On the differences of functions of a real variable

Tiberiu Popoviciu
(original), Approximation Theory, not-at-ICTP, paper

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T. Popoviciu, Sur les diffèrences des fonctions d’une variable réelle, Comptes Rendus des séances de l’Académie des Sciences de Roumanie, 2 (1938) no. 2, pp. 112-114 (in Romanian).

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1938 f -Popoviciu- Comptes Rendus Seances Acad. Sci. Roum. – Sur les differences des fonctions d’une variable reelle

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1938 f -Popoviciu- Reports of the Sessions of the Acad. Sci. Roum. - On the differences in the functions of a
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28. On the differences of functions of a real variable

by TIBERIU POPOVICIU

Presented by Miron Nicolesco, Mc. ASR

(Session of + June 1937).

  1. We consider real functions of the real variable x x xxx, defined and uniform in a finite and closed interval which can be assumed to be ( 0,1 ). Let us set, as usual,
Δ h n f ( x ) = ∑ i = o n ( − I ) n − i ( i i ) / ( x + i h ) , x + i h ⊂ ( O , I ) . Δ h n f ( x ) = ∑ i = o n   ( − I ) n − i ( i i ) / ( x + i h ) , x + i h ⊂ ( O , I ) . Delta_(h)^(n)f(x)=sum_(i=o)^(n)(-I)^(ni)((i)/(i))//(x+ih),quad x+ih sub(O,I).\Delta_{h}^{n} f(x)=\sum_{i=o}^{n}(-\mathrm{I})^{ni}\binom{i}{i} /(x+ih), \quad x+ih \subset(\mathrm{O}, \mathrm{I}) .Δhnf(x)=∑i=on(−I)n−i(ii)/(x+ih),x+ih⊂(O,I).
MA Marchaud demonstrated 1 1 ^(1){ }^{1}1) that if f ( x ) f ( x ) f(x)f(x)f(x)is bounded and if Δ h n f ( x ) Δ h n f ( x ) Delta_(h)^(n)f(x)\Delta_{h}^{n} f(x)Δhnf(x)tends uniformly to zero when h → 0 h → 0 h rarr0h \rightarrow 0h→0, the function f ( x ) f ( x ) f(x)f(x)f(x)is continuous in (0,1).
We have the more precise property:
If f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)is bounded in a subinterval, however small it may be, and if Δ h n f ( x ) Δ h n f ( x ) Delta_(h)^(n)f(x)\Delta_{h}^{n} \mathrm{f}(\mathrm{x})Δhnf(x)tends uniformly to zero for h → 0 h → 0 hrarr0\mathrm{h} \rightarrow 0h→0, the function f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)is continuous in (0,1).
It is enough to demonstrate that if f ( x ) f ( x ) f(x)f(x)f(x)is bounded in ( c ; I c ; I c;Ic ; \mathrm{I}c;I) it is also in ( 0 , 1 ) ( 0 , 1 ) (0.1)(0.1)(0,1). Either | f ( x ) | < M , x ⊂ ( c , 1 ) | f ( x ) | < M , x ⊂ ( c , 1 ) |f(x)| < M,x sub(c,1)|f(x)|<M, x \subset(c, 1)|f(x)|<M,x⊂(c,1)And o < η < I − c n − 1 o < η < I − c n − 1 o < eta < (Ic)/(n-1)o<\eta<\frac{Ic}{n-1}o<η<I−cn−1such as | Δ h n f ( x ) | < M Δ h n f ( x ) < M |Delta_(h)^(n)f(x)| < M\left|\Delta_{h}^{n} f(x)\right|<\mathrm{M}|Δhnf(x)|<MFor | h | < η | h | < η |h| < eta|h|<\eta|h|<η. We have | f ( x ) | < 2 n M | f ( x ) | < 2 n M |f(x)| < 2^(n)M|f(x)|<2^{n} \mathrm{M}|f(x)|<2nMFor x ⩾ c − η x ⩾ c − η x >= c-etax \geqslant c-\etax⩾c−η, from which we deduce that
| f ( x ) | < 2 n s M , x ⊂ ( 0 , 1 ) , s > c η . | f ( x ) | < 2 n s M , x ⊂ ( 0 , 1 ) , s > c η . |f(x)| < 2^(ns)M,quad x sub(0,1),quad s > (c)/( eta).|f(x)|<2^{ns} \mathrm{M}, \quad x \subset(0,1), \quad s>\frac{c}{\eta} .|f(x)|<2nsM,x⊂(0,1),s>cη.
We will now demonstrate that MA Marchaud's theorem extends to measurable functions:
If f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)is measurable in ( O , I ) ( O , I ) (O,I)(\mathrm{O}, \mathrm{I})(O,I)and if Δ h n f ( x ) Δ h n f ( x ) Delta_(h)^(n)f(x)\Delta_{h}^{n} \mathrm{f}(\mathrm{x})Δhnf(x)tends uniformly to zero for h → 0 h → 0 hrarr0\mathrm{h} \rightarrow 0h→0, the function f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)is continuous in ( 0 , I ) ( 0 , I ) (0,I)(0, \mathrm{I})(0,I).
Indeed, if f ( x ) f ( x ) f(x)f(x)f(x)is not bounded, nor is it bounded in the interval ( 0 , I 2 ) 0 , I 2 (0,(I)/(2))\left(0, \frac{I}{2}\right)(0,I2). Either 0 < η < I 2 n 0 < η < I 2 n 0 < eta < (I)/(2n)0<\eta<\frac{I}{2 n}0<η<I2nsuch as | Δ h n f ( x ) | < I Δ h n f ( x ) < I |Delta_(h)^(n)f(x)| < I\left|\Delta_{h}^{n} f(x)\right|<I|Δhnf(x)|<IFor | h | ⩽ η | h | ⩽ η |h| <= eta|h| \leqslant \eta|h|⩽η. Given a number HAS > 0 HAS > 0 A > 0\mathrm{A}>0HAS>0, there is a ξ ⊂ ( 0 , 1 2 ) ξ ⊂ 0 , 1 2 xi sub(0,(1)/(2))\xi \subset\left(0, \frac{1}{2}\right)ξ⊂(0,12)such as | f ( ξ ) | > I + ( 2 n − I ) | f ( ξ ) | > I + 2 n − I |f(xi)| > I+(2^(n)-I)|f(\xi)|>\mathrm{I}+\left(2^{n}-\mathrm{I}\right)|f(ξ)|>I+(2n−I)A. But we have | Δ h n f ( ξ ) | < I Δ h n f ( ξ ) < I |Delta_(h)^(n)f(xi)| < I\left|\Delta_{h}^{n} f(\xi)\right|<\mathrm{I}|Δhnf(ξ)|<IFor n − I h η < h < η n − I h η < h < η (nI)/(h)eta < h < eta\frac{n-\mathrm{I}}{h} \eta<h<\etan−Ihη<h<η,
112
therefore | f ( ξ + i h ) | > | f ( ξ + i h ) | > |f(xi+ih)| >|f(\xi+ih)|>|f(ξ+ih)|>. A for at least one i , i ⩽ i ⩽ n i , i ⩽ i ⩽ n i,i <= i <= ni, i \leqslant i \leqslant ni,i⩽i⩽n. We deduce that the set E | | f ( x ) | > HAS | | | f ( x ) | > HAS | ||f(x)| > A|||f(x)|>\mathrm{A}|||f(x)|>HAS|is measuring ⩾ η n ⩾ η n >= (eta )/(n)\geqslant \frac{\eta}{n}⩾ηn. The number A being arbitrary, this is in contradiction with a theorem of ME B ore 1 on measurable functions. The stated property results from this. The case of the equation Δ h n f ( x ) = O Δ h n f ( x ) = O Delta_(h)^(n)f(x)=O\Delta_{h}^{n} f(x)=\mathrm{O}Δhnf(x)=Owas studied by MW Sierpinski ( n = 2 ) 1 ) ( n = 2 ) 1 {:(n=2)^(1))\left.(n=2)^{1}\right)(n=2)1)and by ourselves ( n > 2 ) 3 ) ( n > 2 ) 3 {:(n > 2)^(3))\left.(n>2)^{3}\right)(n>2)3).
2. Consider the more general expression 3 3 ^(3){ }^{3}3).
(1)
Δ h ( n ) f ( x ) = ∑ i = o n has i f ( x + i h ) Δ h ( n ) f ( x ) = ∑ i = o n   has i f ( x + i h ) Delta_(h)^((n))f(x)=sum_(i=o)^(n)a_(i)f(x+ih)\Delta_{h}^{(n)} f(x)=\sum_{i=o}^{n} a_{i} f(x+ih)Δh(n)f(x)=∑i=onhasif(x+ih)
THE has i has i a_(i)a_{i}hasibeing given constants. This expression is of order k if ∑ i = o n has i i I = 0 , I = 0 , I , … , k − I , ∑ i = o n has i i k ≠ 0 ∑ i = o n   has i i I = 0 , I = 0 , I , … , k − I , ∑ i = o n   has i i k ≠ 0 sum_(i=o)^(n)a_(i)i^(j)=0,j=0,I,dots,kI,sum_(i=o)^(n)a_(i)i^(k)!=0\sum_{i=o}^{n} a_{i} i^{j}=0, j=0, \mathrm{I}, \ldots, k-\mathrm{I}, \sum_{i=o}^{n} a_{i} i^{k} \neq 0∑i=onhasiiI=0,I=0,I,…,k−I,∑i=onhasiik≠0, so if the polynomial F ( x ) = ∑ i = o n has i x i F ( x ) = ∑ i = o n   has i x i F(x)=sum_(i=o)^(n)a_(i)x^(i)\mathrm{F}(x)=\sum_{i=o}^{n} a_{i} x^{i}F(x)=∑i=onhasixihas the root I of order k k kkkof multiplicity. In this case we can determine a positive integer p p pppand two polynomials φ ( x ) , ψ ( x ) φ ( x ) , ψ ( x ) varphi(x),psi(x)\varphi(x), \psi(x)φ(x),ψ(x), so that we have 4 4 ^(4){ }^{4}4) :
φ ( x ) F ( x ) + ψ ( λ ) F ( x P ) ≡ ( I − λ P ) k . φ ( x ) F ( x ) + ψ ( λ ) F x P ≡ I − λ P k . varphi(x)F(x)+psi(lambda)F(x^(P))-=(I-lambda^(P))^(k).\varphi(x) \mathrm{F}(x)+\psi(\lambda) \mathrm{F}\left(x^{\mathrm{P}}\right) \equiv\left(\mathrm{I}-\lambda^{\mathrm{P}}\right)^{k} .φ(x)F(x)+ψ(λ)F(xP)≡(I−λP)k.
It follows that we have a relation of the form
(2) Δ p h k f ( x ) = ∑ i = o p n − h − i α i Δ h ( n ) f ( x + i h ) + ∑ i = o n − k − i β i Δ p h ( v ) f ( x + i h ) , (2) Δ p h k f ( x ) = ∑ i = o p n − h − i   α i Δ h ( n ) f ( x + i h ) + ∑ i = o n − k − i   β i Δ p h ( v ) f ( x + i h ) , {:(2)Delta_(ph)^(k)f(x)=sum_(i=o)^(pn-h-i)alpha_(i)Delta_(h)^((n))f(x+ih)+sum_(i=o)^(n-k-i)beta_(i)Delta_(ph)^((v))f(x+ih)",":}\begin{equation*} \Delta_{p h}^{k} f(x)=\sum_{i=o}^{p n-h-i} \alpha_{i} \Delta_{h}^{(n)} f(x+i h)+\sum_{i=o}^{n-k-i} \beta_{i} \Delta_{p h}^{(v)} f(x+i h), \tag{2} \end{equation*}(2)Δphkf(x)=∑i=opn−h−iαiΔh(n)f(x+ih)+∑i=on−k−iβiΔph(v)f(x+ih),
THE α i , β i α i , β i alpha_(i),beta_(i)\alpha_{i}, \beta_{i}αi,βibeing constants independent of x , h x , h x,hx, hx,hand function f ( x ) f ( x ) f(x)f(x)f(x).
We see that the properties of No. I remain true for expression (1). In particular, if ∑ i = o n a i = 0 ∑ i = o n   a i = 0 sum_(i=o)^(n)a_(i)=0\sum_{i=o}^{n} a_{i}=0∑i=onhasi=0and if (1) tends uniformly to zero, the function f ( x ) f ( x ) f(x)f(x)f(x)is zero identically in ( 0 , I 0 , I 0,I0, \mathrm{I}0,I). If
  1. W. Sierpinski: On measurable convex functions. Fund. Math. t. I (1920), pp. 125-129). M.W. Sierpinski also proved this theorem without using Zermelo's axiom, see: On the functional equation f ( x + y ) = f ( x ) + f ( y ) f ( x + y ) = f ( x ) + f ( y ) f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y)id. p. 116-122.
  2. T. Popoviciu: On some properties of functions of one or two real variables. Mathematica vol. VIII (1934) p. I— 85.
  3. T. Pcpoviciu: On certain functional equations defining polynomials". Mathematica t. X, (1934), p. 197-211.
  4. We will give the demonstration in a work which will appear in vol. XIV of the journal "Mathematica".
    ∑ i = o n a i = 0 , ∑ i = o n a i i = 0 ∑ i = o n   a i = 0 , ∑ i = o n   a i i = 0 sum_(i=o)^(n)a_(i)=0,sum_(i=o)^(n)a_(i)i=0\sum_{i=o}^{n} a_{i}=0, \sum_{i=o}^{n} a_{i} i=0∑i=onhasi=0,∑i=onhasii=0and if (1) tends to zero for all x x xxx, the function is continuous in ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1).
  5. MA Marchaud also demonstrated that if f ( x ) f ( x ) f(x)f(x)f(x)is bounded and if the ratio h − h Δ h h f ( x ) h − h Δ h h f ( x ) h^(-h)Delta_(h)^(h)f(x)h^{-h} \Delta_{h}^{h} f(x)h−hΔhhf(x)tends uniformly towards a determined function g ( x ) g ( x ) g(x)g(x)g(x), the function f ( x ) f ( x ) f(x)f(x)f(x)has a derivative k e ̀ m e k e ̀ m e k^(ème)k^{e ̀ m e}kèmecontinues which is (obviously) equal to g ( x ) g ( x ) g(x)g(x)g(x). Let us demonstrate that this theorem is still true for measurable functions, so
If f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)is measurable and if h − k Δ h k f ( x ) h − k Δ h k f ( x ) h^(-k)Delta_(h)^(k)f(x)\mathrm{h}^{-k} \Delta_{h}^{k} \mathrm{f}(\mathrm{x})h−kΔhkf(x)tends uniformly to a function g ( x ) g ( x ) g(x)\mathrm{g}(\mathrm{x})g(x), the derivative k ème k ème  k^("ème ")k^{\text {ème }}kth of f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)exists, is continuous and we have f ( k ) ( x ) = g ( x ) f ( k ) ( x ) = g ( x ) f^((k))(x)=g(x)\mathrm{f}^{(k)}(\mathrm{x})=\mathrm{g}(\mathrm{x})f(k)(x)=g(x)In ( 0 , I ) ( 0 , I ) (0,I)(0, \mathrm{I})(0,I).
Indeed, for everything h , h − k Δ h k f ( x ) h , h − k Δ h k f ( x ) h,h^(-k)Delta_(h)^(k)f(x)h, h^{-k} \Delta_{h}^{k} f(x)h,h−kΔhkf(x)is a measurable function, so g ( x ) g ( x ) g(x)g(x)g(x)is also. Note that Δ 2 h h f ( x ) = ∑ i = o h ( k i ) Δ h k f ( x + i h ) Δ 2 h h f ( x ) = ∑ i = o h   ( k i ) Δ h k f ( x + i h ) Delta_(2h)^(h)f(x)=sum_(i=o)^(h)((k)/(i))Delta_(h)^(k)f(x+ih)\Delta_{2 h}^{h} f(x)=\sum_{i=o}^{h}\binom{k}{i} \Delta_{h}^{k} f(x+i h)Δ2hhf(x)=∑i=oh(ki)Δhkf(x+ih), SO
| I 2 k ∑ i = o k ( k i ) g ( x + i h ) − g ( x ) | ⩽ ⩽ | Δ 2 h k f ( x ) ( 2 h ) k − g ( x ) | + I 2 k ∑ i = o h ( k i ) | Δ h h f ( x + i h ) h k − g ( x + i h ) | I 2 k ∑ i = o k   ( k i ) g ( x + i h ) − g ( x ) ⩽ ⩽ Δ 2 h k f ( x ) ( 2 h ) k − g ( x ) + I 2 k ∑ i = o h   ( k i ) Δ h h f ( x + i h ) h k − g ( x + i h ) {:[|(I)/(2^(k))sum_(i=o)^(k)((k)/(i))g(x+ih)-g(x)| <= ],[ <= |(Delta_(2h)^(k)f(x))/((2h)^(k))-g(x)|+(I)/(2^(k))sum_(i=o)^(h)((k)/(i))|(Delta_(h)^(h)f(x+ih))/(h^(k))-g(x+ih)|]:}\begin{aligned} & \left|\frac{\mathrm{I}}{2^{k}} \sum_{i=o}^{k}\binom{k}{i} g(x+i h)-g(x)\right| \leqslant \\ & \leqslant\left|\frac{\Delta_{2 h}^{k} f(x)}{(2 h)^{k}}-g(x)\right|+\frac{\mathrm{I}}{2^{k}} \sum_{i=o}^{h}\binom{k}{i}\left|\frac{\Delta_{h}^{h} f(x+i h)}{h^{k}}-g(x+i h)\right| \end{aligned}|I2k∑i=ok(ki)g(x+ih)−g(x)|⩽⩽|Δ2hkf(x)(2h)k−g(x)|+I2k∑i=oh(ki)|Δhhf(x+ih)hk−g(x+ih)|
The expression of order I I III,
I 2 h ∑ i = o k ( k i ) g ( x + i h ) − g ( x ) , I 2 h ∑ i = o k   ( k i ) g ( x + i h ) − g ( x ) , (I)/(2^(h))sum_(i=o)^(k)((k)/(i))g(x+ih)-g(x),\frac{\mathrm{I}}{2^{h}} \sum_{i=o}^{k}\binom{k}{i} g(x+i h)-g(x),I2h∑i=ok(ki)g(x+ih)−g(x),
therefore tends (moreover uniformly) towards zero. It follows that g ( x ) g ( x ) g(x)g(x)g(x)is continuous and, therefore, bounded in ( 0 , I 0 , I 0,I0, \mathrm{I}0,I). SO, Δ h h f ( x ) Δ h h f ( x ) Delta_(h)^(h)f(x)\Delta_{h}^{h} f(x)Δhhf(x)tends uniformly towards zero, so f ( x ) f ( x ) f(x)f(x)f(x)is also continuous. The stated property results from this.
If we pose f ( x ) = x h f ( x ) = x h f(x)=x^(h)f(x)=x^{h}f(x)=xhin (2), we find
I p k ∑ i = o p n − k − 1 α i + ∑ i = o n − k − 1 β i = k ! ∑ i = o k a i i k I p k ∑ i = o p n − k − 1   α i + ∑ i = o n − k − 1   β i = k ! ∑ i = o k   a i i k (I)/(p^(k))sum_(i=o)^(pn-k-1)alpha_(i)+sum_(i=o)^(n-k-1)beta_(i)=(k!)/(sum_(i=o)^(k)a_(i)i^(k))\frac{\mathrm{I}}{p^{k}} \sum_{i=o}^{p n-k-1} \alpha_{i}+\sum_{i=o}^{n-k-1} \beta_{i}=\frac{k!}{\sum_{i=o}^{k} a_{i} i^{k}}Ipk∑i=opn−k−1αi+∑i=on−k−1βi=k!∑i=okhasiik
and we deduce the following property:
If f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)is measurable or if f ( x ) f ( x ) f(x)\mathrm{f}(x)f(x)is bounded and if h − k Δ h ( n ) f ( x ) h − k Δ h ( n ) f ( x ) h^(-k)Delta_(h)^((n))f(x)\mathrm{h}^{-k} \Delta_{h}^{(n)} \mathrm{f}(\mathrm{x})h−kΔh(n)f(x)tends uniformly to a function g ( x ) g ( x ) g(x)\mathrm{g}(\mathrm{x})g(x), the function f ( x ) f ( x ) f(x)\mathrm{f}(\mathrm{x})f(x)has a derivative k ème k ème  k^("ème ")\mathrm{k}^{\text {ème }}kth continue and we have
f ( k ) ( x ) = k ! ∑ i = o n a i i k g ( x ) , x ⊂ ( 0 , I ) . f ( k ) ( x ) = k ! ∑ i = o n   a i i k g ( x ) , x ⊂ ( 0 , I ) . f^((k))(x)=(k!)/(sum_(i=o)^(n)a_(i)i^(k))g(x),quad x sub(0,I).f^{(k)}(x)=\frac{k!}{\sum_{i=o}^{n} a_{i} i^{k}} g(x), \quad x \subset(0, \mathrm{I}) .f(k)(x)=k!∑i=onhasiikg(x),x⊂(0,I).
    1. A. Marchaud: On the derivatives and on the differences of functions of real variables. Journ, de Math. t. 6 (1927), p. 337-425.
1938

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