On the approximation of higher-order convex functions (II)

Tiberiu Popoviciu
(original), Approximation Theory, convexity, Mathematical Analysis, paper

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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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1938 b -Popoviciu- Bull. Sect. Sci. Acad. Roum. – Sur l’approximation des fonctions convexes d’ordre superieur (II)

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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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1938 b -Popoviciu- Bull. Sect. Sci. Acad. Roum. - On the approximation of convex functions of order
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BULLETIN

No. 7

SUMMARY

ON THE APPROXIMATION OF HIGHER ORDER CONVEX FUNCTIONS

BYTIBERIU POPOVICIUNote presented by MG Tzitzéica, MAK

I. - In a previous work 1 1 ^(1){ }^{1}1), I have demonstrated that any function ∫ ( x ) ∫ ( x ) int(x)\int(x)∫(x)continues in a finite and closed interval ( has , b has , b a,ba, bhas,b) is the limit of a uniformly convergent stite of polynomials that preserve all the convexity properties of f ( x ) f ( x ) f(x)f(x)f(x)This property is realized by MS Bernstein polynomials.
(I) P m ( x ) = 1 ( b − has ) m ∑ i = 0 m ( m i ) f ( has + i b − has m ) ( x − has ) i ( b − x ) m − i . (I) P m ( x ) = 1 ( b − has ) m ∑ i = 0 m   ( m i ) f has + i b − has m ( x − has ) i ( b − x ) m − i . {:(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*}(I)Pm(x)=1(b−has)m∑i=0m(mi)f(has+ib−hasm)(x−has)i(b−x)m−i.
But, in general, nothing can be said about the convexity characteristics of polynomials ( I I III), outside the interval ( has , b has , b a,ba, bhas,b). In the following we will demonstrate the following property:
Any continuous, non-concave function of order n n nnnin the meantime ( has , b has , b a,ba, bhas,b) is the limit of a uniformly convergent series in this interval of polynomials which are convex of order n n nnneverywhere, so in ( − ∞ , + ∞ ) ( − ∞ , + ∞ ) (-oo,+oo)(-\infty,+\infty)(−∞,+∞).
We assume, of course, that f ( x ) f ( x ) f(x)f(x)f(x)does not reduce to a polynomial of degree n n nnn.
2. - If we say that a non-negative function is non-concave of order - 1 , the previous property is also true for n = − 1 n = − 1 n=-1n=-1n=−1. Indeed, in this case there exists a continuous function g ( x ) g ( x ) g(x)g(x)g(x)such that we have f ( x ) = g 2 ( x ) f ( x ) = g 2 ( x ) f(x)=g^(2)(x)f(x)=g^{2}(x)f(x)=g2(x)In ( has , b ) ( has , b ) (a,b)(a, b)(has,b). So then M = max | g ( x ) | M = max | g ( x ) | M=max|g(x)|M=\max |g(x)|M=max|g(x)|And ε ε epsi\varepsilonεyour positive number. ( has , b ) ( has , b ) (a,b)(a, b)(has,b)
According to Weierstrass's theorem for all 0 < ε ′ < min ( I , ε 2 M + I ) 0 < ε ′ < min I , ε 2 M + I 0 < epsi^(') < min(I,(epsi)/(2M+I))0<\varepsilon^{\prime}<\min \left(\mathrm{I}, \frac{\varepsilon}{2 M+\mathrm{I}}\right)0<ε′<min(I,ε2M+I)there exists a polynomial P ( x ) P ( x ) P(x)P(x)P(x)such as | g − P | < ε ′ | g − P | < ε ′ |gP| < epsi^(')|gP|<\varepsilon^{\prime}|g−P|<ε′In ( has , b ) ( has , b ) (a,b)(a, b)(has,b). We have
| P | < M + I , | g + P | < 2 M + I In ( has , b ) , | P | < M + I , | g + P | < 2 M + I  In  ( has , b ) , |P| < M+I,|g+P| < 2M+I" in "(a,b),|P|<M+I,|g+P|<2 M+I \text { in }(a, b),|P|<M+I,|g+P|<2M+I In (has,b),
SO | f − P 2 | = | g 2 − P 2 | = | g − P | | g + P | < ε ′ ( 2 M + 1 ) < ε f − P 2 = g 2 − P 2 = | g − P | | g + P | < ε ′ ( 2 M + 1 ) < ε quad|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|f−P2|=|g2−P2|=|g−P||g+P|<ε′(2M+1)<ε, In ( has , b ) ( has , b ) (a,b)(a, b)(has,b), which demonstrates the property.
We can now demonstrate the property for n n nnnany. Let us therefore suppose that f ( x ) f ( x ) f(x)f(x)f(x)be non-concave of order n n nnnIn ( has , b ) ( has , b ) (a,b)(a, b)(has,b)and either ε ε epsi\varepsilonεany positive number. We can determine a m m mmmsuch that we have
(2) | f − P m | < ε 2 , In ( has , b ) (2) f − P m < ε 2 ,  In  ( has , b ) {:(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*}(2)|f−Pm|<ε2, In (has,b)
P m P m P_(m)P_{m}Pmbeing the polynomial ( I ), generally of degree > n > n > n>n>n. In this case P m ( n + 1 ) ≧ 0 P m ( n + 1 ) ≧ 0 P_(m)^((n+1)) >= 0P_{m}^{(n+1)} \geqq 0Pm(n+1)≧0In ( has , b ) ( has , b ) (a,b)(a, b)(has,b), so there exists a polynomial that is everywhere non-negative P ( x ) P ( x ) P(x)P(x)P(x)such that the origin has
| P m ( n + 1 ) − P | < ε ( n + 1 ) ! 2 ( b − has ) n + 1 , In ( has , b ) P m ( n + 1 ) − P < ε ( n + 1 ) ! 2 ( b − has ) n + 1 ,  In  ( has , b ) |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)|Pm(n+1)−P|<ε(n+1)!2(b−has)n+1, In (has,b)
The polynomial
Q ( x ) = ∫ has x ( x − t ) n n ! P ( t ) d t + ∑ i = 0 n ( x − has ) i i ! P n ( i ) ( has ) Q ( x ) = ∫ has x   ( x − t ) n n ! P ( t ) d t + ∑ i = 0 n   ( x − has ) i i ! P n ( i ) ( has ) 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)Q(x)=∫hasx(x−t)nn!P(t)dt+∑i=0n(x−has)ii!Pn(i)(has)
is convex of order n ( − ∞ , + ∞ ) n ( − ∞ , + ∞ ) n(-oo,+oo)n(-\infty,+\infty)n(−∞,+∞)and we have
P m ( x ) − Q ( x ) = ∫ has x ( x − t ) n n ! [ P m ( n + 1 ) ( t ) − P ( t ) ] d t P m ( x ) − Q ( x ) = ∫ has x   ( x − t ) n n ! P m ( n + 1 ) ( t ) − P ( t ) d t 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] dtPm(x)−Q(x)=∫hasx(x−t)nn![Pm(n+1)(t)−P(t)]dt
from where
(3) | P m − Q | < ε ( n + 1 ) ! 2 ( b − has ) n + 1 ⋅ ( x − has ) n + 1 ( n + 1 ) ! ≤ ε 2 P m − Q < ε ( n + 1 ) ! 2 ( b − has ) n + 1 ⋅ ( x − has ) n + 1 ( n + 1 ) ! ≤ ε 2 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}|Pm−Q|<ε(n+1)!2(b−has)n+1⋅(x−has)n+1(n+1)!≤ε2, In ( a , b ) ( a , b ) (a,b)(a, b)(has,b).
From (2) and (3) we deduce that
| f − Q | ≤ | f − P m | + | P m − Q | < ε 2 + ε 2 = ε , dans ( a , b ) | f − Q | ≤ f − P m + P m − Q < ε 2 + ε 2 = ε ,  dans  ( a , b ) |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)|f−Q|≤|f−Pm|+|Pm−Q|<ε2+ε2=ε, In (has,b)
which demonstrates the property.
3. We will give another demonstration for n = 1 n = 1 n=1n=1n=1, using MS Bernstein polynomials directly. Let us divide the interval ( a , b ) ( a , b ) (a,b)(a, b)(has,b)in r r rrrequal parts by points
x i = a + i b − a r , i = 0 , 1 , … , r x i = a + i b − a r , i = 0 , 1 , … , r 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, rxi=has+ib−hasr,i=0,1,…,r
and either L γ ( x ) L γ ( x ) L_(gamma)(x)L_{\gamma}(x)Lγ(x)the function represented by the polygonal line of vertices ( x i , f ( x i ) ) x i , f x i (x_(i),f(x_(i)))\left(x_{i}, f\left(x_{i}\right)\right)(xi,f(xi)). We know that L γ ( x ) → f ( x ) L γ ( x ) → f ( x ) L_(gamma)(x)rarr f(x)L_{\gamma}(x) \rightarrow f(x)Lγ(x)→f(x)For r → ∞ r → ∞ r rarr oor \rightarrow \inftyr→∞, uniformly in ( a , b a , b a,ba, bhas,b). If we consider continuous functions
φ i ( x ) = { 0 , dans ( a , x i ) x − x i , dans ( x i , b ) i = 1 , 2 , … , r − 1 φ i ( x ) = 0 ,  dans  a , x i x − x i ,  dans  x i , b i = 1 , 2 , … , r − 1 {:[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}φi(x)={0, In (has,xi)x−xi, In (xi,b)i=1,2,…,r−1
we can write 2 2 ^(2){ }^{2}2)
L r ( x ) = f ( a ) + [ a , x 1 ; f ] ( x − a ) + ∑ i = 0 γ ( x i + 2 − x i ) [ x i , x i + 1 , x i + 2 ; f ] φ i + 1 ( x ) L r ( x ) = f ( a ) + a , x 1 ; f ( x − a ) + ∑ i = 0 γ   x i + 2 − x i x i , x i + 1 , x i + 2 ; f φ i + 1 ( x ) 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)Lr(x)=f(has)+[has,x1;f](x−has)+∑i=0γ(xi+2−xi)[xi,xi+1,xi+2;f]φi+1(x).
We now see that it is sufficient to demonstrate the property for functions φ i ( x ) φ i ( x ) varphi_(i)(x)\varphi_{i}(x)φi(x). Let us extend this function in the interval
( x i − b + a , x i + b − a ) x i − b + a , x i + b − a (x_(i)-b+a,x_(i)+b-a)\left(x_{i}-b+a, x_{i}+b-a\right)(xi−b+has,xi+b−has)
by
φ i ( x ) = { 0 , dans ( x i − b + a , x i ) x → x i , dans ( x i , x i + b − a ) φ i ( x ) = 0 ,       dans  x i − b + a , x i x → x i ,       dans  x i , x i + b − a 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}φi(x)={0, In (xi−b+has,xi)x→xi, In (xi,xi+b−has)
and either P m , i ( x ) P m , i ( x ) P_(m,i)(x)P_{m, i}(x)Pm,i(x)the MS Bernstein polynomial of degree m m mmmof this function in the interval ( x i − b + a , x i + b − a ) x i − b + a , x i + b − a (x_(i)-b+a,x_(i)+b-a)\left(x_{i}-b+a, x_{i}+b-a\right)(xi−b+has,xi+b−has). We verify, by a direct calculation, that P m , i ′ ′ ( x ) ≥ 0 P m , i ′ ′ ( x ) ≥ 0 P_(m,i)^('')(x) >= 0P_{m, i}^{\prime \prime}(x) \geq 0Pm,i′′(x)≥0partnership, therefore P m , i P m , i P_(m,i)P_{m, i}Pm,iis convex (of order I I III) In ( − ∞ , − ∞ ) ( − ∞ , − ∞ ) (-oo,-oo)(-\infty,-\infty)(−∞,−∞)if m m mmmis even. We now see that if ε ε epsi\varepsilonεis any positive number, if we determine the positive integers γ γ gamma\gammaγAnd s s sssso that
| f − L y | < ε 2 , dans ( a , b ) | φ i − P 2 s , i | < ε 2 ( r − 1 ) ( x i + 2 + x i ) [ x i , x i + 1 , x i + 2 ; f ] f − L y < ε 2 , dans ( a , b ) φ i − P 2 s , i < ε 2 ( r − 1 ) x i + 2 + x i x i , x i + 1 , x i + 2 ; f {:[|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}|f−Ly|<ε2,In(has,b)|φi−P2s,i|<ε2(r−1)(xi+2+xi)[xi,xi+1,xi+2;f]
dan
( x i − b + a , x i + b − a ) , i = 1 , 2 … , r − 1 3 ) , x i − b + a , x i + b − a , i = 1 , 2 … , r − 1 3 , {:(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),(xi−b+has,xi+b−has),i=1,2…,r−13),
and if we ask
Q ( x ) = f ( a ) + [ a , x 1 ; f ] ( x − a ) + ∑ i = 0 r − 2 ( x i + 2 − x i ) [ x i , x i + 1 , x i + 2 ; f ] P 2 s , i + 1 ( x ) Q ( x ) = f ( a ) + a , x 1 ; f ( x − a ) + ∑ i = 0 r − 2   x i + 2 − x i x i , x i + 1 , x i + 2 ; f P 2 s , i + 1 ( x ) 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)Q(x)=f(has)+[has,x1;f](x−has)+∑i=0r−2(xi+2−xi)[xi,xi+1,xi+2;f]P2s,i+1(x).
the polynomial Q ( x ) Q ( x ) Q(x)Q(x)Q(x)is everywhere convex (of order I) and we have
| f − Q | < ε , dans ( a , b ) | f − Q | < ε ,  dans  ( a , b ) |f-Q| < epsi," dans "(a,b)|f-Q|<\varepsilon, \text { dans }(a, b)|f−Q|<ε, In (has,b)
    • Note that for m m mmmeven polynomials P m , i P m , i P_(m,i)P_{m, i}Pm,iare increasing in the interval ( x i − b + a , + ∞ ) x i − b + a , + ∞ (x_(i)-b+a,+oo)\left(x_{i}-b+a,+\infty\right)(xi−b+has,+∞). So we also have the following property:
Any continuous, non-decreasing, non-concave (order I) function in the interval ( a , b a , b a,ba, bhas,b) is the limit of a uniformly convergent sequence of polynomials which are increasing in ( a , + ∞ a , + ∞ a,+ooa,+\inftyhas,+∞) 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 , + ∞ c , + ∞ c,+ooc,+\inftyc,+∞), c c cccbeing a number ≤ a ≤ a <= a\leq a≤hasbut, it is obvious, that we cannot take c = − ∞ c = − ∞ c=-ooc=-\inftyc=−∞. A similar property holds for continuous non-increasing and non-concave functions. In this case the polynomials are decreasing in
( − ∞ , d ) , où d ≥ b . ( − ∞ , d ) , où  d ≥ b .  (-oo,d)", où "d >= b". "(-\infty, d) \text {, où } d \geq b \text {. }(−∞,d), Or d≥b. 
We can also obtain sequences of polynomials of indefinite approximation preserving several convexity properties at the same time, everywhere or in intervals ( c + ∞ ) , ( − ∞ , d ) ( c + ∞ ) , ( − ∞ , d ) (c+oo),(-oo,d)(c+\infty),(-\infty, d)(c+∞),(−∞,d), but we do not dwell on these questions further here.
Cernăuti, September 27, 1938.
  1. 1 1 ^(1){ }^{1}1) See: Tiberiu Popoviciu, On the approximation of convex functions of higher order, «Mathematica», vol. X (1935), pp. 49-54.
  2. 2 ) 2 ) ^(2)){ }^{2)}2)For notations, see my previous work.
    3 ) 3 ) ^(3)){ }^{3)}3)If [ x i , x i + 1 , x i + 2 ; f ] = 0 x i , x i + 1 , x i + 2 ; f = 0 [x_(i),x_(i+1),x_(i+2);f]=0\left[x_{i}, x_{i+1}, x_{i+2} ; f\right]=0[xi,xi+1,xi+2;f]=0, we can remove from our considerations the function ϕ i ϕ i phi_(i)\phi_{i}ϕicorresponding.
1938

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