),k=1,dots,n-1c_{k}, k=1, \ldots, n-1$c_k,k=1,\dots ,n-1$, can be chosen so that ${\psi }_n(x)\le f(x)\le {\psi }_n(x)+(f(b)-f(a))/n$${\psi }_n(x)\le f(x)\le {\psi }_n(x)+(f(b)-f(a))/n$psi_(n)(x) <= f(x) <= psi_(n)(x)+(f(b)-f(a))//n\psi_{n}(x) \leq f(x) \leq \psi_{n}(x)+(f(b)-f(a)) / n${\psi }_n(x)\le f(x)\le {\psi }_n(x)+(f(b)-f(a))/n$ for every $x\in [a,b]$$x\in [a,b]$x in[a,b]x \in[a, b]$x\in [a,b]$.
theorem 1. Every $f\in K_l[a,b],l⩾2$$f\in K_l[a,b],l⩾2$f inK_(l)[a,b],l >= 2f \in K_{l}[a, b], l \geqslant 2$f\in K_l[a,b],l⩾2$, can be approximated uniformly on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ by spline functions of the form

where $C(f,m)⩾0,m⩾l,n⩾2,c_k\in (a,b),k=1,\dots ,n-1$$C(f,m)⩾0,m⩾l,n⩾2,c_k\in (a,b),k=1,\dots ,n-1$C(f,m) >= 0,m >= l,n >= 2,c_(k)in(a,b),k=1,dots,n-1C(f, m) \geqslant 0, m \geqslant l, n \geqslant 2, c_{k} \in(a, b), k=1, \ldots, n-1$C(f,m)⩾0,m⩾l,n⩾2,c_k\in (a,b),k=1,\dots ,n-1$ and

In view of the preceding remark, Theorem 1 is true also if $l=1$$l=1$l=1l=1$l=1$. The coefficient $C(f,m)$$C(f,m)$C(f,m)C(f, m)$C(f,m)$ is defined by

and

As an application of this theorem we shall mention two results which characterize convexity preserving continuous linear operators defined on the space $C[a,b]$$C[a,b]$C[a,b]C[a, b]$C[a,b]$ of continuous functions on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$, with values in $F[a,b]$$F[a,b]$F[a,b]F[a, b]$F[a,b]$, the space of bounded real-valued functions on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ endowed with the supremum norm.

The first of these results gives necessary and sufficient conditions for a continuous linear operator to transform every continuous convex function of order $l⩾2$$l⩾2$l >= 2l \geqslant 2$l⩾2$ into a convex function of order $r⩾0$$r⩾0$r >= 0r \geqslant 0$r⩾0$.
theorem 2. Let $\mathrm{\Psi }:C[a,b]\to F[a,b]$$\mathrm{\Psi }:C[a,b]\to F[a,b]$Psi:C[a,b]rarr F[a,b]\Psi: C[a, b] \rightarrow F[a, b]$\mathrm{\Psi }:C[a,b]\to F[a,b]$ be a continuous linear operator. In order that for every $f\in K_l[a,b],l⩾2$$f\in K_l[a,b],l⩾2$f inK_(l)[a,b],l >= 2f \in K_{l}[a, b], l \geqslant 2$f\in K_l[a,b],l⩾2$, we have $\mathrm{\Psi }(f,\cdot )\in K_r[a,b]$$\mathrm{\Psi }(f,\cdot )\in K_r[a,b]$Psi(f,*)inK_(r)[a,b]\Psi(f, \cdot) \in K_{r}[a, b]$\mathrm{\Psi }(f,\cdot )\in K_r[a,b]$, $r⩾0$$r⩾0$r >= 0r \geqslant 0$r⩾0$, it is necessary and sufficient that
(i) $[\mathrm{\Psi }(P,x_0),\dots ,\mathrm{\Psi }(P,x_r)]=0$$\left[\mathrm{\Psi }\left(P,x_0\right),\dots ,\mathrm{\Psi }\left(P,x_r\right)\right]=0$[Psi(P,x_(0)),dots,Psi(P,x_(r))]=0\left[\Psi\left(P, x_{0}\right), \ldots, \Psi\left(P, x_{r}\right)\right]=0$[\mathrm{\Psi }(P,x_0),\dots ,\mathrm{\Psi }(P,x_r)]=0$ for every polynomial $P$$P$PP$P$ of degree $\le l-1$$\le l-1$<= l-1\leq l-1$\le l-1$ and every set of $r+1$$r+1$r+1r+1$r+1$ points $x_0<\dots <x_p$$x_0<\dots <x_p$x_(0) < dots < x_(p)x_{0}<\ldots<x_{p}$x_0<\dots <x_p$ in $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$.
(ii) $\mathrm{\Psi }((t-c{)}_{+}^{l-1},\cdot )\in K_r[a,b]$$\mathrm{\Psi }\left((t-c{)}_{+}^{l-1},\cdot \right)\in K_r[a,b]$Psi((t-c)_(+)^(l-1),*)inK_(r)[a,b]\Psi\left((t-c)_{+}^{l-1}, \cdot\right) \in K_{r}[a, b]$\mathrm{\Psi }((t-c{)}_{+}^{l-1},\cdot )\in K_r[a,b]$ for every $c\in (a,b)$$c\in (a,b)$c in(a,b)c \in(a, b)$c\in (a,b)$.

Clearly, condition (i) means that for every polynomial $P$$P$PP$P$ of degree $\le l-1,\mathrm{\Psi }(P,\cdot )$$\le l-1,\mathrm{\Psi }(P,\cdot )$<= l-1,Psi(P,*)\leq l-1, \Psi(P, \cdot)$\le l-1,\mathrm{\Psi }(P,\cdot )$ should be identically zero if $y=0$$y=0$y=0y=0$y=0$, or a polynomial of degree $\le r-1$$\le r-1$<= r-1\leq r-1$\le r-1$ if $r⩾1$$r⩾1$r >= 1r \geqslant 1$r⩾1$.

The second result gives necessary and sufficient conditions for a continuous linear operator to transform a convex function which is in every class $K_i[a,b],i=0,1,\dots ,l(l⩾2)$$K_i[a,b],i=0,1,\dots ,l(l⩾2)$K_(i)[a,b],i=0,1,dots,l(l >= 2)K_{i}[a, b], i=0,1, \ldots, l(l \geqslant 2)$K_i[a,b],i=0,1,\dots ,l(l⩾2)$ into an element of an arbitrary closed, convex cone $M[a,b]\subseteq C[a,b]$$M[a,b]\subseteq C[a,b]$M[a,b]sube C[a,b]M[a, b] \subseteq C[a, b]$M[a,b]\subseteq C[a,b]$.

THEOREM 3. Let $\mathrm{\Psi }:C[a,b]\to F[a,b]$$\mathrm{\Psi }:C[a,b]\to F[a,b]$Psi:C[a,b]rarr F[a,b]\Psi: C[a, b] \rightarrow F[a, b]$\mathrm{\Psi }:C[a,b]\to F[a,b]$ be a continuous linear operator and let $M[a,b]$$M[a,b]$M[a,b]M[a, b]$M[a,b]$ be a cloeed convex cone in $C[a,b]$$C[a,b]$C[a,b]C[a, b]$C[a,b]$. In order that for every $f\in \bigcap _{i=0}^l{K}_i[a,b],l⩾2$$f\in \bigcap _{i=0}^l K_i[a,b],l⩾2$f innnn_(i=0)^(l)K_(i)[a,b],l >= 2f \in \bigcap_{i=0}^{l} K_{i}[a, b], l \geqslant 2$f\in \bigcap _{i=0}^l{K}_i[a,b],l⩾2$, we have $\mathrm{\Psi }(f,\cdot )\in M[a,b]$$\mathrm{\Psi }(f,\cdot )\in M[a,b]$Psi(f,*)in M[a,b]\Psi(f, \cdot) \in M[a, b]$\mathrm{\Psi }(f,\cdot )\in M[a,b]$, it is necessary and sufficient that
(i) $\mathrm{\Psi }((t-a{)}^r,\cdot )\in M[a,b]$$\mathrm{\Psi }\left((t-a{)}^r,\cdot \right)\in M[a,b]$Psi((t-a)^(r),*)in M[a,b]\Psi\left((t-a)^{r}, \cdot\right) \in M[a, b]$\mathrm{\Psi }((t-a{)}^r,\cdot )\in M[a,b]$ for every $0\le r\le l-1$$0\le r\le l-1$0 <= r <= l-10 \leq r \leq l-1$0\le r\le l-1$
(ii) $\mathrm{\Psi }\cdot ((t-c{)}_{+}^{t-1},\cdot )\in M[a,b]$$\mathrm{\Psi }\cdot \left((t-c{)}_{+}^{t-1},\cdot \right)\in M[a,b]$Psi*((t-c)_(+)^(t-1),*)in M[a,b]\Psi \cdot\left((t-c)_{+}^{t-1}, \cdot\right) \in M[a, b]$\mathrm{\Psi }\cdot ((t-c{)}_{+}^{t-1},\cdot )\in M[a,b]$ for every $c\in (a,b)$$c\in (a,b)$c in(a,b)c \in(a, b)$c\in (a,b)$.

Theorems 2 and 3 are true also if $l=1$$l=1$l=1l=1$l=1$ if one assumes that $\mathrm{\Psi }((t-c{)}_{+}^0,\cdot )$$\mathrm{\Psi }\left((t-c{)}_{+}^0,\cdot \right)$Psi((t-c)_(+)^(0),*)\Psi\left((t-c)_{+}^{0}, \cdot\right)$\mathrm{\Psi }((t-c{)}_{+}^0,\cdot )$ has a meaning since $(t-c{)}_{+}^0={\chi }_{[c,\mathrm{\infty })}(t)$$(t-c{)}_{+}^0={\chi }_{[c,\mathrm{\infty })}(t)$(t-c)_(+)^(0)=chi_([c,oo))(t)(t-c)_{+}^{0}=\chi_{[c, \infty)}(t)$(t-c{)}_{+}^0={\chi }_{[c,\mathrm{\infty })}(t)$ is not continuous on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ if $c\in (a,b)$$c\in (a,b)$c in(a,b)c \in(a, b)$c\in (a,b)$.

Problems such as these have been studied by several authors. First results of this type were obtained by T. popoviciu [1]. He has studied monotonicity preserving operators of the form

where $a\le {\xi }_1<\dots <{\xi }_n\le b$$a\le {\xi }_1<\dots <{\xi }_n\le b$a <= xi_(1) < dots < xi_(n) <= ba \leq \xi_{1}<\ldots<\xi_{n} \leq b$a\le {\xi }_1<\dots <{\xi }_n\le b$ and ${\phi }_i,i=1,\dots ,n$${\phi }_i,i=1,\dots ,n$varphi_(i),i=1,dots,n\varphi_{i}, i=1, \ldots, n${\phi }_i,i=1,\dots ,n$ are differentiable functions satisfying the condition

Under these hypotheses, in view of Theorem 2, a necessary and sufficient condition for $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi$\mathrm{\Phi }$ to transform a non-decreasing function into a non-decreasing function is that for every $c\in (a,b)$$c\in (a,b)$c in(a,b)c \in(a, b)$c\in (a,b)$ the function

be a non-decreasing function on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$. Since the functions ${\phi }_i,i=1,\dots ,n$${\phi }_i,i=1,\dots ,n$varphi_(i),i=1,dots,n\varphi_{i}, i=1, \ldots, n${\phi }_i,i=1,\dots ,n$ are differentiable functions, this will be true if

which is equivalent to Popoviciu's condition

Similar results were obtained by J. A. ROULIER [2] for operators of the form

where $K$$K$KK$K$ is a continuous function on $[a,b]\times [a,b]$$[a,b]\times [a,b]$[a,b]xx[a,b][a, b] \times[a, b]$[a,b]\times [a,b]$. He has proved that $\mathrm{\Phi }(f,\cdot )$$\mathrm{\Phi }(f,\cdot )$Phi(f,*)\Phi(f, \cdot)$\mathrm{\Phi }(f,\cdot )$ is a non-negative increasing function on $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$ for every continuous, non-negative and increasing function $f$$f$ff$f$ if and only if

is non-negative and increasing for every $c\in [a,b]$$c\in [a,b]$c in[a,b]c \in[a, b]$c\in [a,b]$. This result is clearly a simple consequence of Theorem 3.

Using a more general definition of convexity and an essentially different approach, Z. ZIEGLER [3] and S. KARLIN and W. J. STUDDEN [4, Ch. 11] have developed a theory of generalized convex functions which contains results similar to Theorems 1-3. Our proof of Theorem 1 is based on the fact, observed first by т. popoviciu [5], that Bernstein polynomial $B_m(f,\cdot )$$B_m(f,\cdot )$B_(m)(f,*)B_{m}(f, \cdot)$B_m(f,\cdot )$ of a convex function $f$$f$ff$f$ of order $l$$l$ll$l$ is also a convex function of order $l$$l$ll$l$, and on the well-known fact that $B_{m}f,x)\to f(x)$$\left.B_{m}f,x\right)\to f(x)${:B_(m)f,x)rarr f(x)\left.B_{m} f, x\right) \rightarrow f(x)$B_{m}f,x)\to f(x)$ uniformly as $m\to \mathrm{\infty }$$m\to \mathrm{\infty }$m rarr oom \rightarrow \infty$m\to \mathrm{\infty }$. These two facts. for which, apparently, there is no obvious analogy in the theory of generalized convex functions, make our proof of Theorem 1 very simple. Theorems 2 and 3 follow then easily from the Theorem 1.

We shall, for simplicity, prove Theorem 1 only for the special interval $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$; the general case then follows by simple substitutions. The proof of Theorem 1 is based on the following lemma.

Lemma 1 . Let $g$$g$gg$g$ be l-times differentiable on $[0,1],l⩾2$$[0,1],l⩾2$[0,1],l >= 2[0,1], l \geqslant 2$[0,1],l⩾2$, and let $g^{(l)}(x)⩾0$$g^{(l)}(x)⩾0$g^((l))(x) >= 0g^{(l)}(x) \geqslant 0$g^{(l)}(x)⩾0$ for $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$. Let

be the Taylor polynomial of $g$$g$gg$g$ of degree $\le l-1$$\le l-1$<= l-1\leq l-1$\le l-1$. Then for every $n\ge 2$$n\ge 2$n >= 2n \geq 2$n\ge 2$ there exist points $c_1<\dots <c_{n-1}$$c_1<\dots <c_{n-1}$c_(1) < dots < c_(n-1)c_{1}<\ldots<c_{n-1}$c_1<\dots <c_{n-1}$ in $(0,1)$$(0,1)$(0,1)(0,1)$(0,1)$ such that

where

Proof of Lemma 1. We have first

Since $g^{(l)}(x)⩾0$$g^{(l)}(x)⩾0$g^((l))(x) >= 0g^{(l)}(x) \geqslant 0$g^{(l)}(x)⩾0$ on $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$, the function $g^{(l-1)}(x)-g^{(l-1)}(0)$$g^{(l-1)}(x)-g^{(l-1)}(0)$g^((l-1))(x)-g^((l-1))(0)g^{(l-1)}(x)-g^{(l-1)}(0)$g^{(l-1)}(x)-g^{(l-1)}(0)$ is non-negative, non-decreasing and continuous on [ 0,1 ]. Hence we can find $c_1<\dots <c_{n-1}$$c_1<\dots <c_{n-1}$c_(1) < dots < c_(n-1)c_{1}<\ldots<c_{n-1}$c_1<\dots <c_{n-1}$ in ( 0,1 ) such that

and it follows that, for every $t\in [0,1]$$t\in [0,1]$t in[0,1]t \in[0,1]$t\in [0,1]$,

where

The proof of Lemma 1 is finally completed by substituting (2.2) into (2.1) and using inequality (2.3).

Proof of Theorem 1. Let $B_m(f,\cdot )$$B_m(f,\cdot )$B_(m)(f,*)B_{m}(f, \cdot)$B_m(f,\cdot )$ be the Bernstein polynomial of $f$$f$ff$f$ of degree $m⩾l$$m⩾l$m >= lm \geqslant l$m⩾l$ :

Since