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T. Popoviciu, Sur l’approximation des fonctions convexes d’ordre supérieur (II), Bull. de la Sect. sci. de l’Acad. Roum., 20 (1938) no. 7, pp. 50-53; 192-195 (in French).
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Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie
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Romanian Society of Mathematical Sciences
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[Zbl 0021.11701,JFM 64.1025.01]
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BULLETIN
No. 7
SUMMARY
ON THE APPROXIMATION OF HIGHER ORDER CONVEX FUNCTIONS
I. - In a previous work^(1){ }^{1}), I have demonstrated that any functionint(x)\int(x)continues in a finite and closed interval (a,ba, b) is the limit of a uniformly convergent stite of polynomials that preserve all the convexity properties off(x)f(x)This property is realized by MS Bernstein polynomials.
{:(I)P_(m)(x)=(1)/((ba)^(m))sum_(i=0)^(m)((m)/(i))f(a+i(ba)/(m))(xa)^(i)(bx)^(mi).:}\begin{equation*} P_{m}(x)=\frac{1}{(ba)^{m}} \sum_{i=0}^{m}\binom{m}{i} f\left(a+i \frac{ba}{m}\right)(xa)^{i}(bx)^{mi} . \tag{I} \end{equation*}
But, in general, nothing can be said about the convexity characteristics of polynomials (II), outside the interval (a,ba, b). In the following we will demonstrate the following property:
Any continuous, non-concave function of ordernnin the meantime (a,ba, b) is the limit of a uniformly convergent series in this interval of polynomials which are convex of ordernneverywhere, so in(-oo,+oo)(-\infty,+\infty).
We assume, of course, thatf(x)f(x)does not reduce to a polynomial of degreenn.
2. - If we say that a non-negative function is non-concave of order - 1 , the previous property is also true forn=-1n=-1. Indeed, in this case there exists a continuous functiong(x)g(x)such that we havef(x)=g^(2)(x)f(x)=g^{2}(x)In(a,b)(a, b). So thenM=max|g(x)|M=\max |g(x)|Andepsi\varepsilonyour positive number.(a,b)(a, b)
According to Weierstrass's theorem for all0 < epsi^(') < min(I,(epsi)/(2M+I))0<\varepsilon^{\prime}<\min \left(\mathrm{I}, \frac{\varepsilon}{2 M+\mathrm{I}}\right)there exists a polynomialP(x)P(x)such as|gP| < epsi^(')|gP|<\varepsilon^{\prime}In(a,b)(a, b). We have
2. - If we say that a non-negative function is non-concave of order - 1 , the previous property is also true forn=-1n=-1. Indeed, in this case there exists a continuous functiong(x)g(x)such that we havef(x)=g^(2)(x)f(x)=g^{2}(x)In(a,b)(a, b). So thenM=max|g(x)|M=\max |g(x)|Andepsi\varepsilonyour positive number.(a,b)(a, b)
According to Weierstrass's theorem for all0 < epsi^(') < min(I,(epsi)/(2M+I))0<\varepsilon^{\prime}<\min \left(\mathrm{I}, \frac{\varepsilon}{2 M+\mathrm{I}}\right)there exists a polynomialP(x)P(x)such as|gP| < epsi^(')|gP|<\varepsilon^{\prime}In(a,b)(a, b). We have
|P| < M+I,|g+P| < 2M+I" in "(a,b),|P|<M+I,|g+P|<2 M+I \text { in }(a, b),
SOquad|fP^(2)|=|g^(2)-P^(2)|=|gP||g+P| < epsi^(')(2M+1) < epsi\quad\left|fP^{2}\right|=\left|g^{2}-P^{2}\right|=|gP||g+P|<\varepsilon^{\prime}(2 M+1)<\varepsilon, In(a,b)(a, b), which demonstrates the property.
We can now demonstrate the property fornnany. Let us therefore suppose thatf(x)f(x)be non-concave of ordernnIn(a,b)(a, b)and eitherepsi\varepsilonany positive number. We can determine ammsuch that we have
{:(2)|f-P_(m)| < (epsi)/(2)","" in "(a","b):}\begin{equation*} \left|f-P_{m}\right|<\frac{\varepsilon}{2}, \text { in }(a, b) \tag{2} \end{equation*}
P_(m)P_{m}being the polynomial ( I ), generally of degree> n>n. In this caseP_(m)^((n+1)) >= 0P_{m}^{(n+1)} \geqq 0In(a,b)(a, b), so there exists a polynomial that is everywhere non-negativeP(x)P(x)such that the origin has
|P_(m)^((n+1))-P| < (epsi(n+1)!)/(2(ba)^(n+1)),quad" in "(a,b)\left|P_{m}^{(n+1)}-P\right|<\frac{\varepsilon(n+1)!}{2(ba)^{n+1}}, \quad \text { in }(a, b)
The polynomial
Q(x)=int_(a)^(x)((xt)^(n))/(n!)P(t)dt+sum_(i=0)^(n)((xa)^(i))/(i!)P_(n)^((i))(a)Q(x)=\int_{a}^{x} \frac{(xt)^{n}}{n!} P(t) d t+\sum_{i=0}^{n} \frac{(xa)^{i}}{i!} P_{n}^{(i)}(a)
is convex of ordern(-oo,+oo)n(-\infty,+\infty)and we have
P_(m)(x)-Q(x)=int_(a)^(x)((xt)^(n))/(n!)[P_(m)^((n+1))(t)-P(t)]dtP_{m}(x)-Q(x)=\int_{a}^{x} \frac{(xt)^{n}}{n!}\left[P_{m}^{(n+1)}(t)-P(t)\right] dt
from where
(3)quad|P_(m)-Q| < (epsi(n+1)!)/(2(b-a)^(n+1))*((x-a)^(n+1))/((n+1)!) <= (epsi)/(2)\quad\left|P_{m}-Q\right|<\frac{\varepsilon(n+1)!}{2(b-a)^{n+1}} \cdot \frac{(x-a)^{n+1}}{(n+1)!} \leq \frac{\varepsilon}{2}, In(a,b)(a, b).
(3)quad|P_(m)-Q| < (epsi(n+1)!)/(2(b-a)^(n+1))*((x-a)^(n+1))/((n+1)!) <= (epsi)/(2)\quad\left|P_{m}-Q\right|<\frac{\varepsilon(n+1)!}{2(b-a)^{n+1}} \cdot \frac{(x-a)^{n+1}}{(n+1)!} \leq \frac{\varepsilon}{2}, In(a,b)(a, b).
From (2) and (3) we deduce that
|f-Q| <= |f-P_(m)|+|P_(m)-Q| < (epsi)/(2)+(epsi)/(2)=epsi," dans "(a,b)|f-Q| \leq\left|f-P_{m}\right|+\left|P_{m}-Q\right|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon, \text { dans }(a, b)
which demonstrates the property.
3. We will give another demonstration forn=1n=1, using MS Bernstein polynomials directly. Let us divide the interval(a,b)(a, b)inrrequal parts by points
3. We will give another demonstration forn=1n=1, using MS Bernstein polynomials directly. Let us divide the interval(a,b)(a, b)inrrequal parts by points
x_(i)=a+i(b-a)/(r),quad i=0,1,dots,rx_{i}=a+i \frac{b-a}{r}, \quad i=0,1, \ldots, r
and eitherL_(gamma)(x)L_{\gamma}(x)the function represented by the polygonal line of vertices(x_(i),f(x_(i)))\left(x_{i}, f\left(x_{i}\right)\right). We know thatL_(gamma)(x)rarr f(x)L_{\gamma}(x) \rightarrow f(x)Forr rarr oor \rightarrow \infty, uniformly in (a,ba, b). If we consider continuous functions
{:[varphi_(i)(x)={[0","," dans "(a,x_(i))],[x-x_(i)","," dans "(x_(i),b)]:}],[i=1","2","dots","r-1]:}\begin{aligned}
\varphi_{i}(x) & = \begin{cases}0, & \text { dans }\left(a, x_{i}\right) \\
x-x_{i}, & \text { dans }\left(x_{i}, b\right)\end{cases} \\
i & =1,2, \ldots, r-1
\end{aligned}
we can write^(2){ }^{2})
L_(r)(x)=f(a)+[a,x_(1);f](x-a)+sum_(i=0)^(gamma)(x_(i+2)-x_(i))[x_(i),x_(i+1),x_(i+2);f]varphi_(i+1)(x)L_{r}(x)=f(a)+\left[a, x_{1} ; f\right](x-a)+\sum_{i=0}^{\gamma}\left(x_{i+2}-x_{i}\right)\left[x_{i}, x_{i+1}, x_{i+2} ; f\right] \varphi_{i+1}(x).
We now see that it is sufficient to demonstrate the property for functionsvarphi_(i)(x)\varphi_{i}(x). Let us extend this function in the interval
L_(r)(x)=f(a)+[a,x_(1);f](x-a)+sum_(i=0)^(gamma)(x_(i+2)-x_(i))[x_(i),x_(i+1),x_(i+2);f]varphi_(i+1)(x)L_{r}(x)=f(a)+\left[a, x_{1} ; f\right](x-a)+\sum_{i=0}^{\gamma}\left(x_{i+2}-x_{i}\right)\left[x_{i}, x_{i+1}, x_{i+2} ; f\right] \varphi_{i+1}(x).
We now see that it is sufficient to demonstrate the property for functionsvarphi_(i)(x)\varphi_{i}(x). Let us extend this function in the interval
(x_(i)-b+a,x_(i)+b-a)\left(x_{i}-b+a, x_{i}+b-a\right)
by
varphi_(i)(x)={[0","," dans "(x_(i)-b+a,x_(i))],[x rarrx_(i)","," dans "(x_(i),x_(i)+b-a)]:}\varphi_{i}(x)= \begin{cases}0, & \text { dans }\left(x_{i}-b+a, x_{i}\right) \\ x \rightarrow x_{i}, & \text { dans }\left(x_{i}, x_{i}+b-a\right)\end{cases}
and eitherP_(m,i)(x)P_{m, i}(x)the MS Bernstein polynomial of degreemmof this function in the interval(x_(i)-b+a,x_(i)+b-a)\left(x_{i}-b+a, x_{i}+b-a\right). We verify, by a direct calculation, thatP_(m,i)^('')(x) >= 0P_{m, i}^{\prime \prime}(x) \geq 0partnership, thereforeP_(m,i)P_{m, i}is convex (of orderII) In(-oo,-oo)(-\infty,-\infty)ifmmis even. We now see that ifepsi\varepsilonis any positive number, if we determine the positive integersgamma\gammaAndssso that
{:[|f-L_(y)| < (epsi)/(2)","dans(a","b)],[|varphi_(i)-P_(2s,i)| < (epsi)/(2(r-1)(x_(i+2)+x_(i))[x_(i),x_(i+1),x_(i+2);f])]:}\begin{gathered}
\left|f-L_{y}\right|<\frac{\varepsilon}{2}, \operatorname{dans}(a, b) \\
\left|\varphi_{i}-P_{2 s, i}\right|<\frac{\varepsilon}{2(r-1)\left(x_{i+2}+x_{i}\right)\left[x_{i}, x_{i+1}, x_{i+2} ; f\right]}
\end{gathered}
dan
{:(x_(i)-b+a,x_(i)+b-a)quad,quad i=1,2dots,r-1^(3)),\left.\left(x_{i}-b+a, x_{i}+b-a\right) \quad, \quad i=1,2 \ldots, r-1{ }^{3}\right),
and if we ask
Q(x)=f(a)+[a,x_(1);f](x-a)+sum_(i=0)^(r-2)(x_(i+2)-x_(i))[x_(i),x_(i+1),x_(i+2);f]P_(2s,i+1)(x)Q(x)=f(a)+\left[a, x_{1} ; f\right](x-a)+\sum_{i=0}^{r-2}\left(x_{i+2}-x_{i}\right)\left[x_{i}, x_{i+1}, x_{i+2} ; f\right] P_{2 s, i+1}(x).
the polynomialQ(x)Q(x)is everywhere convex (of order I) and we have
Q(x)=f(a)+[a,x_(1);f](x-a)+sum_(i=0)^(r-2)(x_(i+2)-x_(i))[x_(i),x_(i+1),x_(i+2);f]P_(2s,i+1)(x)Q(x)=f(a)+\left[a, x_{1} ; f\right](x-a)+\sum_{i=0}^{r-2}\left(x_{i+2}-x_{i}\right)\left[x_{i}, x_{i+1}, x_{i+2} ; f\right] P_{2 s, i+1}(x).
the polynomialQ(x)Q(x)is everywhere convex (of order I) and we have
|f-Q| < epsi," dans "(a,b)|f-Q|<\varepsilon, \text { dans }(a, b)
-
- Note that formmeven polynomialsP_(m,i)P_{m, i}are increasing in the interval(x_(i)-b+a,+oo)\left(x_{i}-b+a,+\infty\right). So we also have the following property:
Any continuous, non-decreasing, non-concave (order I) function in the interval (a,ba, b) is the limit of a uniformly convergent sequence of polynomials which are increasing in (a,+ooa,+\infty) and convex (of order 1) in(-oo,+oo)(-\infty,+\infty).
It is clear that we can find approximation polynomials such that they are increasing in the interval (c,+ooc,+\infty),ccbeing a number<= a\leq abut, it is obvious, that we cannot takec=-ooc=-\infty. A similar property holds for continuous non-increasing and non-concave functions. In this case the polynomials are decreasing in
(-oo,d)", où "d >= b". "(-\infty, d) \text {, où } d \geq b \text {. }ù
We can also obtain sequences of polynomials of indefinite approximation preserving several convexity properties at the same time, everywhere or in intervals(c+oo),(-oo,d)(c+\infty),(-\infty, d), but we do not dwell on these questions further here.
Cernăuti, September 27, 1938.
- ^(1){ }^{1}) See: Tiberiu Popoviciu, On the approximation of convex functions of higher order, «Mathematica», vol. X (1935), pp. 49-54.
- ^(2)){ }^{2)}For notations, see my previous work.
^(3)){ }^{3)}If[x_(i),x_(i+1),x_(i+2);f]=0\left[x_{i}, x_{i+1}, x_{i+2} ; f\right]=0, we can remove from our considerations the functionphi_(i)\phi_{i}corresponding.