Notes on generalizations of the higher-order convex functions (III)

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

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T. Popoviciu, Notes sur les généralisations des fonctions convexes d’ordre supérieur (III), Bull. de la Sect. sci. de l’Acad. Roum., 24 (1942) no. 6, pp. 409-416 (in French)

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1942 g -Popoviciu- Bull. Sect. Sci. Acad. Roum. – Notes sur les generalisations des fonctions convexes d’ordre superieur (III)

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1942 g -Popoviciu- Bull. Sect. Sci. Acad. Roum. - Notes on the generalizations of convex functions
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NOTES ON THE GENERALISATIONS OF HIGHER-ORDER CONVEX FUNCTIONS (III) 1 1 ^(1){ }^{1}1)

BY

TIBERIU POPOVICIU

Note presented by Mr. S. Stoilow, Mc. AR at the meeting of January 9, 1912

ORDER FUNCTIONS ( n ∣ k n ∣ k n|kn \mid kn∣k) AND ORDER FUNCTIONS n n nnnBY SEGMENTS

  1. In the previous note 2 2 ^(2){ }^{2}2) we have defined the order functions n n nnnby segments. We will now show that there is a close connection between these functions and the order functions ( n ∣ k n ∣ k n|kn \mid kn∣k) which we studied in note I of this series 3 3 ^(3){ }^{3}3). We will see, in fact, that any function of order n n nnnby segments is of a certain order ( n ∣ k ) ( n ∣ k ) (n|k)(n \mid k)(n∣k)determined and, conversely, any order function ( n ∣ k n ∣ k n|kn \mid kn∣k) is of order n n nnnby segments.

2. Let's first demonstrate the

Theorem 1. Any function of order n by segments and of characteristic h h hhhis at most of order ( n ∣ ( h − 1 ) ( n + 2 ) n ∣ ( h − 1 ) ( n + 2 ) n|(h-1)(n+2)n \mid(h-1)(n+2)n∣(h−1)(n+2)).
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 (and ordered) sequence of the set E of definition of the function. We can assume that among these points at most 2 ( n + 1 ) 2 ( n + 1 ) 2(n+1)2(n+1)2(n+1)belong to each of the subsets E i ∗ E i ∗ E_(i)^(**)E_{i}^{*}Ei∗of canonical decomposition. Otherwise, indeed, e e eeeis certainly reducible.
If we have s > 1 , x I − 1 ⋅ E E i ∗ , x I , x I + 1 , … , x I + n + s ∈ E ∗ s > 1 , x I − 1 ⋅ E E i ∗ , x I , x I + 1 , … , x I + n + s ∈ E ∗ s > 1,x_(j-1)*EE_(i)^(**),x_(j),x_(j+1),dots,x_(j+n+s)inE^(**)s>1, x_{j-1} \cdot \mathcal{E} E_{i}^{*}, x_{j}, x_{j+1}, \ldots, x_{j+n+s} \in E^{*}s>1,xI−1⋅EEi∗,xI,xI+1,…,xI+n+s∈E∗, x I + n + s + 1 ∈ L i + 1 ∗ x I + n + s + 1 ∈ L i + 1 ∗ x_(j+n+s+1)inL_(i+1)^(**)x_{j+n+s+1} \in L_{i+1}^{*}xI+n+s+1∈Li+1∗the sequel
Δ n + 1 I ( f ) , Δ n + 1 I + 1 ( f ) , … , Δ n + 1 I + s − 1 ( f ) Δ n + 1 I ( f ) , Δ n + 1 I + 1 ( f ) , … , Δ n + 1 I + s − 1 ( f ) Delta_(n+1)^(j)(f),Delta_(n+1)^(j+1)(f),dots,Delta_(n+1)^(j+s-1)(f)\Delta_{n+1}^{j}(f), \Delta_{n+1}^{j+1}(f), \ldots, \Delta_{n+1}^{j+s-1}(f)Δn+1I(f),Δn+1I+1(f),…,Δn+1I+s−1(f)
does not present any variations. This shows us that the least advantageous case is if m = h ( n + 2 ) , x ( i − 1 ) ( n + 2 ) + I ≡ E i ∗ , I = 1 , 2 , … m = h ( n + 2 ) , x ( i − 1 ) ( n + 2 ) + I ≡ E i ∗ , I = 1 , 2 , … m=h(n+2),x_((i-1)(n+2)+j)-=E_(i)^(**),j=1,2,dotsm=h(n+2), x_{(i-1)(n+2)+j} \equiv E_{\mathbf{i}}^{*}, j=1,2, \ldotsm=h(n+2),x(i−1)(n+2)+I≡Ei∗,I=1,2,…, n + 2 , i = 1 , 2 , … , h n + 2 , i = 1 , 2 , … , h n+2,i=1,2,dots,hn+2, i=1,2, \ldots, hn+2,i=1,2,…,hand if the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeepresents the number
maximum possible variations. Indeed, if we add more points to such a value, we do not increase the number of variations in the sequence. d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1The sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeethen h ( n − 2 ) − n − 1 h ( n − 2 ) − n − 1 h(n-2)-n-1h(n-2)-n-1h(n−2)−n−1terms and therefore presents at most ( h − − 1 h − − 1 h--1h--1h−−1) ( n + 2 n + 2 n+2n+2n+2) variations.
Let us now prove
Theorem 9. Any function of order n n nnnby segments and characteristics h h hhhis at least in order ( n ∣ h − 1 n ∣ h − 1 n|h-1n \mid h-1n∣h−1).
We demonstrate by induction on the number h h hhh. For h = 1 h = 1 h=1h=1h=1The property is obvious because the function is then of order n n nnnon E E EEE. For h = 2 h = 2 h=2h=2h=2The function is not of order n n nnnon pent done trouver denx differences sépare [ x 1 , x 2 , … , x n + 2 ; i ] x 1 , x 2 , … , x n + 2 ; i [x_(1),x_(2),dots,x_(n+2);i]\left[x_{1}, x_{2}, \ldots, x_{n+2}; i\right][x1,x2,…,xn+2;i]. [ . x ′ 1 , x ′ 2 , … , x ′ n + 2 ; f ] . x ′ 1 , x ′ 2 , … , x ′ n + 2 ; f [.x^(')_(1),x^(')_(2),dots,x^(')_(n+2);f]\left[. x^{\prime}{ }_{1}, x^{\prime}{ }_{2}, \ldots, x^{\prime}{ }_{n+2}; f\right][.x′1,x′2,…,x′n+2;f]not mules and of opposite signs. From the formula for the average of divided differences, it immediately follows that the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1from the meeting of points x i , x i ′ x i , x i ′ x_(i),x_(i)^(')x_{i}, x_{i}^{\prime}xi,xi′presents at least one variation.
Let's examine the eas h > 2 h > 2 h > 2h>2h>2Suppose the property is true up to h − 1 h − 1 h-1h-1h−1and let's demonstrate - the for h h hhhThe lonetion being of characteristic h − 1 h − 1 h-1h-1h−1on E − E h ∗ E − E h ∗ E-E_(h)^(**)E-E_{h}^{*}E−Eh∗We can find, by hypothesis, the following { x 1 , x 2 , … , x r , x r + 1 , … , x r + r ′ } , x i ∈ E − E h ∗ − E h − 1 ∗ , i = 1 x 1 , x 2 , … , x r , x r + 1 , … , x r + r ′ , x i ∈ E − E h ∗ − E h − 1 ∗ , i = 1 {x_(1),x_(2),dots,x_(r),x_(r+1),dots,x_(r+r^('))},x_(i)in E-E_(h)^(**)-E_(h-1)^(**),i=1\left\{x_{1}, x_{2}, \ldots, x_{r}, x_{r+1}, \ldots, x_{r+r^{\prime}}\right\}, x_{i} \in E-E_{h}^{*}-E_{h-1}^{*}, i=1{x1,x2,…,xr,xr+1,…,xr+r′},xi∈E−Eh∗−Eh−1∗,i=1, 2 , … , r , x r + i ε E h − 1 , i = 1 , 2 , … , r ′ 2 , … , r , x r + i ε E h − 1 , i = 1 , 2 , … , r ′ 2,dots,r,x_(r+i)epsiE_(h-1),i=1,2,dots,r^(')2, \ldots, r, x_{r+i} \varepsilon E_{h-1}, i=1,2, \ldots, r^{\prime}2,…,r,xr+iεEh−1,i=1,2,…,r′such as the following d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1corresponding presents at least h − 2 h − 2 h-2h-2h−2variations. It is possible, of course, that r ′ = 0 r ′ = 0 r^(')=0r^{\prime}=0r′=0, so all the x i ε E − E h ∗ − E h − 1 ∗ x i ε E − E h ∗ − E h − 1 ∗ x_(i)epsi E-E_(h)^(**)-E_(h-1)^(**)x_{i} \varepsilon E-E_{h}^{*}-E_{h-1}^{*}xiεE−Eh∗−Eh−1∗The function being of characteristic 2 on E h − 1 ∗ + E h ∗ E h − 1 ∗ + E h ∗ E_(h-1)^(**)+E_(h)^(**)E_{h-1}^{*}+E_{h}^{*}Eh−1∗+Eh∗we can find the rest { x 1 , x 2 , … , x s ′ , x s + 1 ′ , … , x s + s ′ ′ } x 1 , x 2 , … , x s ′ , x s + 1 ′ , … , x s + s ′ ′ {x_(1),x_(2),dots,x_(s)^('),x_(s+1)^('),dots,x_(s+s^('))^(')}\left\{x_{1}, x_{2}, \ldots, x_{s}^{\prime}, x_{s+1}^{\prime}, \ldots, x_{s+s^{\prime}}^{\prime}\right\}{x1,x2,…,xs′,xs+1′,…,xs+s′′}such as x i ′ ε E h − 1 ∗ , i = 1 , 2 , … , s x i ′ ε E h − 1 ∗ , i = 1 , 2 , … , s x_(i)^(')epsiE_(h-1)^(**),i=1,2,dots,sx_{i}^{\prime} \varepsilon E_{h-1}^{*}, i=1,2, \ldots, sxi′εEh−1∗,i=1,2,…,s, x s + i ′ ε E h ∗ , i = 1 , 2 , … , s ′ x s + i ′ ε E h ∗ , i = 1 , 2 , … , s ′ x_(s+i)^(')epsiE_(h)^(**),i=1,2,dots,s^(')x_{s+i}^{\prime} \varepsilon E_{h}^{*}, i=1,2, \ldots, s^{\prime}xs+i′εEh∗,i=1,2,…,s′and the rest d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1The corresponding element presents at least one variation. Here s ≥ 1 , s ′ ≥ 1 , s + s ′ ≥ n + 3 s ≥ 1 , s ′ ≥ 1 , s + s ′ ≥ n + 3 s >= 1,s^(') >= 1,s+s^(') >= n+3s \geq 1, s^{\prime} \geq 1, s+s^{\prime} \geq n+3s≥1,s′≥1,s+s′≥n+3according to the property which characterizes ≥ 0 ≥ 0 >= 0\geq 0≥0, according to the property that characterizes a proper decomposition. Among the points x i , x i x i , x i x_(i),x_(i)x_{i}, x_{i}xi,xibelonging to E h − 1 ∗ E h − 1 ∗ E_(h-1)^(**)E_{h-1}^{*}Eh−1∗There is s ′ ′ s ′ ′ s^('')s^{\prime \prime}s′′distinct, r ′ + s ≧ s ′ ′ ≧ max ( r ′ , s ) r ′ + s ≧ s ′ ′ ≧ max r ′ , s r^(')+s >= s^('') >= max(r^('),s)r^{\prime}+s \geqq s^{\prime \prime} \geqq \max \left(r^{\prime}, s\right)r′+s≧s′′≧max(r′,s). Let us designate by e e eeeall of the x i , x i x i , x i x_(i),x_(i)x_{i}, x_{i}xi,xidistinct belonging to E − E h ∗ E − E h ∗ E-E_(h)^(**)E-E_{h}^{*}E−Eh∗and by e ′ e ′ e^(')e^{\prime}e′all of the x i , x i x i , x i x_(i),x_(i)x_{i}, x_{i}xi,xidistinct belonging to E h − 1 ∗ + E h ∗ E h − 1 ∗ + E h ∗ E_(h-1)^(**)+E_(h)^(**)E_{h-1}^{*}+E_{h}^{*}Eh−1∗+Eh∗The sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of the remuneration e e eeeAnd e ′ e ′ e^(')e^{\prime}e′is
(1) Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) … , Δ n + 1 r + ε ′ ′ + s ′ − n − 1 ( f ) Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) … , Δ n + 1 r + ε ′ ′ + s ′ − n − 1 ( f ) quadDelta_(n+1)^(1)(f),Delta_(n+1)^(2)(f)dots,Delta_(n+1)^(r+epsi^('')+s^(')-n-1)(f)\quad \Delta_{n+1}^{1}(f), \Delta_{n+1}^{2}(f) \ldots, \Delta_{n+1}^{r+\varepsilon^{\prime \prime}+s^{\prime}-n-1}(f)Δn+11(f),Δn+12(f)…,Δn+1r+ε′′+s′−n−1(f)
And then the sequels d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e e eeeand of e ′ e ′ e^(')e^{\prime}e′are
(2) Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) , … , Δ n + 1 r + s ′ ′ − n − 1 ( f ) Δ n + 1 1 ( f ) , Δ n + 1 2 ( f ) , … , Δ n + 1 r + s ′ ′ − n − 1 ( f ) quadDelta_(n+1)^(1)(f),Delta_(n+1)^(2)(f),dots,Delta_(n+1)^(r+s^('')-n-1)(f)\quad \Delta_{n+1}^{1}(f), \Delta_{n+1}^{2}(f), \ldots, \Delta_{n+1}^{r+s^{\prime \prime}-n-1}(f)Δn+11(f),Δn+12(f),…,Δn+1r+s′′−n−1(f),
(3) Δ n + 1 r + 1 ( f ) , Δ n + 1 r + 2 ( f ) , … , Δ n + 1 r + 3 s ′ ′ + 3 ′ − n − 1 ( f ) Δ n + 1 r + 1 ( f ) , Δ n + 1 r + 2 ( f ) , … , Δ n + 1 r + 3 s ′ ′ + 3 ′ − n − 1 ( f ) quadDelta_(n+1)^(r+1)(f),Delta_(n+1)^(r+2)(f),dots,Delta_(n+1)^(r+3s^('')+3'-n-1)quad(f)\quad \Delta_{n+1}^{r+1}(f), \Delta_{n+1}^{r+2}(f), \ldots, \Delta_{n+1}^{r+3 s^{\prime \prime}+3 \prime-n-1} \quad(f)Δn+1r+1(f),Δn+1r+2(f),…,Δn+1r+3s′′+3′−n−1(f)
respectively, The sequence (2) presents at least h − 2 h − 2 h-2h-2h−2variations and sequence (3) at least one variation. If sequences (2) and (3) have no common terms, sequence (1) has at least h − 1 h − 1 h-1h-1h−1variations. If r ′ ′ = s ′ ′ − n − 1 ≥ 1 r ′ ′ = s ′ ′ − n − 1 ≥ 1 r^('')=s^('')-n-1 >= 1r^{\prime \prime}=s^{\prime \prime}-n-1 \geq 1r′′=s′′−n−1≥1, sequences (2), (3) have the common terms Δ n + 1 r + 1 ( f ) , Δ n + 1 r + 2 ( f ) , … , Δ n + 1 r + r n ( j ) Δ n + 1 r + 1 ( f ) , Δ n + 1 r + 2 ( f ) , … , Δ n + 1 r + r n ( j ) Delta_(n+1)^(r+1)(f),Delta_(n+1)^(r+2)(f),dots,Delta_(n+1)^(r+r^(n))(j)\Delta_{n+1}^{r+1}(f), \Delta_{n+1}^{r+2}(f), \ldots, \Delta_{n+1}^{r+r^{n}}(j)Δn+1r+1(f),Δn+1r+2(f),…,Δn+1r+rn(I)and this sequel does not present any

411 (GENERALLIZATIONS OF HIGHER-ORDER CONVEX FUNCTIONS (III) 3

variations since it is the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a sequence of points belonging to E h − 1 ∗ E h − 1 ∗ E_(h-1)^(**)E_{h-1}^{*}Eh−1∗It also appears that (1) presents at least h − 1 h − 1 h-1h-1h−1variations. Theorem 2 is therefore proven.
3. It remains to show that any function of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) is of order n n nnnby segments. It will suffice to demonstrate that if the function r' is not of order n n nnnby segments, it is not of an order ( n ∣ n ∣ n∣n \midn∣k) determined.
Let us first prove
Lemma 1. If the function f is not of order n by segments on E E EEEThis set can be decomposed into two consecutive subsets. E ( 1 ) , E ( 2 ) E ( 1 ) , E ( 2 ) E^((1)),E^((2))E^{(1)}, E^{(2)}E(1),E(2)in such a way that:
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘Over a year, less than the sets E ( 1 ) , E ( 2 ) E ( 1 ) , E ( 2 ) E^((1)),E^((2))E^{(1)}, E^{(2)}E(1),E(2)the function n ′ n ′ n^(')n^{\prime}n′is not in order n n nnnby segments.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘The function is not of order n n nnnon the sets E ( 1 ) , E ( 2 ) E ( 1 ) , E ( 2 ) E^((1)),E^((2))E^{(1)}, E^{(2)}E(1),E(2)
He is flesh that E E EEEcan't be a finite set and each of the sets E ( 1 ) , E ( 2 ) E ( 1 ) , E ( 2 ) E^((1)),E^((2))E^{(1)}, E^{(2)}E(1),E(2)must be at least one year old n + 3 n + 3 n+3n+3n+3points. The first part is obviously true for any decomposition into two consecutive subsets. Let's therefore prove the second part. Let E 1 , E 2 E 1 , E 2 E_(1),E_(2)E_{1}, E_{2}E1,E2a decomposition of E E EEEin two consecutive subsets, each of the subsets having at least n + 3 n + 3 n+3n+3n+3points. If the function is not of order n n nnnon E 1 E 1 E_(1)E_{1}E1and on E 2 E 2 E_(2)E_{2}E2The property is demonstrated and we can take E ( 1 ) = E 1 , E ( 2 ) = E 2 E ( 1 ) = E 1 , E ( 2 ) = E 2 E^((1))=E_(1),E^((2))=E_(2)E^{(1)}=E_{1}, E^{(2)}=E_{2}E(1)=E1,E(2)=E2Let's assume the opposite, therefore that on one of the sets E 1 , E 2 E 1 , E 2 E_(1),E_(2)E_{1}, E_{2}E1,E2the function is of order n n nnnSo, to clarify, E 1 E 1 E_(1)E_{1}E1This set. So E 2 E 2 E_(2)E_{2}E2contains an infinite number of points and the function is not of order n n nnnby segments on this set. That is x 0 x 0 x_(0)x_{0}x0the right end of the set of x ∈ E x ∈ E x in Ex \in Ex∈Esuch as on the intersection of E E EEEwith the interval ( a , c a , c a,ca, chas,c) the function is of order n n nnn, a being the left end of E ( α = min E ) E ( α = min E ) E(alpha=min E)E(\alpha=\min E)E(α=minE)The entire set of x x xxxsuch as x 0 > x ∈ E x 0 > x ∈ E x_(0) > x in Ex_{0}>x \in Ex0>x∈Eis then inlini. It is clear that there is a x 1 > x 0 , x 1 ε E x 1 > x 0 , x 1 ε E x_(1) > x_(0),x_(1)epsi Ex_{1}>x_{0}, x_{1} \varepsilon Ex1>x0,x1εEsuch that the function is not of order n n nnnby segments on the intersection of E E EEEwith the interval ( x 1 , b x 1 , b x_(1),bx_{1}, bx1,b), b b bbbbeing the right end of E ( b = max E ) E ( b = max E ) E(b=max E)E(b=\max E)E(b=maxE)By taking as E ( 2 ) E ( 2 ) E^((2))E^{(2)}E(2)this last set and E ( 1 ) = E − E ( 2 ) E ( 1 ) = E − E ( 2 ) E^((1))=E-E^((2))E^{(1)}=E-E^{(2)}E(1)=E−E(2)Lemma 1 is completely proven.
We can now prove
Theorem 3. If a function is not of order n by segments on E E EEEand if k k kkkis a natural number, we can find a finite sequence of E E EEEincluding the soot d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1presents at least k k kkkparities.
This property demonstrates, obviously, that a function that is not of order n n nnnby segments cannot be of an order ( n ∣ k n ∣ k n∣kn \mid kn∣k) determined.
Let us now proceed to the proof of the theorem. Let E ( 1 ) , E ( 2 ) E ( 1 ) , E ( 2 ) E^((1)),E^((2))E^{(1)}, E^{(2)}E(1),E(2)a decomposition of E E EEEsatisfying Lemma 1. Let us denote by U ( 1 ) U ( 1 ) U^((1))U^{(1)}U(1)one of these subsets on which the function is not of order n n nnnby segments and so forth. U 1 U 1 U_(1)U_{1}U1the other subset. On U 1 U 1 U_(1)U_{1}U1the function is not of order n n nnnWe proceed in the same way with U ( 1 ) U ( 1 ) U(1)U(1)U(1). et nous en déduisons un U 2 ⊂ U ( 1 ) U 2 ⊂ U ( 1 ) U_(2)subU^((1))U_{2} \subset U^{(1)}U2⊂U(1) sur lequel la fonction n'est pas.
d'ordre n n nnn, tel que sur U ( 1 ) − U 2 = U ( 2 ) U ( 1 ) − U 2 = U ( 2 ) U^((1))-U_(2)=U^((2))U^{(1)}-U_{2}=U^{(2)}U(1)−U2=U(2) elle ne soit pas d'ordre n n nnn par segments. De U ( 2 ) U ( 2 ) U^((2))U^{(2)}U(2) nous déduisons, de la même manière U 3 , U ( 3 ) U 3 , U ( 3 ) U_(3),U^((3))U_{3}, U^{(3)}U3,U(3) et ainsi de suite. Si nous faisons k k kkk fois cette opération, nous déduisons les sous-ensembles (sections de E E EEE )
(4)
U 1 , U 2 , … , U k U 1 , U 2 , … , U k U_(1),U_(2),dots,U_(k)U_{1}, U_{2}, \ldots, U_{k}U1,U2,…,Uk
de E E EEE, qui sont disjoints et la fonction n'est d'ordre n n nnn sur aucun de ces ensembles. Les ensembles (4), rangés dans un certain ordre
U 1 ∗ , U 2 ∗ , … , U k ∗ U 1 ∗ , U 2 ∗ , … , U k ∗ U_(1)^(**),U_(2)^(**),dots,U_(k)^(**)U_{1}^{*}, U_{2}^{*}, \ldots, U_{k}^{*}U1∗,U2∗,…,Uk∗
donnent une décomposition en sous-ensembles consécutifs de leur somme U 1 + U 2 + … + U k U 1 + U 2 + … + U k U_(1)+U_(2)+dots+U_(k)U_{1}+U_{2}+\ldots+U_{k}U1+U2+…+Uk.
La fonction n'étant pas d'ordre n n nnn sur les U i ∗ U i ∗ U_(i)^(**)U_{i}^{*}Ui∗, on peut trouver une suite finie e i ≦ U i ∗ e i ≦ U i ∗ e_(i) <= U_(i)^(**)e_{i} \leqq U_{i}^{*}ei≦Ui∗ dont la suite d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1 présente au moins une variation i = 1 , 2 , … , k i = 1 , 2 , … , k i=1,2,dots,ki=1,2, \ldots, ki=1,2,…,k. Il en résulte que la suite d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1 de e = e 1 + e 2 + … + e k e = e 1 + e 2 + … + e k e=e_(1)+e_(2)+dots+e_(k)e=e_{1}+e_{2}+\ldots+e_{k}e=e1+e2+…+ek présente au moins k k kkk variations.
Le théorème 3 est donc démontré.
Il est clair qu'on peut obtenir une suite partielle de e e eee dont la suite d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1 présente exactement k k kkk variations.
Remarque. Dans le cas particulier n = − 1 n = − 1 n=-1n=-1n=−1, il est clair que toute fonction de caractéristique h h hhh est d'ordre ( − 1 ∣ h − 1 − 1 ∣ h − 1 -1∣h-1-1 \mid h-1−1∣h−1 ) et réciproquement.
4. En nous rapportant aux résultats des notes précédentes, remarquons que si n = − 1 n = − 1 n=-1n=-1n=−1, toute suite maximisante et irréductible a h h hhh termes, dont un appartient à chacun des E i ∗ E i ∗ E_(i)^(**)E_{i}^{*}Ei∗ de la décomposition canonique. Ce cas ne présente donc pas beaucoup de particularités. Au contraire pour n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0 nous pouvons faire d'intéressantes remarques sur les fonctions d'ordre n n nnn par segments. Nous allons d'abord examiner le cas n = 0 n = 0 n=0n=0n=0, donc le cas des fonctions monotones par segments.
Considérons une décomposition de E E EEE
(5) E 1 , E 2 , … , E m (5) E 1 , E 2 , … , E m {:(5)E_(1)","E_(2)","dots","E_(m):}\begin{equation*} E_{1}, E_{2}, \ldots, E_{m} \tag{5} \end{equation*}(5)E1,E2,…,Em
pour une fonction f f fff, monotone par segments et soient a i , b i a i , b i a_(i),b_(i)a_{i}, b_{i}ai,bi les extrémités (gauche et droite) de E i , i = 1 , 2 , … , m E i , i = 1 , 2 , … , m E_(i),i=1,2,dots,mE_{i}, i=1,2, \ldots, mEi,i=1,2,…,m.
Nous allons considérer maintenant certaines suites finies e ε e ε e_(epsi)e_{\varepsilon}eε de E E EEE définies de la manière suivante:
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ Si E i E i E_(i)E_{i}Ei a un seul point ce point appartient à e ε e ε e_(epsi)e_{\varepsilon}eε.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ Si E i E i E_(i)E_{i}Ei a au moins deux points, il a en commun avec e ε e ε e_(epsi)e_{\varepsilon}eε exactement deux points x i , y i x i , y i x_(i),y_(i)x_{i}, y_{i}xi,yi. Si a i ε E i a i ε E i a_(i)epsiE_(i)a_{i} \varepsilon E_{i}aiεEi on a x i = a i x i = a i x_(i)=a_(i)x_{i}=a_{i}xi=ai et si a i a i a_(i)a_{i}ai n'appartient pas à E i E i E_(i)E_{i}Ei on a x i ′ − a i < ε x i ′ − a i < ε x_(i)^(')-a_(i) < epsix_{i}^{\prime}-a_{i}<\varepsilonxi′−ai<ε. Si b i ε , E i b i ε , E i b_(i)epsi,E_(i)b_{i} \varepsilon, E_{i}biε,Ei on a y i ′ = b i y i ′ = b i y_(i)^(')=b_(i)y_{i}^{\prime}=b_{i}yi′=bi et si b i b i b_(i)b_{i}bi n'appartient pas à E i E i E_(i)E_{i}Ei on prend b i − y i < ε b i − y i < ε b_(i)-y_(i) < epsib_{i}-y_{i}<\varepsilonbi−yi<ε.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘. Le nombre positif ε ε epsi\varepsilonε est assez petit pour que l'on ait x i ′ < y i ′ x i ′ < y i ′ x_(i)^(') < y_(i)^(')x_{i}^{\prime}<y_{i}^{\prime}xi′<yi′ et de phus f ( x i ′ ) ≠ f ( y i ′ ) f x i ′ ≠ f y i ′ f(x_(i)^('))!=f(y_(i)^('))f\left(x_{i}^{\prime}\right) \neq f\left(y_{i}^{\prime}\right)f(xi′)≠f(yi′) si la fonction ne se réduit pas à une constante sur E i ( i = 1 , 2 , … , m ) E i ( i = 1 , 2 , … , m ) E_(i)(i=1,2,dots,m)E_{i}(i=1,2, \ldots, m)Ei(i=1,2,…,m).
Si E E EEE n'est pas borné à gauche ( a 1 = − ∞ a 1 = − ∞ a_(1)=-ooa_{1}=-\inftya1=−∞ ), la condition x 1 − a 1 < ε x 1 − a 1 < ε x_(1)-a_(1) < epsix_{1}-a_{1}<\varepsilonx1−a1<ε doit ètre remplaçée par x ˙ 1 < − 1 ε x ˙ 1 < − 1 ε x^(˙)_(1) < -(1)/(epsi)\dot{x}_{1}<-\frac{1}{\varepsilon}x˙1<−1ε et si E E EEE n'est pas borné à droite ( b m = + ∞ b m = + ∞ b_(m)=+oob_{m}=+\inftybm=+∞ ), la condition b m − y m ′ < ε b m − y m ′ < ε b_(m)-y_(m)^(') < epsib_{m}-y_{m}^{\prime}<\varepsilonbm−ym′<ε doit être remplaçée par y m ′ > 1 ε y m ′ > 1 ε y_(m)^(') > (1)/(epsi)y_{m}^{\prime}>\frac{1}{\varepsilon}ym′>1ε. Il peut, bien entendu, arriver qu'il n'y ait qu'un seul e ε e ε e_(epsi)e_{\varepsilon}eε. Ceciarrive si d i , b i ∈ E i , i = 1 , 2 , … , m d i , b i ∈ E i , i = 1 , 2 , … , m d_(i),b_(i)inE_(i),i=1,2,dots,md_{i}, b_{i} \in E_{i}, i=1,2, \ldots, mdi,bi∈Ei,i=1,2,…,m et, en particulier, si E E EEE est fini.
On voit done que e ε e ε e_(epsi)e_{\varepsilon}eε contient deux sortes de points. Les points fixes, qui coincident avec une extrémité a i , b i a i , b i a_(i),b_(i)a_{i}, b_{i}ai,bi et les points variables qui sont à une distance moindre que ε ε epsi\varepsilonε de l'une des extrémités a i , b i a i , b i a_(i),b_(i)a_{\boldsymbol{i}}, b_{\boldsymbol{i}}ai,bi.
Démontrons maintenant le
Lemme 2. Si les points pariables de es s'approchent des extrémités a i , b i a i , b i a_(i),b_(i)a_{i}, b_{i}ai,bi correspondantes, le nombre des variations de la suite d 1 d 1 d_(1)d_{1}d1 de e ε e ε e_(epsi)e_{\varepsilon}eε ne peut pas diminuer.
Il suffit de démontrer la propriété lorsque un de ces points varie. Si ce point est x 1 ′ x 1 ′ x_(1)^(')x_{1}^{\prime}x1′ ou y m ′ y m ′ y_(m)^(')y_{m}^{\prime}ym′, la propriété est immédiate et le nombre des variations de la suite d 1 d 1 d_(1)d_{1}d1 ne change pas. Supposons maintenant, pour fixer les idées, qu'un y i ′ y i ′ y_(i)^(')y_{i}^{\prime}yi′ varie. Si la fonction est constante sur E i E i E_(i)E_{\mathbf{i}}EiThe number of variations does not change. Otherwise, there is only a possible decrease in the number of variations if f ( x i ′ ) f x i ′ f(x_(i)^('))f\left(x_{\mathbf{i}}^{\prime}\right)f(xi′)- − f ( y i ′ ) , f ( y i ′ ) − f ( x i + 1 ′ ) − f y i ′ , f y i ′ − f x i + 1 ′ -f(y_(i)^(')),f(y_(i)^('))-f(x_(i+1)^('))-f\left(y_{i}^{\prime}\right), f\left(y_{i}^{\prime}\right)-f\left(x_{i+1}^{\prime}\right)−f(yi′),f(yi′)−f(xi+1′)are of opposite signs. Here x i + 1 ′ x i + 1 ′ x_(i+1)^(')x_{i+1}^{\prime}xi+1′designates the unique point of E i + 1 E i + 1 E_(i+1)E_{i+1}Ei+1if this set is formed by a single point. But if y i ′ y i ′ y_(i)^(')y_{i}^{\prime}yi′believes towards b i , f ( x i ) − f ( y i ′ ) b i , f x i − f y i ′ b_(i),f(x_(i))-f(y_(i)^('))b_{i}, f\left(x_{i}\right)-f\left(y_{i}^{\prime}\right)bi,f(xi)−f(yi′)cannot decrease in absolute value, therefore f ( y i ′ ) − f ( x i + 1 ′ ) f y i ′ − f x i + 1 ′ f(y_(i)^('))-f(x_(i+1)^('))f\left(y_{i}^{\prime}\right)-f\left(x_{i+1}^{\prime}\right)f(yi′)−f(xi+1′)cannot decrease in absolute value. On the other hand, f ( y i ) f y i f(y_(i))f\left(y_{i}\right)f(yi)varying in the same direction, we see that we do not lose any variations in the sequence d 1 d 1 d_(1)d_{1}d1We demonstrate Ta in the same way if a x i ′ x i ′ x_(i)^(')x_{i}^{\prime}xi′decreases towards a i a i a_(i)a_{i}hasi.
We can deduce that if ε ε epsi\varepsilonεAs the number of variations of the sequence tends towards zero, the number of variations of the sequence d 1 d 1 d_(1)d_{1}d1of e ε e ε e_(epsi)e_{\varepsilon}eεtends towards a limit k k kkkwhich is obviously over.
We can also say that there is a positive number ε 1 ε 1 epsi_(1)\varepsilon_{1}ε1such as for ε < ε 1 ε < ε 1 epsi < epsi_(1)\varepsilon<\varepsilon_{1}ε<ε1the sequel d 1 d 1 d_(1)d_{1}d1of e ε e ε e_(epsi)e_{\varepsilon}eεpresents k k kkkvariations. If e ε = { x 1 , x 2 , … , x r } e ε = x 1 , x 2 , … , x r e_(epsi)={x_(1),x_(2),dots,x_(r)}e_{\varepsilon}=\left\{x_{1}, x_{2}, \ldots, x_{r}\right\}eε={x1,x2,…,xr}Moreover, we can replace the sequence d 1 d 1 d_(1)d_{1}d1of e ε e ε e_(epsi)e_{\varepsilon}eεsubsequently
(6) f ( x 2 ) − f ( x 1 ) , f ( x 3 ) − f ( x 2 ) , … , f ( x r ) − f ( x r − 1 ) f x 2 − f x 1 , f x 3 − f x 2 , … , f x r − f x r − 1 quad f(x_(2))-f(x_(1)),f(x_(3))-f(x_(2)),dots,f(x_(r))-f(x_(r-1))\quad f\left(x_{2}\right)-f\left(x_{1}\right), f\left(x_{3}\right)-f\left(x_{2}\right), \ldots, f\left(x_{r}\right)-f\left(x_{r-1}\right)f(x2)−f(x1),f(x3)−f(x2),…,f(xr)−f(xr−1)
In this way each decomposition (5) is characterized by a certain number k k kkkWe have the
Theorem 4. The function f is of order ( 0 ∣ k 0 ∣ k 0∣k0 \mid k0∣k
Indeed, there are finite sequences . e e eeeof E E EEEincluding the sequel d 1 d 1 d_(1)d_{1}d1presents k variations. These are, in particular, the sequences e for sufficiently small e.
Be it now e e eeeany finite sequence of E E EEEand consider a e e e e  e_("e ")e_{\text {e }}ee so that:
1 ∘ ε < ε 1 , ε 1 1 ∘ ε < ε 1 , ε 1 1^(@)epsi < epsi_(1),epsi_(1)1^{\circ} \varepsilon<\varepsilon_{1}, \varepsilon_{1}1∘ε<ε1,ε1being the positive number defined above.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘If E i E i E_(i)E_{i}Eicontains more than one point and if the common part e i e i e_(i)e_{i}eiof e e eeeand of E i E i E_(i)E_{i}Eiis not empty, we have e i ⊂ e i ⊂ e_(i)sube_{i} \subsetei⊂closed interval ( x i ′ , y i ′ ) , x i ′ , y i ′ x i ′ , y i ′ , x i ′ , y i ′ (x_(i)^('),y_(i)^(')),x_(i)^('),y_(i)^(')\left(x_{i}^{\prime}, y_{i}^{\prime}\right), x_{i}^{\prime}, y_{i}^{\prime}(xi′,yi′),xi′,yi′being the points of e ε e ε e_(epsi)e_{\varepsilon}eεbelonging to E i E i E_(i)E_{i}Ei.
Let e ∗ e ∗ e^(**)e^{*}e∗the meeting of the sequels e , e ε e , e ε e,e_(epsi)e, e_{\varepsilon}e,eεIf of e ∗ e ∗ e^(**)e^{*}e∗We remove the points that do not belong to e ε e ε e_(epsi)e_{\varepsilon}eε, we do not decrease the number of variations of the sequence d 1 d 1 d_(1)d_{1}d1, which results from the fact that the function is monotonic on each of the sets E i E i E_(i)E_{i}EiIt follows that the following d 1 d 1 d_(1)d_{1}d1of e ∗ e ∗ e^(**)e^{*}e∗presents exactly k k kkkvariations, therefore the sequel d 1 d 1 d_(1)d_{1}d1of e presents at most k k kkkvariations, from which theorem 4.5 follows
. Let's return to the decomposition (5). The function f f fffis monotonous on E i E i E_(i)E_{i}Ei. If a i , b i a i , b i a_(i),b_(i)a_{i}, b_{i}hasi,biare always the extremities of E E EEEwe have or a i ε E i a i ε E i a_(i)epsiE_(i)a_{i} \varepsilon E_{i}hasiεEiand then we take c 2 i − 1 = f ( a i ) c 2 i − 1 = f a i c_(2i-1)=f(a_(i))c_{2 i-1}=f\left(a_{i}\right)c2i−1=f(hasi), or the limit
lim f ( x ) = c 2 i − 1 E i ⩾ x → a i lim f ( x ) = c 2 i − 1 E i ⩾ x → a i {:[lim f(x)=c_(2i-1)],[E_(i) >= x rarra_(i)]:}\begin{array}{r} \lim f(x)=c_{2 i-1} \\ E_{i} \geqslant x \rightarrow a_{i} \end{array}limitf(x)=c2i−1Ei⩾x→hasi
exists in the literal sense or is + ∞ + ∞ +oo+\infty+∞Or − ∞ − ∞ -oo-\infty−∞Likewise, or h i ε E i h i ε E i h_(i)epsiE_(i)h_{i} \varepsilon E_{i}hiεEiand then we take c 2 i = f ( b i ) c 2 i = f b i c_(2i)=f(b_(i))c_{2 i}=f\left(b_{i}\right)c2i=f(bi), or the limit
lim f ( x ) = c 2 i , lim f ( x ) = c 2 i , lim f(x)=c_(2i),\lim f(x)=c_{2 i},limitf(x)=c2i,
E i ℑ x → b i E i ℑ x → b i E_(i)ℑx rarrb_(i)E_{i} \Im x \rightarrow b_{i}EiNo.x→bi
exists in the literal sense or is + ∞ + ∞ +oo+\infty+∞Or − ∞ − ∞ -oo-\infty−∞.
In particular, if E i E i E_(i)E_{i}Eiis formed by a single point we have a i = b i a i = b i a_(i)=b_(i)a_{i}=b_{i}hasi=biAnd c 2 i − 1 = c 2 i = f ( a i ) c 2 i − 1 = c 2 i = f a i c_(2i-1)=c_(2i)=f(a_(i))c_{2 i-1}=c_{2 i}=f\left(a_{i}\right)c2i−1=c2i=f(hasi).
Let's consider the following
(7) c 2 − c 1 , c 3 − c 2 , … , c 2 m − c 2 m − 1 . (7) c 2 − c 1 , c 3 − c 2 , … , c 2 m − c 2 m − 1 . {:(7)c_(2)-c_(1)","c_(3)-c_(2)","dots","c_(2m)-c_(2m-1).:}\begin{equation*} c_{2}-c_{1}, c_{3}-c_{2}, \ldots, c_{2 m}-c_{2 m-1} . \tag{7} \end{equation*}(7)c2−c1,c3−c2,…,c2m−c2m−1.
In this continuation we agree, as usual, that ( + ∞ ) − − u = u − ( − ∞ ) = ( + ∞ ) − ( − ∞ ) = + ∞ > 0 , ( − ∞ ) − u = u − D → ( + ∞ ) = ( − ∞ ) − ( + ∞ ) = − ∞ < 0 ( + ∞ ) − − u = u − ( − ∞ ) = ( + ∞ ) − ( − ∞ ) = + ∞ > 0 , ( − ∞ ) − u = u − D → ( + ∞ ) = ( − ∞ ) − ( + ∞ ) = − ∞ < 0 (+oo)--u=u-(-oo)=(+oo)-(-oo)=+oo > 0,(-oo)-u=u- vec(D)(+oo)=(-oo)-(+oo)=-oo < 0(+\infty)- -u=u-(-\infty)=(+\infty)-(-\infty)=+\infty>0,(-\infty)-u=u- \vec{D}(+\infty)=(-\infty)-(+\infty)=-\infty<0(+∞)−−u=u−(−∞)=(+∞)−(−∞)=+∞>0,(−∞)−u=u−D→(+∞)=(−∞)−(+∞)=−∞<0if u u uuuis a finite number. Furthermore, we will follow the conventions ( + ∞ ) − ( + ∞ ) = ( − ∞ ) − ( + ∞ ) − ( + ∞ ) = ( − ∞ ) − (+oo)-(+oo)=(-oo)-(+\infty)-(+\infty)=(-\infty)-(+∞)−(+∞)=(−∞)−- − ( − ∞ ) = 0 − ( − ∞ ) = 0 -(-oo)=0-(-\infty)=0−(−∞)=0Then each term of the sequence (7) is either zero or has a determined sign. The sequence (7) can be viewed as the limit, for ε → 0 ε → 0 epsi rarr0\varepsilon \rightarrow 0ε→0, of the sequence (6) corresponding to a e ε e ε e_(epsi)e_{\varepsilon}eε, possibly by removing certain null terms arising from the fact that some E i E i E_(i)E_{i}Eican have only one point. The sequence (7) therefore presents k k kkkvariations.
From the foregoing, it follows that
Theorem 5. The number of variations of the sequence (7), corresponding to the decomposition (5), is independent of this decomposition. If k k kkkis this number, the function is of order ( 0 ∣ k 0 ∣ k 0∣k0 \mid k0∣k) on E E EEE.
We can establish the invariance of the number of variations of the sequence (7), independently of the definition, already given, of the order of a function. We thus obtain a new definition of the order of a segment-monotone function.
6. We will now extend the previous results to the case n > 0 n > 0 n > 0n>0n>0First, we will construct the sequences es in this case. To do this, let's specify the points of es that belong to a E i E i E_(i)E_{i}EiFirst of all i ≠ 1 , m i ≠ 1 , m i!=1,mi \neq 1, mi≠1,m, SO E i E i E_(i)E_{i}Eiis neither the first nor the last term of the decomposition (5) of E E EEEfor the function f f fff, of order n n nnnby segments. If E i E i E_(i)E_{i}Eiwithin 2 ( n + 1 ) 2 ( n + 1 ) 2(n+1)2(n+1)2(n+1)all these points belong to e ε e ε e_(epsi)e_{\varepsilon}eε. If E i E i E_(i)E_{i}Eihas at least 2 ( n + 1 ) 2 ( n + 1 ) 2(n+1)2(n+1)2(n+1)points it has in common with e ε e ε e_(epsi)e_{\varepsilon}eεExactly 2 ( n + 1 ) 2 ( n + 1 ) 2(n+1)2(n+1)2(n+1)points x i ′ , x i ′ ′ , … , x i ( n + 1 ) ; y i ′ , y i ′ ′ , … , y i ( n + 1 ) x i ′ , x i ′ ′ , … , x i ( n + 1 ) ; y i ′ , y i ′ ′ , … , y i ( n + 1 ) x_(i)^('),x_(i)^(''),dots,x_(i)^((n+1));y_(i)^('),y_(i)^(''),dots,y_(i)^((n+1))x_{i}^{\prime}, x_{i}^{\prime \prime}, \ldots, x_{i}^{(n+1)} ; y_{i}^{\prime}, y_{i}^{\prime \prime}, \ldots, y_{i}^{(n+1)}xi′,xi′′,…,xi(n+1);yi′,yi′′,…,yi(n+1)Let's agree that x i ′ < x i ′ ′ < … < x i ( n + 1 ) ⋅ y i ′ > y i ′ ′ > … > y i ( n + 1 ) x i ′ < x i ′ ′ < … < x i ( n + 1 ) ⋅ y i ′ > y i ′ ′ > … > y i ( n + 1 ) x_(i)^(') < x_(i)^('') < dots < x_(i)^((n+1))*y_(i)^(') > y_(i)^('') > dots > y_(i)^((n+1))x_{i}^{\prime}<x_{i}^{\prime \prime}<\ldots<x_{i}^{(n+1)} \cdot y_{i}^{\prime}>y_{i}^{\prime \prime}>\ldots>y_{i}^{(n+1)}xi′<xi′′<…<xi(n+1)⋅yi′>yi′′>…>yi(n+1). If a a aahasdoes not belong to E i E i E_(i)E_{i}Eiwe take x i − u i < ε , x i − x i < ε , … , i x i ( n + 1 ) − x i ( n ) < ε x i − u i < ε , x i − x i < ε , … , i x i ( n + 1 ) − x i ( n ) < ε x_(i)-u_(i) < epsi,x_(i)-x_(i) < epsi,dots,^(i)x_(i)^((n+1))-x_(i)^((n)) < epsix_{i}-u_{i}<\varepsilon, x_{i}-x_{i}<\varepsilon, \ldots,{ }^{i} x_{i}^{(n+1)}-x_{i}^{(n)}<\varepsilonxi−ui<ε,xi−xi<ε,…,ixi(n+1)−xi(n)<ε. If a i ε E a i ε E a_(i)epsi Ea_{i} \varepsilon EhasiεE, let us designate by a i a i a_(i)a_{i}hasithe left end of E − a i E − a i E-a_(i)E-a_{i}E−hasi, by a i ′ ′ a i ′ ′ a_(i)^('')a_{i}^{\prime \prime}hasi′′the left end of E − ( a + a i ) E − a + a i E-(a+a_(i))E-\left(a+a_{i}\right)E−(has+hasi)and so on. The general case is that a i , a i , … , a i ( r − 1 ) a i , a i , … , a i ( r − 1 ) a_(i),a_(i),dots,a_(i)^((r-1))a_{i}, a_{i}, \ldots, a_{i}^{(r-1)}hasi,hasi,…,hasi(r−1)are isolated points of E i E i E_(i)E_{i}EiAnd a i < a i < … < a i ( r − 1 ) < a i ( r ) = a i ( r + 1 ) = … a i < a i < … < a i ( r − 1 ) < a i ( r ) = a i ( r + 1 ) = … a_(i) < a_(i) < dots < a_(i)^((r-1)) < a_(i)^((r))=a_(i)^((r+1))=dotsa_{i}<a_{i}<\ldots<a_{i}^{(r-1)}<a_{i}^{(r)}=a_{i}^{(r+1)}=\ldotshasi<hasi<…<hasi(r−1)<hasi(r)=hasi(r+1)=…Two scenarios are possible: 1 ∘ a i ( r ) ε E i 1 ∘ a i ( r ) ε E i 1^(@)a_(i)^((r))epsiE_(i)1^{\circ} a_{i}^{(r)} \varepsilon E_{i}1∘hasi(r)εEiand we take x i ′ = a i , x i ′ ′ = a i ′ x i ′ = a i , x i ′ ′ = a i ′ x_(i)^(')=a_(i),x_(i)^('')=a_(i)^(')x_{i}^{\prime}=a_{i}, x_{i}^{\prime \prime}=a_{i}^{\prime}xi′=hasi,xi′′=hasi′... x i ( r + 1 ) = u i ( r ) , x i ( r + 2 ) − x i ( r + 1 ) < ε , x i ( r + 3 ) − x i ( r + 2 ) < ε , … x i ( r + 1 ) = u i ( r ) , x i ( r + 2 ) − x i ( r + 1 ) < ε , x i ( r + 3 ) − x i ( r + 2 ) < ε , … x_(i)^((r+1))=u_(i)^((r)),x_(i)^((r+2))-x_(i)^((r+1)) < epsi,x_(i)^((r+3))-x_(i)^((r+2)) < epsi,dotsx_{i}^{(r+1)}=u_{i}^{(r)}, x_{i}^{(r+2)}-x_{i}^{(r+1)}<\varepsilon, x_{i}^{(r+3)}-x_{i}^{(r+2)}<\varepsilon, \ldotsxi(r+1)=ui(r),xi(r+2)−xi(r+1)<ε,xi(r+3)−xi(r+2)<ε,…. a i ( n + 1 ) − x i ( n ) < ε , 2 ∘ a i ( r ) a i ( n + 1 ) − x i ( n ) < ε , 2 ∘ a i ( r ) a_(i)^((n+1))-x_(i)^((n)) < epsi,2^(@)a_(i)^((r))a_{i}^{(n+1)}-x_{i}^{(n)}<\varepsilon, 2^{\circ} a_{i}^{(r)}hasi(n+1)−xi(n)<ε,2∘hasi(r)does not belong to i E i E i E_(i)E_{i}Eiand then we take x i ′ = a i , x i ′ ′ = a i ′ , … , x i ( r ) = a i ( r − 1 ) , x i ( r + 1 ) − x i ( r ) < ε , x i ( r + 2 ) − a i ( r + 1 ) < ε , … , x i ( n + 1 ) − x i ( n ) < ε .11 x i ′ = a i , x i ′ ′ = a i ′ , … , x i ( r ) = a i ( r − 1 ) , x i ( r + 1 ) − x i ( r ) < ε , x i ( r + 2 ) − a i ( r + 1 ) < ε , … , x i ( n + 1 ) − x i ( n ) < ε .11 x_(i)^(')=a_(i),x_(i)^('')=a_(i)^('),dots,x_(i)^((r))=a_(i)^((r-1)),x_(i)^((r+1))-x_(i)^((r)) < epsi,x_(i)^((r+2))-a_(i)^((r+1)) < epsi,dots,x_(i)^((n+1))-x_(i)^((n)) < epsi.11x_{i}^{\prime}=a_{i}, x_{i}^{\prime \prime}=a_{i}^{\prime}, \ldots, x_{i}^{(r)}=a_{i}^{(r-1)}, x_{i}^{(r+1)}-x_{i}^{(r)}<\varepsilon, x_{i}^{(r+2)}-a_{i}^{(r+1)}<\varepsilon , \ldots, x_{i}^{(n+1)}-x_{i}^{(n)}<\varepsilon .11xi′=hasi,xi′′=hasi′,…,xi(r)=hasi(r−1),xi(r+1)−xi(r)<ε,xi(r+2)−hasi(r+1)<ε,…,xi(n+1)−xi(n)<ε.11It is possible, of course, that r ≥ n + 1 r ≥ n + 1 r >= n+1r \geq n+1r≥n+1, then all the points x i x i x_(i)x_{i}xihave a fixed position. The points y i ′ , y i ′ ′ , … , y i ( n + 1 ) y i ′ , y i ′ ′ , … , y i ( n + 1 ) y_(i)^('),y_(i)^(''),dots,y_(i)^((n+1))y_{i}^{\prime}, y_{i}^{\prime \prime}, \ldots, y_{i}^{(n+1)}yi′,yi′′,…,yi(n+1)are distributed in the same way in the vicinity of the end b i b i b_(i)b_{i}biThe points of e ε e ε e_(epsi)e_{\varepsilon}eεbelonging to E 1 E 1 E_(1)E_{1}E1and to E m E m E_(m)E_{m}Em. If E 1 E 1 E_(1)E_{1}E1within n + 2 n + 2 n+2n+2n+2all these points belong to e ε e ε e_(epsi)e_{\varepsilon}eε. If E 1 E 1 E_(1)E_{1}E1has at least n + 2 n + 2 n+2n+2n+2points it has in common with e ε e ε e_(epsi)e_{\varepsilon}eεExactly n + 2 n + 2 n+2n+2n+2points x 1 ′ , y 1 ′ , y 1 ′ ′ , … , y 1 ( n + 1 ) x 1 ′ , y 1 ′ , y 1 ′ ′ , … , y 1 ( n + 1 ) x_(1)^('),y_(1)^('),y_(1)^(''),dots,y_(1)^((n+1))x_{1}^{\prime}, y_{1}^{\prime}, y_{1}^{\prime \prime}, \ldots, y_{1}^{(n+1)}x1′,y1′,y1′′,…,y1(n+1), where the n + 1 n + 1 n+1n+1n+1the last points are distributed in the neighborhood of b 1 b 1 b_(1)b_{1}b1as above. The point x 1 ′ x 1 ′ x_(1)^(')x_{1}^{\prime}x1′coincides with a 1 a 1 a_(1)a_{1}has1if a 1 ε E a 1 ε E a_(1)epsi Ea_{1} \varepsilon Ehas1εEand we have x 1 ′ − a 1 < ε x 1 ′ − a 1 < ε x_(1)^(')-a_(1) < epsix_{1}^{\prime}-a_{1}<\varepsilonx1′−has1<εif a 1 a 1 a_(1)a_{1}has1does not belong to E 1 E 1 E_(1)E_{1}E1. When a 1 = − ∞ a 1 = − ∞ a_(1)=-ooa_{1}=-\inftyhas1=−∞we take x 1 ′ x 1 ′ x_(1)^(')x_{1}^{\prime}x1′such as x 1 ′ < − 1 x 1 ′ < − 1 x_(1)^(') < -^(1)x_{1}^{\prime}<-{ }^{1}x1′<−1
The same is true for E m E m E_(m)E_{m}Emexcept that here we will have n + 1 n + 1 n+1n+1n+1points in the vicinity of a m a m a_(m)a_{m}hasmand a point in the vicinity of b m b m b_(m)b_{m}bm.
In this way, the set es is perfectly characterized. If the positive number ε ε epsi\varepsilonεis quite small, we have x i ( n + 1 ) < y i ( n + 1 ) , i = 2 , 3 , … x i ( n + 1 ) < y i ( n + 1 ) , i = 2 , 3 , … x_(i)^((n+1)) < y_(i)^((n+1)),i=2,3,dotsx_{i}^{(n+1)}<y_{i}^{(n+1)}, i=2,3, \ldotsxi(n+1)<yi(n+1),i=2,3,…, m − 1 , x 1 < y 1 ( n + 1 ) , x m ( n + 1 ) < y m m − 1 , x 1 < y 1 ( n + 1 ) , x m ( n + 1 ) < y m m-1,x_(1) < y_(1)^((n+1)),x_(m)^((n+1)) < y_(m)m-1, x_{1}<y_{1}^{(n+1)}, x_{m}^{(n+1)}<y_{m}m−1,x1<y1(n+1),xm(n+1)<ymIt can still happen that ′ ε ′ ε ^(')epsi{ }^{\prime} \varepsilon′εbe
completely determined. This is what happens, for example, if E E EEEis finished and ε ε epsi\varepsilonεis small enough. In general, the points of e ε e ε e_(epsi)e_{\varepsilon}eεare some fixed and others variable, decreasing towards ai or increasing towards b i b i b_(i)b_{i}bi.
We still have
Theorem 6. There exists a positive number ε 1 ε 1 epsi_(1)\varepsilon_{1}ε1such as, for ε < ε 1 ε < ε 1 epsi < epsi_(1)\varepsilon<\varepsilon_{1}ε<ε1the sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e ε ε epsi\varepsilonεpresents the same number k k kkkof bets.
For the demonstration, we will follow a slightly different path than in the case n = 0 n = 0 n=0n=0n=0The function being of order n n nnnby segments, the number of variations of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of a sequel e e eeehas a maximum, in other words the function is of a certain order ( n ∣ k ) ( n ∣ k ) (n∣k)(n \mid k)(n∣k). Either e e eeea maximizing sequence and e i e i e_(i)e_{i}eithe part of e e eeebelonging to E i E i E_(i)E_{i}EiLet us then consider a sequence e ε e ε e_(epsi)e_{\varepsilon}eε. If ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0is quite small all points of e i e i e_(i)e_{i}eithat do not belong to e ε e ε e_(epsi)e_{\varepsilon}eεare in the interval ( x i ( n + 1 ) , y i ( n + 1 ) ) [ x i ( n + 1 ) , y i ( n + 1 ) (x_(i)^((n+1)),y_(i)^((n+1)))[:}\left(x_{i}^{(n+1)}, y_{i}^{(n+1)}\right)\left[\right.(xi(n+1),yi(n+1))[Or ( x i ′ , y i ( n + 1 ) ) x i ′ , y i ( n + 1 ) (x_(i)^('),y_(i)^((n+1)))\left(x_{i}^{\prime}, y_{i}^{(n+1)}\right)(xi′,yi(n+1))if i = 1 , ( x m ( n + 1 ) , y m ′ ) i = 1 , x m ( n + 1 ) , y m ′ i=1,(x_(m)^((n+1)),y_(m)^('))i=1,\left(x_{m}^{(n+1)}, y_{m}^{\prime}\right)i=1,(xm(n+1),ym′)if i = m ] i = m {:i=m]\left.i=m\right]i=m]The result is that the consequences d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e ε e ε e_(epsi)e_{\varepsilon}eεand the meeting e ∗ e ∗ e^(**)e^{*}e∗of e e eeeAnd e ε e ε e_(epsi)e_{\varepsilon}eεpresent the same number of variations. But since e is maximizing, e ∗ e ∗ e^(**)e^{*}e∗is also maximizing, therefore the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e ε e ε e_(epsi)e_{\varepsilon}eεpresents k k kkkvariations and the theorem is proven.
The previous property can also be stated in the following form:
Theorem 7. If ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0As the number of variations of the sequence tends towards zero, the number of variations of the sequence d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e ε e ε e_(epsi)e_{\varepsilon}eεtends towards a limit. If k k kkkis this limit, the function is of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) on E E EEE.
Either e ε = { x 1 , x 2 , … , x r } e ε = x 1 , x 2 , … , x r e_(epsi)={x_(1),x_(2),dots,x_(r)}e_{\varepsilon}=\left\{x_{1}, x_{2}, \ldots, x_{r}\right\}eε={x1,x2,…,xr}The sequel d n + 1 d n + 1 d_(n+1)d_{n+1}dn+1of e ε e ε e_(epsi)e_{\varepsilon}eεcan be replaced later
Δ n 2 ( f ) − Δ n 1 ( f ) , Δ n 3 ( f ) − Δ n 2 ( f ) , … , Δ n r − n ( f ) − Δ n r − n − 1 ( f ) . Δ n 2 ( f ) − Δ n 1 ( f ) , Δ n 3 ( f ) − Δ n 2 ( f ) , … , Δ n r − n ( f ) − Δ n r − n − 1 ( f ) . Delta_(n)^(2)(f)-Delta_(n)^(1)(f),Delta_(n)^(3)(f)-Delta_(n)^(2)(f),dots,Delta_(n)^(r-n)(f)-Delta_(n)^(r-n-1)(f).\Delta_{n}^{2}(f)-\Delta_{n}^{1}(f), \Delta_{n}^{3}(f)-\Delta_{n}^{2}(f), \ldots, \Delta_{n}^{r-n}(f)-\Delta_{n}^{r-n-1}(f) .Δn2(f)−Δn1(f),Δn3(f)−Δn2(f),…,Δnr−n(f)−Δnr−n−1(f).
So if ε → 0 ε → 0 epsi rarr0\varepsilon \rightarrow 0ε→0, the number of variations of this sequence tends towards k k kkkWe can also
introduce a sequence analogous to (7), using derivatives up to the order n n nnnorder functions n n nnnand the limits of these derivatives as we approach an extremity a i a i a_(i)a_{i}hasiOr b i b i b_(i)b_{i}biand which still exist in the literal or literal sense. We will leave aside this generalization.
  1. 1 ) 1 ) ^(1)){ }^{1)}1)This note was in press in the Bulletin of the Faculty of Sciences of Cernăuți in June 1940. Having managed to find the manuscript, I am now publishing it without modifications.
    2 2 ^(2){ }^{2}2) Bulletin of the Scientific Section of the Romanian Academy, vol. 22.
    2 2 ^(2){ }^{2}2Disquisitiones Mathematicae et Physicae, 1, 35-42, 1940.
1942

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