Notes on generalizations of higher-order convex functions (I)

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

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T. Popoviciu

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T. Popoviciu, Notes sur les généralisations des fonctions convexes d’ordre supérieur (I), Disquisitiones mathematicae et physicae, 1 (1940), pp. 35-42 (in French).

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1940 e -Popoviciu- Disquisit. Math. Phys. – Notes sur les generalisations des fonctions convexes d’ordre superieur (I)

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1940 e -Popoviciu- Disquisit. Math. Phys. - Notes on the generalizations of convex functions d_o
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NOTES ON GENERALIZATIONS OF HIGHER ORDER CONVEX FUNCTIONS (I)

BYTIBERIU POPOVICIU

In a series of notes entitled "Notes on Higher Order Convex Functions" we continue the study of higher order functions n n nnn. In this new series of notes we propose to examine the various classes of functions which generalize the functions of order n n nnnof a variable.

The order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k).

  1. Let us first recall the definition of functions of order n. The function f = f ( x ) f = f ( x ) f=f(x)f=f(x)f=f(x), real, finite, uniform and defined on any linear set E E EEEis said to be of order n n nnnon E E EEEif its difference divided by order n + 2 , [ x 1 , x 2 , … , x n + 2 ; f ] n + 2 , x 1 , x 2 , … , x n + 2 ; f n+2,[x_(1),x_(2),dots,x_(n+2);f]n+2,\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right]n+2,[x1,x2,…,xn+2;f]does not change sign on E E EEE. More precisely we have the following definition
The function f is consex, non-concave, polynomial, non-consex resp. concape of order n n nnnon E E EEE, following that inequality
(1)
[ x 1 , x 2 , … , x n + 2 ; f ] > , ≧ , = , ≦ resp. < 0 x 1 , x 2 , … , x n + 2 ; f > , ≧ , = , ≦  resp.  < 0 [x_(1),x_(2),dots,x_(n+2);f] > , >= ,=, <= " resp. " < 0\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right]>, \geqq,=, \leqq \text { resp. }<0[x1,x2,…,xn+2;f]>,≧,=,≦ resp. <0
is serified, whatever the points x 1 , x 2 , … , x n + 2 ε E x 1 , x 2 , … , x n + 2 ε E x_(1),x_(2),dots,x_(n+2)epsi Ex_{1}, x_{2}, \ldots, x_{n+2} \varepsilon Ex1,x2,…,xn+2εE.
Convexity and polynomiality of order n n nnnare special cases of non-concavity of order n n nnn. If f f fffis convex, non-concave, etc. of order n n nnn, the function − f − f -f-f−fconcave, non-convex, etc., of order n n nnnand vice versa. We can therefore take as type of order function n n nnnthe non-concave function of order n n nnn.
In particular, the definition applies to n = − 1 n = − 1 n=-1n=-1n=−1and then we have the functions which do not change sign on E E EEE, more precisely the positive, non-negative, identically zero, non-positive resp. negative functions. For n = 0 n = 0 n=0n=0n=0we have monotonic, increasing, non-decreasing, constant, non-increasing resp. decreasing functions
. Finally, for n = 1 n = 1 n=1n=1n=1, we have the usual convex, non-concave, linear, non-convex resp. concave functions.
It is clear that the order functions n n nnnare thus only defined on sets E E EEEhaving at least n + 2 n + 2 n+2n+2n+2points. However, in some statements it is useful to assume that any function defined on less than n + 2 n + 2 n+2n+2n+2points is of order n n nnnand indifferently convex or concave of order n n nnn.
We will designate by { x 1 , x 2 , … , x m } x 1 , x 2 , … , x m {x_(1),x_(2),dots,x_(m)}\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}{x1,x2,…,xm}an ordered sequence of points, therefore of points such that we have x 1 < x 2 < … < x n x 1 < x 2 < … < x n x_(1) < x_(2) < dots < x_(n)x_{1}<x_{2}<\ldots<x_{n}x1<x2<…<xn. For the function f f fffeither convex, non-concave, polynomial, non-convex, or concave of order n n nnnon the ordered sequence { x 1 , x 2 , … , x m } ( m ≥ n + 2 ) x 1 , x 2 , … , x m ( m ≥ n + 2 ) {x_(1),x_(2),dots,x_(m)}(m >= n+2)\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}(m \geq n+2){x1,x2,…,xm}(m≥n+2), it is necessary and sufficient that we have
Δ n + 2 i ( f ) > , ≥ , = , ≦ resp . < 0 , i = 1 , 2 , … , m − n − 1 , Δ n + 2 i ( f ) > , ≥ , = , ≦ resp . < 0 , i = 1 , 2 , … , m − n − 1 , Delta_(n+2)^(i)(f) > , >= ,=, <= resp. < 0,i=1,2,dots,m-n-1,\Delta_{n+2}^{i}(f)>, \geq,=, \leqq \mathrm{resp} .<0, i=1,2, \ldots, m-n-1,Δn+2i(f)>,≥,=,≦resp.<0,i=1,2,…,m−n−1,
by posing
(2) Δ j i ( f ) = [ x i , x i + 1 , … , x i + j ; f ] , Δ 0 i ( f ) = f ( x i ) i = 1 , 2 , … , m − j , j = 0 , 1 , … , m − 1 (2) Δ j i ( f ) = x i , x i + 1 , … , x i + j ; f , Δ 0 i ( f ) = f x i i = 1 , 2 , … , m − j , j = 0 , 1 , … , m − 1 {:[(2)Delta_(j)^(i)(f)=[x_(i),x_(i+1),dots,x_(i+j);f]","Delta_(0)^(i)(f)=f(x_(i))],[i=1","2","dots","m-j","j=0","1","dots","m-1]:}\begin{gather*} \Delta_{j}^{i}(f)=\left[x_{i}, x_{i+1}, \ldots, x_{i+j} ; f\right], \Delta_{0}^{i}(f)=f\left(x_{i}\right) \tag{2}\\ i=1,2, \ldots, m-j, j=0,1, \ldots, m-1 \end{gather*}(2)ΔIi(f)=[xi,xi+1,…,xi+I;f],Δ0i(f)=f(xi)i=1,2,…,m−I,I=0,1,…,m−1
This property is an immediate consequence of what we can call the theorem of the mean of divided differences. This theorem expresses the property that any divided difference [ x i 1 , x i 2 , … , x i n + 2 ; f ] x i 1 , x i 2 , … , x i n + 2 ; f [x_(i_(1)),x_(i_(2)),dots,x_(i_(n+2));f]\left[x_{i_{1}}, x_{i_{2}}, \ldots, x_{i_{n+2}} ; f\right][xi1,xi2,…,xin+2;f]on n + 2 n + 2 n+2n+2n+2points of the ordered sequence { x 1 , x 2 , … , x m } x 1 , x 2 , … , x m {x_(1),x_(2),dots,x_(m)}\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}{x1,x2,…,xm}, is a (generalized) arithmetic mean of the divided differences
Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) , … , Δ n + 1 m − n − 1 ( f ) . Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) , … , Δ n + 1 m − n − 1 ( f ) . Delta_(n+1)^(1)(f),Delta_(n+1)^(2)(f),dots,Delta_(n+1)^(m-n-1)(f).\Delta_{n+1}^{1}(f), \Delta_{n+1}^{2}(f), \ldots, \Delta_{n+1}^{m-n-1}(f) .Δn+11(f),Δn+12(f),…,Δn+1m−n−1(f).
  1. We will now generalize the order functions n n nnnby introducing the order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k).
Either e = { x 1 , x 2 , … , x m } e = x 1 , x 2 , … , x m e={x_(1),x_(2),dots,x_(m)}e=\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}e={x1,x2,…,xm}a finite subset (a sequence) of E E EEE.
Definition 1. We will say that the sequence (with the notations (2))
(3) Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) , … , Δ n + 1 m − n − 1 ( f ) , (3) Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) , … , Δ n + 1 m − n − 1 ( f ) , {:(3)Delta_(n+1)^(1)(f)","Delta_(n+1)^(2)(f)","dots","Delta_(n+1)^(m-n-1)(f)",":}\begin{equation*} \Delta_{n+1}^{1}(f), \Delta_{n+1}^{2}(f), \ldots, \Delta_{n+1}^{m-n-1}(f), \tag{3} \end{equation*}(3)Δn+11(f),Δn+12(f),…,Δn+1m−n−1(f),
is the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1corresponding to the following e e eee, or simply the continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee. In particular, the following d 0 d 0 d_(0)d_{0}d0of e e eeeEast
f ( x 1 ) , f ( x 2 ) , … , f ( x m ) f x 1 , f x 2 , … , f x m f(x_(1)),f(x_(2)),dots,f(x_(m))f\left(x_{1}\right), f\left(x_{2}\right), \ldots, f\left(x_{m}\right)f(x1),f(x2),…,f(xm)
Let us now introduce the following definition.
Definition 2. We will say that the function f f fffis of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) on E E EEEif the maximum number of variations of the sequences d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1, of all finite sequences e of E E EEE, is equal to k k kkk.
The number k k kkkis equal to 0 or a natural number. The definition therefore requires that k k kkkis finished, so that the number of variations of the sequences d n + n d n + n d_(n+n)d_{n+n}dn+nis bounded. It is clear that there then exists at least one sequence e e eeewhose
continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1presents exactly k k kkkvariations. The number n n nnncan take the values − 1 , 0 , 1 , 2 , … − 1 , 0 , 1 , 2 , … -1,0,1,2,dots-1,0,1,2, \ldots−1,0,1,2,…
The order functions ( n ∣ 0 n ∣ 0 n∣0n \mid 0n∣0) coincide with the order functions n n nnn.
To simplify the language, we will say that the order ( n ∣ k n ∣ k n∣kn \mid kn∣k) is less than, at most equal, equal, at least equal resp. greater than the order ( n ∣ k ′ n ∣ k ′ n∣k^(')n \mid k^{\prime}n∣k′) depending on whether k < , ⩽ , = , ⩾ k < , ⩽ , = , ⩾ k < , <= ,=, >=k<, \leqslant,=, \geqslantk<,⩽,=,⩾resp. > k ′ > k ′ > k^(')>k^{\prime}>k′.
It is clear that if f f fffis of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) on E E EEE, it is at the most orderly ( n ∣ k ) ( n ∣ k ) (n∣k)(n \mid k)(n∣k)Above all E 1 ⊂ E E 1 ⊂ E E_(1)sub EE_{1} \subset EE1⊂E. If f f fffis in order ( n ∣ k ) ( n ∣ k ) (n∣k)(n \mid k)(n∣k), the function c f c f cfc fcf, Or c c cccis a non-zero constant and is also of order ( n ∣ k n ∣ k n∣kn \mid kn∣k). The same is true of the function f + P f + P f+Pf+Pf+P, Or P P PPPis any polynomial of degree n n nnn.
3. The order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k) are therefore characterized by a property of sequences d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1finite subsets of E E EEE. We must therefore first study more closely the structure of these sequences. d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1.
From a more general theorem of MI Schoenberg 1 1 ^(1){ }^{1}1
), Lemma 1 results . If γ i , μ i , i = 1 , 2 , … , m − 1 γ i , μ i , i = 1 , 2 , … , m − 1 gamma_(i),mu_(i),i=1,2,dots,m-1\gamma_{i}, \mu_{i}, i=1,2, \ldots, m-1γi,μi,i=1,2,…,m−1are non-negative numbers, the number of variations of the sequence
(4) λ 1 c 1 + μ 1 c 2 , λ 2 c 2 + μ 2 c 3 , … , λ m − 1 c m − 1 + μ m − 1 c m λ 1 c 1 + μ 1 c 2 , λ 2 c 2 + μ 2 c 3 , … , λ m − 1 c m − 1 + μ m − 1 c m lambda_(1)c_(1)+mu_(1)c_(2),lambda_(2)c_(2)+mu_(2)c_(3),dots,lambda_(m-1)c_(m-1)+mu_(m-1)c_(m)\lambda_{1} c_{1}+\mu_{1} c_{2}, \lambda_{2} c_{2}+\mu_{2} c_{3}, \ldots, \lambda_{m-1} c_{m-1}+\mu_{m-1} c_{m}λ1c1+μ1c2,λ2c2+μ2c3,…,λm−1cm−1+μm−1cm
is at most equal to the number of bets in the sequence
(5)
c 1 , c 2 , … , c m c 1 , c 2 , … , c m c_(1),c_(2),dots,c_(m)c_{1}, c_{2}, \ldots, c_{m}c1,c2,…,cm
We complete this property with
Lemma 2. If the sequences (4) and (5) have the same number k k kkkof variations, the first non-zero terms in these sequences have the same sign and the last non-zero terms also have the same sign.
In a sequence (5) the first non-zero term is the term c s c s c_(s)c_{s}cssuch as c 1 = c 2 = … = c s − 1 = 0 , c s ≠ 0 c 1 = c 2 = … = c s − 1 = 0 , c s ≠ 0 c_(1)=c_(2)=dots=c_(s-1)=0,c_(s)!=0c_{1}=c_{2}=\ldots=c_{s-1}=0, c_{s} \neq 0c1=c2=…=cs−1=0,cs≠0and the last non-zero term is defined in an analogous way. The proof of Lemma 2 is easily done by induction on the number k k kkk. It is of course assumed that, if k = 0 k = 0 k=0k=0k=0, none of the sequences (4), (5) is identically zero.
Now consider a finite sequence e = { x 1 , x 2 , … , x m } e = x 1 , x 2 , … , x m e={x_(1),x_(2),dots,x_(m)}e=\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}e={x1,x2,…,xm}of E E EEE. We have the
Theorem 1. The number of variations of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a partial sequence of e is at most equal to the number of pariations of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e.
It is obviously sufficient to demonstrate the property for partial sequences e l = { x 1 , x 2 , … , x j − 1 , x j + 1 … , x m } , l ≦ j ≦ m e l = x 1 , x 2 , … , x j − 1 , x j + 1 … , x m , l ≦ j ≦ m e_(l)={x_(1),x_(2),dots,x_(j-1),x_(j+1)dots,x_(m)},l <= j <= me_{l}=\left\{x_{1}, x_{2}, \ldots, x_{j-1}, x_{j+1} \ldots, x_{m}\right\}, l \leqq j \leqq meL={x1,x2,…,xI−1,xI+1…,xm},L≦I≦mof e e eee. Let (3) be the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeeand
(6)
Δ n + 1 ∗ 1 ( f ) , Δ n + 1 ∗ 2 ( f ) , … , Δ n + 1 ∗ m − n − 2 ( f ) Δ n + 1 ∗ 1 ( f ) , Δ n + 1 ∗ 2 ( f ) , … , Δ n + 1 ∗ m − n − 2 ( f ) Delta_(n+1)^(**1)(f),Delta_(n+1)^(**2)(f),dots,Delta_(n+1)^(**m-n-2)(f)\Delta_{n+1}^{* 1}(f), \Delta_{n+1}^{* 2}(f), \ldots, \Delta_{n+1}^{* m-n-2}(f)Δn+1∗1(f),Δn+1∗2(f),…,Δn+1∗m−n−2(f)
the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e j e j e_(j)e_{j}eI. We have
(7) { Δ n + 1 ∗ i ( f ) = Δ n + 1 i ( f ) , Δ n + 1 ∗ i ( f ) = ( x j − x i ) Δ n + 1 i ( f ) + ( x i + n + 2 − x j ) Δ n + 1 i + 1 ( f ) x i + n + 2 − x i i = j − n − 1 , j − n , … , j − 1 Δ n + 1 ∗ i ( f ) = Δ n + 1 i + 1 ( f ) , i = j , j + 1 , … , m − n − 2 (7) Δ n + 1 ∗ i ( f ) = Δ n + 1 i ( f ) , Δ n + 1 ∗ i ( f ) = x j − x i Δ n + 1 i ( f ) + x i + n + 2 − x j Δ n + 1 i + 1 ( f ) x i + n + 2 − x i i = j − n − 1 , j − n , … , j − 1 Δ n + 1 ∗ i ( f ) = Δ n + 1 i + 1 ( f ) , i = j , j + 1 , … , m − n − 2 {:(7){[Delta_(n+1)^(^(**)i),(f)=Delta_(n+1)^(i)(f)","],[Delta_(n+1)^(^(**)i)(f)=((x_(j)-x_(i))Delta_(n+1)^(i)(f)+(x_(i+n+2)-x_(j))Delta_(n+1)^(i+1)(f))/(x_(i+n+2)-x_(i)),],[,i=j-n-1","j-n","dots","j-1],[Delta_(n+1)^(^(**)i)(f)=Delta_(n+1)^(i+1)(f)",",i=j","j+1","dots","m-n-2]:}:}\begin{cases}\Delta_{n+1}^{{ }^{*} i} & (f)=\Delta_{n+1}^{i}(f), \tag{7}\\ \Delta_{n+1}^{{ }^{*} i}(f)=\frac{\left(x_{j}-x_{i}\right) \Delta_{n+1}^{i}(f)+\left(x_{i+n+2}-x_{j}\right) \Delta_{n+1}^{i+1}(f)}{x_{i+n+2}-x_{i}} & \\ & i=j-n-1, j-n, \ldots, j-1 \\ \Delta_{n+1}^{{ }^{*} i}(f)=\Delta_{n+1}^{i+1}(f), & i=j, j+1, \ldots, m-n-2\end{cases}(7){Δn+1∗i(f)=Δn+1i(f),Δn+1∗i(f)=(xI−xi)Δn+1i(f)+(xi+n+2−xI)Δn+1i+1(f)xi+n+2−xii=I−n−1,I−n,…,I−1Δn+1∗i(f)=Δn+1i+1(f),i=I,I+1,…,m−n−2
and the property results from Lemma 1. In formulas (7) the first two groups are to be deleted if j = 1 j = 1 j=1j=1I=1. The first group is to be deleted and in the second i i iiivaries from 1 to j − 1 j − 1 j-1j-1I−1if 1 < j ≤ n + 2 1 < j ≤ n + 2 1 < j <= n+21<j \leq n+21<I≤n+2. The same applies to the last two groups if i ≧ m − n − 1 i ≧ m − n − 1 i >= m-n-1i \geqq m-n-1i≧m−n−1.
It also follows that if the following d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a partial sequence of e e eeehas exactly as many variations as the sequence (3), the first non-zero terms in both sequences d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1have the same sign and the last non-zero terms also have the same sign.
4. Consider a function f f fffset to E E EEEand an integer n ≥ − 1 n ≥ − 1 n >= -1n \geq-1n≥−1.
Definition 3. We will say that the finite sequence e = { x 1 , x 2 , … , x m } e = x 1 , x 2 , … , x m e={x_(1),x_(2),dots,x_(m)}e=\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}e={x1,x2,…,xm}of E E EEEis irreducible if we cannot remove any point of e without reducing the number of variations in its sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1.
Otherwise we will say that the sequence e is reducible.
It is clear that reducibility depends on n n nnn, of the function / / ////and the number k k kkkvariations of the suite d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee. We will see that if n n nnnAnd k k kkkare given, the number of terms of an irreducible sequence has a maximum independent of the function f f fff.
In the case k = 0 k = 0 k=0k=0k=0, so that e e eeeto be irreducible it is necessary and sufficient that it be formed by n + 2 n + 2 n+2n+2n+2points. In the demonstrations which follow we will assume k > 0 k > 0 k > 0k>0k>0. The results for k = 0 k = 0 k=0k=0k=0can be easily deduced from this.
So that e e eeeis irreducible it is necessary and sufficient, according to Theorem 1, that the sequences d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1partial sequences e j = { x 1 , x 2 , … , x i − 1 , x j + 1 , … e j = x 1 , x 2 , … , x i − 1 , x j + 1 , … e_(j)={x_(1),x_(2),dots,x_(i-1),x_(j+1),dots:}e_{j}=\left\{x_{1}, x_{2}, \ldots, x_{i-1}, x_{j+1}, \ldots\right.eI={x1,x2,…,xi−1,xI+1,…, x m } , j = 1 , 2 , … , m x m , j = 1 , 2 , … , m {:x_(m)},j=1,2,dots,m\left.x_{m}\right\}, j=1,2, \ldots, mxm},I=1,2,…,mall present fewer variations than the following d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee. So that the continuation e e eeebe reducible it is necessary and it sulfites that the continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of at least one of the suites c j c j c_(j)c_{j}cIhas exactly as many variations as the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee.
Let (3) always be the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeeand (6) the following d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1, of e j e j e_(j)e_{j}eI. Formulas (7) show us that if the partial sequence
Δ n + 1 j − n − 1 ( f ) , Δ n + 1 j − n ( f ) , … , Δ n + 1 j ( f ) Δ n + 1 j − n − 1 ( f ) , Δ n + 1 j − n ( f ) , … , Δ n + 1 j ( f ) Delta_(n+1)^(j-n-1)(f),Delta_(n+1)^(j-n)(f),dots,Delta_(n+1)^(j)(f)\Delta_{n+1}^{j-n-1}(f), \Delta_{n+1}^{j-n}(f), \ldots, \Delta_{n+1}^{j}(f)Δn+1I−n−1(f),Δn+1I−n(f),…,Δn+1I(f)
does not present any variations, the number of variations of the sequences (3),
(6) is the same. It immediately follows that if the partial sequence of (3).
(8) Δ n + 1 α − n ( f ) , Δ n + 1 α − n + 1 ( f ) , … , Δ n + 1 β ( f ) , a < β Δ n + 1 α − n ( f ) , Δ n + 1 α − n + 1 ( f ) , … , Δ n + 1 β ( f ) , a < β quadDelta_(n+1)^(alpha-n)(f),Delta_(n+1)^(alpha-n+1)(f),dots,Delta_(n+1)^(beta)(f),a < beta\quad \Delta_{n+1}^{\alpha-n}(f), \Delta_{n+1}^{\alpha-n+1}(f), \ldots, \Delta_{n+1}^{\beta}(f), a<\betaΔn+1α−n(f),Δn+1α−n+1(f),…,Δn+1β(f),has<β,
does not present any variations, the following d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1, from the following { x 1 , x 2 , … x 1 , x 2 , … {x_(1),x_(2),dots:}\left\{x_{1}, x_{2}, \ldots\right.{x1,x2,…, x α , x β + 1 , … , x m } x α , x β + 1 , … , x m {:x_(alpha),xbeta_(+1),dots,x_(m)}\left.x_{\alpha}, x \beta_{+1}, \ldots, x_{m}\right\}xα,xβ+1,…,xm}presents exactly as many variations as the sequence (3). We can see easily how the property must be modified if
a − n < 1 ou β > m − n − 1 a − n < 1  ou  β > m − n − 1 a-n < 1" ou "beta > m-n-1a-n<1 \text { ou } \beta>m-n-1has−n<1 Or β>m−n−1
We deduce that we can always find a e ⋆ ≦ e e ⋆ ≦ e e^(***) <= ee^{\star} \leqq ee⋆≦ewhose continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1has as many variations as the sequence (3) and in which the first two terms are non-zero and of opposite signs and the last two terms are also non-zero and of opposite signs. This results in:
Theorem 2. If (3) is the following d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of an irreducible sequence e, we have
Δ n + 1 1 ( f ) ⋅ Δ n + 1 2 ( f ) < 0 , Δ n + 1 m − n − 2 ( f ) ⋅ Δ n + 1 m − n − 1 ( f ) < 0 . Δ n + 1 1 ( f ) ⋅ Δ n + 1 2 ( f ) < 0 , Δ n + 1 m − n − 2 ( f ) ⋅ Δ n + 1 m − n − 1 ( f ) < 0 . Delta_(n+1)^(1)(f)*Delta_(n+1)^(2)(f) < 0,Delta_(n+1)^(m-n-2)(f)*Delta_(n+1)^(m-n-1)(f) < 0.\Delta_{n+1}^{1}(f) \cdot \Delta_{n+1}^{2}(f)<0, \Delta_{n+1}^{m-n-2}(f) \cdot \Delta_{n+1}^{m-n-1}(f)<0 .Δn+11(f)⋅Δn+12(f)<0,Δn+1m−n−2(f)⋅Δn+1m−n−1(f)<0.
Theorem 3. When k = 0 , 1 k = 0 , 1 k=0,1k=0,1k=0,1, any irreducible sequence ea n + k + 2 n + k + 2 n+k+2n+k+2n+k+2points.
Especially if k = 1 k = 1 k=1k=1k=1, the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a e e eeeirreducible is formed by two non-zero terms with opposite signs.
We also see, easily:
Theorem 4. When n = − 1 , 0 n = − 1 , 0 n=-1,0n=-1,0n=−1,0, any irreducible sequence ea n + k + 2 n + k + 2 n+k+2n+k+2n+k+2points.
It is clear that the terms of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeeare then all different from zero and alternately positive and negative.
Let's examine the case n ≧ 1 n ≧ 1 n >= 1n \geqq 1n≧1. Suppose that in formulas (7) we have α > n , m − n − 1 ≥ β ≥ α + 2 α > n , m − n − 1 ≥ β ≥ α + 2 alpha > n,m-n-1 >= beta >= alpha+2\alpha>n, m-n-1 \geq \beta \geq \alpha+2α>n,m−n−1≥β≥α+2and that the partial sequence (8) presents a variation. We can then find a γ γ gamma\gammaγsuch as α − n < γ ≦ β α − n < γ ≦ β alpha-n < gamma <= beta\alpha-n<\gamma \leqq \betaα−n<γ≦β, Δ n + 1 γ ( f ) ≠ 0 Δ n + 1 γ ( f ) ≠ 0 Delta_(n+1)^(gamma)(f)!=0\Delta_{n+1}^{\gamma}(f) \neq 0Δn+1γ(f)≠0and that the consequences
(9) Δ n + 1 α − n ( f ) , Δ n + 1 α − n + 1 ( f ) , … , Δ n + 1 γ − 1 ( f ) (10) Δ n + 1 γ ( f ) , Δ n + 1 γ + 1 ( f ) , … , Δ n + 1 β ( f ) (9) Δ n + 1 α − n ( f ) , Δ n + 1 α − n + 1 ( f ) , … , Δ n + 1 γ − 1 ( f ) (10) Δ n + 1 γ ( f ) , Δ n + 1 γ + 1 ( f ) , … , Δ n + 1 β ( f ) {:[(9)Delta_(n+1)^(alpha-n)(f)","Delta_(n+1)^(alpha-n+1)(f)","dots","Delta_(n+1)^(gamma-1)(f)],[(10)Delta_(n+1)^(gamma)(f)","Delta_(n+1)^(gamma+1)(f)","dots","Delta_(n+1)^(beta)(f)]:}\begin{gather*} \Delta_{n+1}^{\alpha-n}(f), \Delta_{n+1}^{\alpha-n+1}(f), \ldots, \Delta_{n+1}^{\gamma-1}(f) \tag{9}\\ \Delta_{n+1}^{\gamma}(f), \Delta_{n+1}^{\gamma+1}(f), \ldots, \Delta_{n+1}^{\beta}(f) \tag{10} \end{gather*}(9)Δn+1α−n(f),Δn+1α−n+1(f),…,Δn+1γ−1(f)(10)Δn+1γ(f),Δn+1γ+1(f),…,Δn+1β(f)
do not present any variations. In (9) there is, moreover, at least one non-zero term of opposite sign with Δ n + 1 γ ( f ) Δ n + 1 γ ( f ) Delta_(n+1)^(gamma)(f)\Delta_{n+1}^{\gamma}(f)Δn+1γ(f). The previous results show us that we can assume that the sequences (9), (10) each have at most n + 1 n + 1 n+1n+1n+1terms. We can, in fact, return to this case by removing a certain number of points x i x i x_(i)x_{i}xiwithout changing the number of variations in the suite d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee. It also follows that the sequences (9) and (10) each have at least two terms. If now the sequence (9) has at most n n nnnterms, by removing the period x a + 1 x a + 1 x_(a+1)x_{a+1}xhas+1we do not reduce the number of variations of the sequence (8) and consequently we do not reduce the number
of variations of the sequence (3). If the sequence (9) has n + 1 n + 1 n+1n+1n+1terms we arrive at the same result by removing the point x a + 2 x a + 2 x_(a+2)x_{a+2}xhas+2. We deduce that if the sequence (3) presents k k kkkvariations, we can find a e ⋆ ≦ e e ⋆ ≦ e e^(***) <= ee^{\star} \leqq ee⋆≦ewhose continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1presents k k kkkvariations and which is formed by at most [ k 2 ] n + k + 1 k 2 n + k + 1 [(k)/(2)]n+k+1\left[\frac{k}{2}\right] n+k+1[k2]n+k+1terms 1 1 ^(1){ }^{1}1)

This results in the

Theorem 5. If n ≧ 0 n ≧ 0 n >= 0n \geqq 0n≧0, any irreducible sequence, whose sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1presents k k kkkpariations, at most [ k + 6 2 ] n + k + 2 k + 6 2 n + k + 2 [(k+6)/(2)]n+k+2\left[\frac{k+6}{2}\right] n+k+2[k+62]n+k+2points.
We can notice that by removing the point x α + 1 x α + 1 x_(alpha+1)x_{\alpha+1}xα+1, we do not reduce the number of variations of the sequence (3) if β = α + 1 β = α + 1 beta=alpha+1\beta=\alpha+1β=α+1And Δ n + 1 γ − 1 ( f ) = 0 Δ n + 1 γ − 1 ( f ) = 0 Delta_(n+1)^(gamma-1)(f)=0\Delta_{n+1}^{\gamma-1}(f)=0Δn+1γ−1(f)=0. Applying this property to the case n = 1 n = 1 n=1n=1n=1, we can easily see that if n = − 0 , 1 n = − 0 , 1 n=-0,1n=-0,1n=−0,1and if the sequence (3) presents k k kkkvariations, we can find a e ⋆ ⩽ e e ⋆ ⩽ e e^(***) <= ee^{\star} \leqslant ee⋆⩽ewhose continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1presents k k kkkvariations and has all its non-zero terms. Done
Theorem 6. When n = − 1 , 0 , 1 n = − 1 , 0 , 1 n=-1,0,1n=-1,0,1n=−1,0,1, all the terms of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of an irreducible sequence e are different from zero.
We can easily see that in the suites d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1, of all irreducible partial sequences of e e eee, whose sequels d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1have as many variations as the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee, the first terms have the same sign and the last terms also have the same sign.
It is obvious, moreover, that if e e eeeis reducible, we can find a e ⋆ < e e ⋆ < e e^(***) < ee^{\star}<ee⋆<eirreducible whose continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1has exactly as many variations as the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eee.
4a. The maximum number of points in an irreducible sequence can actually be reached. It is sufficient to consider the case n > 0 n > 0 n > 0n>0n>0. Let's continue e = { 1 , 2 , … , m } , m = [ k + 2 2 ] n + k + 2 e = { 1 , 2 , … , m } , m = k + 2 2 n + k + 2 e={1,2,dots,m},m=[(k+2)/(2)]n+k+2e=\{1,2, \ldots, m\}, m=\left[\frac{k+2}{2}\right] n+k+2e={1,2,…,m},m=[k+22]n+k+2and choose the function f f fffso that we have
Δ n + 1 i ( n + 2 ) + 1 ( f ) = 1 , i = 0 , 1 , … , [ k 2 ] Δ n + 1 i ( n + 2 ) ( f ) = Δ n + 1 j ( n + 2 ) + 2 ( f ) = − n − 2 i = 1 , 2 , … , [ k 2 ] , i = 0 , 1 , … , [ k − 1 2 ] Δ n + 1 i ( f ) ≦ 0 , i ≠ 0 , 1 , 2 , ( mod n + 2 ) Δ n + 1 i ( n + 2 ) + 1 ( f ) = 1 , i = 0 , 1 , … , k 2 Δ n + 1 i ( n + 2 ) ( f ) = Δ n + 1 j ( n + 2 ) + 2 ( f ) = − n − 2 i = 1 , 2 , … , k 2 , i = 0 , 1 , … , k − 1 2 Δ n + 1 i ( f ) ≦ 0 , i ≠ 0 , 1 , 2 , ( mod n + 2 ) {:[Delta_(n+1)^(i(n+2)+1)(f)=1","quad i=0","1","dots","[(k)/(2)]],[Delta_(n+1)^(i(n+2))(f)=Delta_(n+1)^(j(n+2)+2)quad(f)=-n-2],[i=1","2","dots","[(k)/(2)]","quad i=0","1","dots","[(k-1)/(2)]],[Delta_(n+1)^(i)(f) <= 0","i!=0","1","2","(mod n+2)]:}\begin{array}{r} \Delta_{n+1}^{i(n+2)+1}(f)=1, \quad i=0,1, \ldots,\left[\frac{k}{2}\right] \\ \Delta_{n+1}^{i(n+2)}(f)=\Delta_{n+1}^{j(n+2)+2} \quad(f)=-n-2 \\ i=1,2, \ldots,\left[\frac{k}{2}\right], \quad i=0,1, \ldots,\left[\frac{k-1}{2}\right] \\ \Delta_{n+1}^{i}(f) \leqq 0, i \neq 0,1,2,(\bmod n+2) \end{array}Δn+1i(n+2)+1(f)=1,i=0,1,…,[k2]Δn+1i(n+2)(f)=Δn+1I(n+2)+2(f)=−n−2i=1,2,…,[k2],i=0,1,…,[k−12]Δn+1i(f)≦0,i≠0,1,2,(modn+2)
The sequel e e eeeis then irreducible. This example shows us that if n > 1 n > 1 n > 1n>1n>1the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a sequel e e eeeirreducible can actually have zero terms.
5. Let us return to the order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k). Let us first introduce Definition 4. We will say that the finite sequence e of E ′ E ′ E^(')E^{\prime}E′is a maximizing sequence, for the function f f fffof order ( n ∣ k n ∣ k n∣kn \mid kn∣k), if its continuation d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1presents exactly k k kkkbets.
There are obviously maximizing sequences. Any finite sequence of E E EEEwhich contains a maximizing partial sequence is still maximizing. We immediately deduce that if the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeepresents k k kkkvariations, the function is of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) on e e eee. It also follows that in the sequels d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1maximizing sequences, the first non-zero terms have the same sign and the last non-zero terms also have the same sign. The sign of the last non-zero term (or the first non-zero term) is therefore characteristic for the function.
Definition 5. We will say that the function f of order (n ∣ k ∣ k ∣k\mid k∣k) is of order ( n ∣ k ) + ( n ∣ k ) + (n∣k)^(+)(n \mid k)^{+}(n∣k)+or order ( n ∣ k n ∣ k n∣kn \mid kn∣k) − − ^(-){ }^{-}−following that the last non-zero term of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a maximizing sequence of E E EEEis positive or negative.
Order functions ( n ∣ 0 ) + ( n ∣ 0 ) + (n∣0)^(+)(n \mid 0)^{+}(n∣0)+are the non-concave functions of order n n nnnand the order functions ( n ∣ 0 n ∣ 0 n∣0n \mid 0n∣0)- non-convex functions of order n n nnn. Polynomial functions of order n n nnnthus escape this definition, but we can agree that such a function is indifferently of order ( n ∣ 0 ) + ( n ∣ 0 ) + (n∣0)^(+)(n \mid 0)^{+}(n∣0)+or order ( n ∣ 0 ) − ( n ∣ 0 ) − (n∣0)^(-)(n \mid 0)^{-}(n∣0)−.
If f f fffis in order ( n ∣ k ) + ( n ∣ k ) + (n∣k)^(+)(n \mid k)^{+}(n∣k)+order manager ( n ∣ k ) − ( n ∣ k ) − (n∣k)^(-)(n \mid k)^{-}(n∣k)−, it is the same for c f c f cfc fcf, Or c c cccis a positive constant and of the function f + . P f + . P f+.Pf+. Pf+.P, Or P P PPPis a polynomial of degree n n nnn. The function - f f fffis of order ( n ∣ k n ∣ k n∣kn \mid kn∣k)- resp. order ( n ∣ k ) + ( n ∣ k ) + (n∣k)^(+)(n \mid k)^{+}(n∣k)+.
If the function / / ////is not in order ≦ ( n ∣ k ) ≦ ( n ∣ k ) <= (n∣k)\leqq(n \mid k)≦(n∣k)on E E EEE, we can find a subset of E E EEEon which f f fffeither of order ( n ∣ k + 1 n ∣ k + 1 n∣k+1n \mid k+1n∣k+1). We also see, easily, that we can find a finite subset of E E EEEon which f f fffeither of order ( n ∣ k ) + ( n ∣ k ) + (n∣k)^(+)(n \mid k)^{+}(n∣k)+and a finite subassembly of E E EEEon which f f fffeither of order ( n ∣ k ) − ( n ∣ k ) − (n∣k)^(-)(n \mid k)^{-}(n∣k)−.
6. Let us recall the well-known property, expressed by Lomme 3. If the sequence
(11) c 2 − c 1 , c 3 − c 2 , … , c m − c m − 1 (11) c 2 − c 1 , c 3 − c 2 , … , c m − c m − 1 {:(11)c_(2)-c_(1)","c_(3)-c_(2)","dots","c_(m)-c_(m-1):}\begin{equation*} c_{2}-c_{1}, c_{3}-c_{2}, \ldots, c_{m}-c_{m-1} \tag{11} \end{equation*}(11)c2−c1,c3−c2,…,cm−cm−1
presents k k kkkvariations, the sequel
(12) c 1 , c 2 , … , c m (12) c 1 , c 2 , … , c m {:(12)c_(1)","c_(2)","dots","c_(m):}\begin{equation*} c_{1}, c_{2}, \ldots, c_{m} \tag{12} \end{equation*}(12)c1,c2,…,cm
present at most k + 1 k + 1 k+1k+1k+1variations.
We can complete this lemma with the following
Lemma 4. If the sequence (11) presents k ( ≥ 0 ) k ( ≥ 0 ) k( >= 0)k(\geq 0)k(≥0)sariations and the continuation (12) exactly k + 1 k + 1 k+1k+1k+1pariations, the last non-zero terms in the two sequences have the same sign.
Consider the function f ( x i ) = c i , i = 1 , 2 , … , m f x i = c i , i = 1 , 2 , … , m f(x_(i))=c_(i),i=1,2,dots,mf\left(x_{i}\right)=c_{i}, i=1,2, \ldots, mf(xi)=ci,i=1,2,…,mon the whole e = { x 1 , x 2 , … , x m } e = x 1 , x 2 , … , x m e={x_(1),x_(2),dots,x_(m)}e=\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}e={x1,x2,…,xm}. If m = k + 2 m = k + 2 m=k+2m=k+2m=k+2and the following (12) presents k + 1 k + 1 k+1k+1k+1variations, the sequence (11) necessarily presents k k kkkvariations and the lemma is easily proven. If m > k + 2 m > k + 2 m > k+2m>k+2m>k+2, the function being of order ( − 1 ∣ k + 1 ) ( − 1 ∣ k + 1 ) (-1∣k+1)(-1 \mid k+1)(−1∣k+1)on e e eee, we can find a partial sequence e ∗ = { x 1 ∗ , x 2 ∗ … , x k + 2 ∗ } e ∗ = x 1 ∗ , x 2 ∗ … , x k + 2 ∗ e^(**)={x_(1)^(**),x_(2)^(**)dots,x_(k+2)^(**)}e^{*}=\left\{x_{1}^{*}, x_{2}^{*} \ldots, x_{k+2}^{*}\right\}e∗={x1∗,x2∗…,xk+2∗}of k + 2 k + 2 k+2k+2k+2. terms of e e eee, maximizing and irreducible. f f fffis also of order ( 0 ∣ k ) ( 0 ∣ k ) (0∣k)(0 \mid k)(0∣k)and in relation to this order e e eeeAnd e ∗ e ∗ e^(**)e^{*}e∗are maximizing. Lemma 4 follows immediately.
Given the equalities
Δ n + 1 i ( f ) = Δ n i + 1 ( f ) − Δ n i ( f ) x n + i + 1 − x i , i = 1 , 2 , … , m − n − 1 Δ n + 1 i ( f ) = Δ n i + 1 ( f ) − Δ n i ( f ) x n + i + 1 − x i , i = 1 , 2 , … , m − n − 1 Delta_(n+1)^(i)(f)=(Delta_(n)^(i+1)(f)-Delta_(n)^(i)(f))/(x_(n+i+1)-x_(i)),quad i=1,2,dots,m-n-1\Delta_{n+1}^{i}(f)=\frac{\Delta_{n}^{i+1}(f)-\Delta_{n}^{i}(f)}{x_{n+i+1}-x_{i}}, \quad i=1,2, \ldots, m-n-1Δn+1i(f)=Δni+1(f)−Δni(f)xn+i+1−xi,i=1,2,…,m−n−1
relating to a e = { x 1 , x 2 , … , x m } e = x 1 , x 2 , … , x m e={x_(1),x_(2),dots,x_(m)}e=\left\{x_{1}, x_{2}, \ldots, x_{m}\right\}e={x1,x2,…,xm}, Lemma 3 gives us
Theorem 7. Any function of order ( n ∣ k n ∣ k n∣kn \mid kn∣k), n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0, is at most of order ( n − 1 ∣ k + 1 ) ( n − 1 ∣ k + 1 ) (n-1∣k+1)(n-1 \mid k+1)(n−1∣k+1). In general, any function of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) is at most of order ( n − 1 ∣ k + i ) ( n − 1 ∣ k + i ) (n-1∣k+i)(n-1 \mid k+i)(n−1∣k+i), regardless of i = 1 , 2 , … , n + 1 i = 1 , 2 , … , n + 1 i=1,2,dots,n+1i=1,2, \ldots, n+1i=1,2,…,n+1.
From Lemma 4 the following more complete property results.
Theorem 8. If a function of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) + + ^(+){ }^{+}+is of order ( n − 1 ∣ k + 1 n − 1 ∣ k + 1 n-1∣k+1n-1 \mid k+1n−1∣k+1), it is necessarily of order ( n − 1 ∣ k + 1 ) + ( n − 1 ∣ k + 1 ) + (n-1∣k+1)^(+)(n-1 \mid k+1)^{+}(n−1∣k+1)+. In general, if the function is of order ( n − i ∣ k + i n − i ∣ k + i n-i∣k+in-i \mid k+in−i∣k+i), it is necessarily of order ( n − i ∣ k + i ) + n − i ∣ k + i ) + n-i∣k+i)^(+)n-i \mid k+i)^{+}n−i∣k+i)+, regardless of i = 1 , 2 , … , n + 1 i = 1 , 2 , … , n + 1 i=1,2,dots,n+1i=1,2, \ldots, n+1i=1,2,…,n+1.
In the following notes we will examine some properties of order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k).
Manuscript received on May 2, 1940.
  1. 1 1 ^(1){ }^{1}1) I. Schoenberg: Uber variationsvernindernde lineare Transformationen, Math, Zeitschrift, 32, 321-328, (1930).
  2. 1 1 ^(1){ }^{1}1) We designate, as usual, by [ a ] [ a ] [a][a][has]the largest integer ≦ a ≦ a <= a\leqq a≦has.
1940

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