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T. Popoviciu, Remarques sur la definition fonctionnelle d’un polynôme d’une variable réelle, Mathematica, 12 (1936), pp. 5-12 (in French).
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REMARKS ON THE FUNCTIONAL DEFINITION OF A POLYNOMIAL OF A REAL VARIABLE
by
Received December 18, 1935.
-
- Eitherf(x)f(x)a uniform function of the real variablexxand consider its difference in ordernn
Delta_(h)^(n)=Delta_(h)^(n)f(x)=sum_(i=0)^(n)(-1)^(i)((n)/(i))f(x+ih)\Delta_{h}^{n}=\Delta_{h}^{n} f(x)=\sum_{i=0}^{n}(-1)^{i}\binom{n}{i} f(x+i h)
Whenf(x)f(x)is a polynomial of degreen-1(^(1))n-1\left({ }^{1}\right)we haveDelta_(h)^(n)=0\Delta_{h}^{n}=0whateverxxAndhh. Conversely we know that if we haveDelta_(h)^(n)=0\Delta_{h}^{n}=0for everythingxxand for everythinghh, the functionf(x)f(x), under fairly general assumptions, is reduced to a polynomial of degreen-1(^(2))n-1\left({ }^{2}\right).
In his lecture given at the University of Cluj in May 1935 first and in a note of the CR then, Mr. Paul Montel demonstrated the following property^((3)){ }^{(3)}:
Any continuous function of a real variable satisfying the functional equations
{:(1)Delta_(omega_(1))^(n)=0","quadDelta_(omega_(2))^(n)=0:}\begin{equation*}
\Delta_{\omega_{1}}^{n}=0, \quad \Delta_{\omega_{2}}^{n}=0 \tag{1}
\end{equation*}
relating to two periodsomega_(4),omega_(2)\omega_{4}, \omega_{2}whose ratio is irrational is a polynomial of degreen-1n-1.
^((1)){ }^{(1)}We say that a polynomial is of degreennwhen its effective degree is at most equal tonn.
(2) See: Th. Anghelutza "On a functional equation characterizing polynomials". Mathematica t. VI (1932), p. 1-7., Tiberiu Popoviciu "On some properties of functions of one and two real variables". Mathematica t. Vlll (1934), p. 1-85, sp. p. 57.
(^(3){ }^{3}) Paul Montel "On a theorem of Jacobi" CR Acad. Sc. Paris, t. 201 (1935), p. 586.
Mr. Paul Montel also noted that in the casen=1n=1continuity at a single point is sufficient to be able to affirm that the function reduces to a constant. In the following we will extend this result to the general system (1).
^((1)){ }^{(1)}We say that a polynomial is of degreennwhen its effective degree is at most equal tonn.
(2) See: Th. Anghelutza "On a functional equation characterizing polynomials". Mathematica t. VI (1932), p. 1-7., Tiberiu Popoviciu "On some properties of functions of one and two real variables". Mathematica t. Vlll (1934), p. 1-85, sp. p. 57.
(^(3){ }^{3}) Paul Montel "On a theorem of Jacobi" CR Acad. Sc. Paris, t. 201 (1935), p. 586.
Mr. Paul Montel also noted that in the casen=1n=1continuity at a single point is sufficient to be able to affirm that the function reduces to a constant. In the following we will extend this result to the general system (1).
Note that if the functionf(x)f(x)check the equationDelta_(h)^(n)=0\Delta_{h}^{n}=0for onehhgiven it reduces to a polynomial of degreen-1n-1at equidistant pointsx,x+-h,x+-2h,dotsx, x \pm h, x \pm 2 h, \ldotsand this for any value ofxx. It follows that we also haveDelta_(ph)^(n)=0\Delta_{p h}^{n}=0whatever the integerpp.
2. - Now suppose that the functionf(x)f(x)checks the system (1),omega_(1),omega_(2)\omega_{1}, \omega_{2}being two real numbers (the periods) given. We: have
2. - Now suppose that the functionf(x)f(x)checks the system (1),omega_(1),omega_(2)\omega_{1}, \omega_{2}being two real numbers (the periods) given. We: have
Delta_(omega_(1))^(n)f(x+pomega_(1)+qomega_(2))=0.\Delta_{\omega_{1}}^{n} f\left(x+p \omega_{1}+q \omega_{2}\right)=0 .
Let's take forxxAndqqfixed values. We see that there exists a polynomialP_(q)(u)\mathrm{P}_{q}(u)of degreen-1n-1compared touusuch that
(2)
(2)
{:[f(x+pomega_(1)+qomega_(2))=P_(q)(x+pomega_(1)+qomega_(2))],[p=0","+-1","+-2","dots]:}\begin{gathered}
f\left(x+p \omega_{1}+q \omega_{2}\right)=P_{q}\left(x+p \omega_{1}+q \omega_{2}\right) \\
p=0, \pm 1, \pm 2, \ldots
\end{gathered}
Similarly, assumingxxAndppfixed we see that there exists a polynomialQ_(p)(v)Q_{p}(v)of degreen-1n-1compared tovvsuch that
(3)
(3)
{:[f(x+pomega_(1)+qomega_(2))=Q_(p)(x+pomega_(1)+qomega_(2))],[q=0","+-1","+-2","dots]:}\begin{gathered}
f\left(x+p \omega_{1}+q \omega_{2}\right)=Q_{p}\left(x+p \omega_{1}+q \omega_{2}\right) \\
q=0, \pm 1, \pm 2, \ldots
\end{gathered}
From (2) and (3) we deduce that we can find a polynomialP(u,v)\mathrm{P}(u, v); of two variablesuuAndvvof degreen-1n-1with respect to each of these variables and such that for axx-given we have
{:(4)f(x+pomega_(1)+qomega_(2))=P(pomega_(1),qomega_(2)):}\begin{equation*}
f\left(x+p \omega_{1}+q \omega_{2}\right)=\mathrm{P}\left(p \omega_{1}, q \omega_{2}\right) \tag{4}
\end{equation*}
The polynomialP(u,v)\mathrm{P}(u, v)can be written in the following form
P(u,v)=sum_(i=0)^(2n-2)A_(2n-2-i)u^(i)\mathrm{P}(u, v)=\sum_{i=0}^{2 n-2} \mathrm{~A}_{2 n-2-i} u^{i}
OrA_(i)\mathbf{A}_{i}are polynomials depending only onu+vu+vand are of degreen-1,A_(0),A_(1),dots,A_(n-1)n-1, \mathrm{~A}_{0}, \mathrm{~A}_{1}, \ldots, \mathrm{~A}_{n-1}being of degree equal to their index.
Let us now suppose that forn-1n-1valuesa_(1),a_(2),dots,a_(n-1)a_{1}, a_{2}, \ldots, a_{n-1}the constant has the polynomialP(u,v)\mathrm{P}(u, v)remains limited tou+v=au+v=a. We immediately see that for these values ​​the coefficientsA_(0),A_(1),dotsA_(2n-3)A_{0}, A_{1}, \ldots A_{2 n-3}must cancel each other out. It follows that
{:[P(u","v)=(u+v-a_(1))(u+v-a_(2))dots(u+v-a_(n-1))(b_(0)u^(n-1)+:}],[{:+b_(1)u^(n-2)+,dots,+b_(n-1))+A_(2n-2)]:}\begin{gathered}
\mathrm{P}(u, v)=\left(u+v-a_{1}\right)\left(u+v-a_{2}\right) \ldots\left(u+v-a_{n-1}\right)\left(b_{0} u^{n-1}+\right. \\
\left.+b_{1} u^{n-2}+, \ldots,+b_{n-1}\right)+\mathrm{A}_{2 n-2}
\end{gathered}
For this polynomial to have the required form, it is necessary thatb_(0)=b_(1)=,dots,=b_(n-1)=0b_{0}=b_{1}=, \ldots,=b_{n-1}=0and we then see that we have the following property:
If the polynomialP(u,v)\mathrm{P}(u, v)remains limited ton-1n-1values ​​ofu+vu+vit remains bounded for any given value ofu+vu+v, so it reduces to a: polynomial of degreen-1n-1compared tou+vu+v.
3. - Let's return to our problem. If the periodsomega^(˙)_(1),omega^(˙)_(2)\dot{\omega}_{1}, \dot{\omega}_{2}herU\mathfrak{U}dependent there are two whole numbersppAndqqsuch asomega_(4)=p omega\omega_{4}=p \omega,omega_(2)=q omega\omega_{2}=q \omegaand we see immediately that the functionf(x)f(x)check the equationDelta_(omega)^(n)=0\Delta_{\omega}^{n}=0. In this case the polynomialsP_(q),Q_(p)P_{q}, Q_{p}coincide at an infinite number of points and are therefore identical. We leave aside: this case.
3. - Let's return to our problem. If the periodsomega^(˙)_(1),omega^(˙)_(2)\dot{\omega}_{1}, \dot{\omega}_{2}herU\mathfrak{U}dependent there are two whole numbersppAndqqsuch asomega_(4)=p omega\omega_{4}=p \omega,omega_(2)=q omega\omega_{2}=q \omegaand we see immediately that the functionf(x)f(x)check the equationDelta_(omega)^(n)=0\Delta_{\omega}^{n}=0. In this case the polynomialsP_(q),Q_(p)P_{q}, Q_{p}coincide at an infinite number of points and are therefore identical. We leave aside: this case.
So let's assume that the periodsomega_(4),omega_(2)\omega_{4}, \omega_{2}are independent(^(4))\left({ }^{4}\right). All pointspomega_(4)+qomega_(2),p,qp \omega_{4}+q \omega_{2}, p, qbeing integers, is then everywhere dense on the real axis.
Now suppose that the functionf(x)f(x)be bounded at a pointx^(')x^{\prime}, which amounts to saying that it is bounded in an interval containing the pointx^(')x^{\prime}. There then exist two infinite and non-bozed sequences of integers
{:[p_(1)","p_(2)","dots","p_(nu)","dots],[q_(1)","q_(2)","dots","q_(nu)","dots]:}\begin{aligned}
& p_{1}, p_{2}, \ldots, p_{\nu}, \ldots \\
& q_{1}, q_{2}, \ldots, q_{\nu}, \ldots
\end{aligned}
such as
{:[p_(nu)omega_(1)+q_(nu)omega_(2) rarrx^(')-x],[nu rarr oo.]:}\begin{aligned}
p_{\nu} \omega_{1}+q_{\nu} \omega_{2} & \rightarrow x^{\prime}-x \\
\nu & \rightarrow \infty .
\end{aligned}
From (4) then follows that the polynomialP(u,v)\mathrm{P}(u, v)must be limited foru+v=x^(')-xu+v=x^{\prime}-x. The function being bounded at a point it is bounded atn-1n-1points. It follows that the polynomialP(u,v)\mathrm{P}(u, v)reduces to a polynomial of degreen-1n-1compared tou+vu+vTonsionf(x)f(x)coincides: so on the pointsx+pomega_(1)+qomega_(2),p,q=0.+-1,+-2,dotsx+p \omega_{1}+q \omega_{2}, p, q=0 . \pm 1, \pm 2, \ldotswith a polynomial of degreen-1n-1In particular, we have
Delta_(p_(omega_(1))+qomega_(2))^(n)=0\Delta_{p_{\omega_{1}}+q \omega_{2}}^{n}=0
regardless of the integersppAndqq. So we have the following property:
If the periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent, if the functionf(x)f(x)
(4) Numbersomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent when equalitypomega_(1)+qomega_(2)=0\boldsymbol{p} \boldsymbol{\omega}_{1}+\boldsymbol{q} \boldsymbol{\omega}_{2}=0cannot be satisfied in whole numbersp\boldsymbol{p}, Andq\boldsymbol{q}, that ifp=q=0p=q=0. It is therefore necessary and sufficient that the ratio of the numbersomega_(1),omega_(2)\omega_{1}, \omega_{2}be irrational.
Check the functional equations
(4) Numbersomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent when equalitypomega_(1)+qomega_(2)=0\boldsymbol{p} \boldsymbol{\omega}_{1}+\boldsymbol{q} \boldsymbol{\omega}_{2}=0cannot be satisfied in whole numbersp\boldsymbol{p}, Andq\boldsymbol{q}, that ifp=q=0p=q=0. It is therefore necessary and sufficient that the ratio of the numbersomega_(1),omega_(2)\omega_{1}, \omega_{2}be irrational.
Check the functional equations
Delta_(omega_(1))^(n)=0,quadDelta_(omega_(2))^(n)=0\Delta_{\omega_{1}}^{n}=0, \quad \Delta_{\omega_{2}}^{n}=0
and is bounded at a point, we also have
Delta_(p_(1)+qomega_(2))^(n)=0\Delta_{p_{1}+q \omega_{2}}^{n}=0
for any value ofxxand whatever the integersppAndqq.
If the function is not bounded at any point the property may not be true. For example the function which is equal topqp qat a point of the formpomega_(1)+qomega_(2)p \omega_{1}+q \omega_{2}and is zero everywhere else, check the system carefully
If the function is not bounded at any point the property may not be true. For example the function which is equal topqp qat a point of the formpomega_(1)+qomega_(2)p \omega_{1}+q \omega_{2}and is zero everywhere else, check the system carefully
Delta_(omega_(1))^(n)=0,quadDelta_(omega_(2))^(n)=0quad(n > 1)\Delta_{\omega_{1}}^{n}=0, \quad \Delta_{\omega_{2}}^{n}=0 \quad(n>1)
but we generally have
Delta_(pomega_(1)+qomega_(2))^(n)!=0\Delta_{p \omega_{1}+q \omega_{2}}^{n} \neq 0
-
- We are in the conditions of the previous case iff(x)f(x)is continuous at a pointx^(')x^{\prime}. In this caseP(u,v)\mathrm{P}(u, v)depends only onu+vu+vand all these polynomials must coincide at the pointx^(')x^{\prime}. These polynomials being of degreen-1n-1We can state the following property:
If the periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent, if the functionf(x)f(x)check the functional equations
Delta_(omega_(1))^(n)=0,quadDelta_(omega_(2))^(n)=0\Delta_{\omega_{1}}^{n}=0, \quad \Delta_{\omega_{2}}^{n}=0
and is continuous at n points, it reduces to a polynomial of degree n-1.
The property may not be true iff(x)f(x)is supposed to be continuous in less thannnpoints. For example, the function that takes the values ​​ofxxin every respect of the formpomega^(˙)_(4)+qomega_(2)p \dot{\omega}_{4}+q \omega_{2}and is zero everywhere else is well continuous forx=0x=0, check the equationsDelta_(omega_(1))^(n)=0\Delta_{\omega_{1}}^{n}=0,Delta_(omega_(3))^(n)=0,quad n > 1\Delta_{\omega_{3}}^{n}=0, \quad n>1and yet does not reduce to a polynomial. We can also give other examples of this nature.
5. - Let us consider the more general equation
(5)
The property may not be true iff(x)f(x)is supposed to be continuous in less thannnpoints. For example, the function that takes the values ​​ofxxin every respect of the formpomega^(˙)_(4)+qomega_(2)p \dot{\omega}_{4}+q \omega_{2}and is zero everywhere else is well continuous forx=0x=0, check the equationsDelta_(omega_(1))^(n)=0\Delta_{\omega_{1}}^{n}=0,Delta_(omega_(3))^(n)=0,quad n > 1\Delta_{\omega_{3}}^{n}=0, \quad n>1and yet does not reduce to a polynomial. We can also give other examples of this nature.
5. - Let us consider the more general equation
(5)
sum_(i=0)^(n)a_(i)f(x+ih)=0\sum_{i=0}^{n} a_{i} f(x+i h)=0
where thea_(i)a_{i}are constants. We can assumea_(0)!=0,a_(n)!=0a_{0} \neq 0, a_{n} \neq 0and that the first member of the characteristic equation
F(z)=a_(0)+a_(1)z+,dots,a_(n)z^(n)=0F(z)=a_{0}+a_{1} z+, \ldots, a_{n} z^{n}=0
does not reduce me to a polynomial with respect to an integer power> 1>1ofzz.
We say that equation (5) is of degree n. If 1 is a root of orderkkof multiplicity of the characteristic equation we say that equation (5) is of orderkk. IfF(1)!=0\mathrm{F}(1) \neq 0the equation is of order 0. An equation of order and degreennis necessarily of the form
Delta_(h)^(n)=0\Delta_{h}^{n}=0
For an order equationkkwe have
F(1)=F^(')(1)=,dots,=F^((k-1))(1)=0,F^((k))(1)!=0F(1)=F^{\prime}(1)=, \ldots,=F^{(k-1)}(1)=0, F^{(k)}(1) \neq 0
If equation (5) remains verified, with the same constantsa_(i)a_{i}, when replacinghhbymh,m=2,3,dotsm h, m=2,3, \ldots, we can find an integerr\boldsymbol{r}such as one has
{:(6)Delta_(rh)^(n)=-0:}\begin{equation*}
\Delta_{r h}^{n}=-0 \tag{6}
\end{equation*}
From the properties of equation (5) it follows that it is sufficient to take formmonly a finite number of values. These values, among whichrrstill exists, are indicated by the way in which equation (6) is obtained and which we have specified in another work (^(5){ }^{5}These values ​​can be chosen.mmin an infinite number of ways: In the following when we write an equation (5) it is implied that this equation remains verified also when we replacehhbymhm hwhere the positive integermmtakes a number of values ​​such that reduction to the form (6) is possible.
The numberhhbeing fixed, we immediately verify that any polynomial solution of equation] (5) is necessarily an arbitrary polynomial of degreek-1k-1(ifk=0k=0any solution of this nature is identically zero).
This property also takes place without the restriction imposed on the substitution ofhhbymhm hIn other words, it takes place by taking formmthe only value 1.
6. As a generalization of the problem studied we propose to examine the system
6. As a generalization of the problem studied we propose to examine the system
{:(7)sum_(i=0)^(n_(1))a_(i)f(x+iomega_(1))=0","quadsum_(i=0)^(n_(2))b_(i)f(x+iomega_(2))=0:}\begin{equation*}
\sum_{i=0}^{n_{1}} a_{i} f\left(x+i \omega_{1}\right)=0, \quad \sum_{i=0}^{n_{2}} b_{i} f\left(x+i \omega_{2}\right)=0 \tag{7}
\end{equation*}
(5) Tib. Popoviciu "On certain functional equations defining polynomials", Mathematica t. X (1935), pp. 197-211. We ask the reader to refer to this work.
formed by equations of degreen_(1),n_(2)n_{1}, n_{2}and orderk_(1),k_(2)k_{1}, k_{2}respect_(TT){ }_{\top}tively. The periods, that is to say the numbersomega_(1),omega_(2)\omega_{1}, \omega_{2}, are assumed to be independent. The numbern=max.(n_(1),n_(2))n=\max .\left(n_{1}, n_{2}\right)will be called the degree of the system "(7) and the numberk=min.(k_(1),k_(2))k=\min .\left(k_{1}, k_{2}\right)the order of this system. To fix the ideas we suppose thatk=k_(1) <= k_(2)k=k_{1} \leq k_{2}.
formed by equations of degreen_(1),n_(2)n_{1}, n_{2}and orderk_(1),k_(2)k_{1}, k_{2}respect_(TT){ }_{\top}tively. The periods, that is to say the numbersomega_(1),omega_(2)\omega_{1}, \omega_{2}, are assumed to be independent. The numbern=max.(n_(1),n_(2))n=\max .\left(n_{1}, n_{2}\right)will be called the degree of the system "(7) and the numberk=min.(k_(1),k_(2))k=\min .\left(k_{1}, k_{2}\right)the order of this system. To fix the ideas we suppose thatk=k_(1) <= k_(2)k=k_{1} \leq k_{2}.
We immediately see that we can find two integersr_(1),r_(2)r_{1}, r_{2}: such as
{:(8)Delta_(omega_(1)^('))^(n)=0","quadDelta_(omega_(2)^('))^(n)=0:}\begin{equation*}
\Delta_{\omega_{1}^{\prime}}^{n}=0, \quad \Delta_{\omega_{2}^{\prime}}^{n}=0 \tag{8}
\end{equation*}
by posingomega_(1)^(')=r_(1)omega_(1),omega_(2)^(')=r_(2)omega_(2)\omega_{1}^{\prime}=r_{1} \omega_{1}, \omega_{2}^{\prime}=r_{2} \omega_{2}. The numbersomega_(1)^('),omega_(2)^(')\omega_{1}^{\prime}, \omega_{2}^{\prime}are still independent and we can therefore state the following property:
If the periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent, if the functionf(x)f(x)checks the system of functional equations
sum_(i=0)^(n_(1))a_(i)f(x+iomega_(1))=0,quadsum_(i=0)^(n_(2))b_(i)f(x+iomega_(2))=0\sum_{i=0}^{n_{1}} a_{i} f\left(x+i \omega_{1}\right)=0, \quad \sum_{i=0}^{n_{2}} b_{i} f\left(x+i \omega_{2}\right)=0
of degreenn, of orderkkand is continuous innnpoints, it reduces to a polynomial of degreek-1k-1.
One may wonder whether one could not reduce the number of points of continuity of the function such that the conclusions remain the same. In particular, the continuity inkkpoints alone is enough to conclude that the function reduces to a polynomial?'
We will demonstrate that this is indeed the case.
7. - Equations (8) show us that if we give toxxa fixed value and if the functionf(x)f(x)is assumed to be bounded at a point. It reduces to the pointsx+pomega_(1)^(')+qomega_(2)^('),p,q=0,+-1,+-2,dotsx+p \omega_{1}^{\prime}+q \omega_{2}^{\prime}, p, q=0, \pm 1, \pm 2, \ldots. to a polynomial of degreen-1n-1. In particular, for axxgiven, the function reduces to the pointsx+somega^(˙)_(1)+pomega_(1)^(')+qomega_(2)^('),p,q=0,+-1,+-2,dotsx+s \dot{\omega}_{1}+p \omega_{1}^{\prime}+q \omega_{2}^{\prime}, p, q=0, \pm 1, \pm 2, \ldotsto a polynomialR_(s)(x)\mathrm{R}_{s}(x)of degreen-1n-1. We have thusr_(1)r_{1}polynomialsR_(s)(x)\mathrm{R}_{s}(x)which can be distinct, but we always haveR_(s)(x)=R_(s^('))(x)\mathrm{R}_{s}(x)=\mathrm{R}_{s^{\prime}}(x)s.s=s^(')(mod.r_(1))s=s^{\prime}\left(\bmod . r_{1}\right).
7. - Equations (8) show us that if we give toxxa fixed value and if the functionf(x)f(x)is assumed to be bounded at a point. It reduces to the pointsx+pomega_(1)^(')+qomega_(2)^('),p,q=0,+-1,+-2,dotsx+p \omega_{1}^{\prime}+q \omega_{2}^{\prime}, p, q=0, \pm 1, \pm 2, \ldots. to a polynomial of degreen-1n-1. In particular, for axxgiven, the function reduces to the pointsx+somega^(˙)_(1)+pomega_(1)^(')+qomega_(2)^('),p,q=0,+-1,+-2,dotsx+s \dot{\omega}_{1}+p \omega_{1}^{\prime}+q \omega_{2}^{\prime}, p, q=0, \pm 1, \pm 2, \ldotsto a polynomialR_(s)(x)\mathrm{R}_{s}(x)of degreen-1n-1. We have thusr_(1)r_{1}polynomialsR_(s)(x)\mathrm{R}_{s}(x)which can be distinct, but we always haveR_(s)(x)=R_(s^('))(x)\mathrm{R}_{s}(x)=\mathrm{R}_{s^{\prime}}(x)s.s=s^(')(mod.r_(1))s=s^{\prime}\left(\bmod . r_{1}\right).
The functionsf(x)f(x)verifies, by hypothesis, the relationship
sum_(i=0)^(n_(1))a_(i)f(x+iomega_(1)^('))=0\sum_{i=0}^{n_{1}} a_{i} f\left(x+i \omega_{1}^{\prime}\right)=0
we therefore have, for eachss,
{:(9)sum_(i=0)^(n_(1))a_(i)R_(s)(x+iomega_(1)^('))=0:}\begin{equation*}
\sum_{i=0}^{n_{1}} a_{i} R_{s}\left(x+i \omega_{1}^{\prime}\right)=0 \tag{9}
\end{equation*}
for an infinity of values ​​ofxx. It immediately follows that the equalities (9) are verified identically inxx. The polynomialR_(s)(x)\mathrm{R}_{s}(x)therefore satisfies an equation of the form (5) and of orderkk. It follows that: the polynomialsR_(s)(x)\mathrm{R}_{s}(x)are of degreek-1k-1.
Likewise, for axxgiven, the function reduces to the points.x+somega_(2)+pomega_(1)^(')+qomega_(2)^('),p,q=0,+-1,+-2,dotsx+s \omega_{2}+p \omega_{1}^{\prime}+q \omega_{2}^{\prime}, p, q=0, \pm 1, \pm 2, \ldotsto a polynomialS_(s)(x)S_{s}(x)of degreen-1n-1. We have thusr_(2)r_{2}polynomialsS_(s)(x)S_{s}(x)which can be distinct, but we always haveS_(s)(x)=S_(s^('))(x)\mathrm{S}_{s}(x)=\mathrm{S}_{s^{\prime}}(x)ifs=s^(')(mod.r_(2))s=s^{\prime}\left(\bmod . r_{2}\right). Now, we also have, for eachss,
sum_(i=0)^(n_(1))a_(i)S_(s)(x+iomega_(1)^('))=0\sum_{i=0}^{n_{1}} a_{i} \mathrm{~S}_{s}\left(x+i \omega_{1}^{\prime}\right)=0
for an infinity of values ​​ofxx. It follows that the polynomialsS_(s)(x)\mathrm{S}_{s}(x)are also of a certain degreek-1k-1.
The property stated in № 6 therefore remains true if we assume that the functionf(x)f(x)is continuous only inllpoints. In this case, in fact, all the polynomialsR_(s)(x),S_(s)(x)\mathrm{R}_{s}(x), \mathrm{S}_{s}(x)coincide.
In particular, we can state the following proposition:
If the periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent, if the functionf(x)f(x)checks: functional equations
If the periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent, if the functionf(x)f(x)checks: functional equations
{:[sum_(i=0)^(n_(1))a_(1)f(x+imomega_(1))=0","quadsum_(i=0)^(n_(2))b_(i)f(x+imomega_(2))=0],[m=1","quad2","dots]:}\begin{gathered}
\sum_{i=0}^{n_{1}} a_{1} f\left(x+i m \omega_{1}\right)=0, \quad \sum_{i=0}^{n_{2}} b_{i} f\left(x+i m \omega_{2}\right)=0 \\
m=1, \quad 2, \ldots
\end{gathered}
of order 7 and is continuous inkkpoints, it reduces to a polynomial of degreek-1k-1.
8. - The restriction relating to the substitution ofhhbymhm hin: (5) is essential.
8. - The restriction relating to the substitution ofhhbymhm hin: (5) is essential.
Consider the system
{:[(10)f(x)+f(x+omega_(1))-f(x+2omega_(1))-f(x+3omega_(4))=0],[f(x)+f(x+omega_(2))-f(x+2omega_(2))-f(x+3omega_(2))=0]:}\begin{align*}
& f(x)+f\left(x+\omega_{1}\right)-f\left(x+2 \omega_{1}\right)-f\left(x+3 \omega_{4}\right)=0 \tag{10}\\
& f(x)+f\left(x+\omega_{2}\right)-f\left(x+2 \omega_{2}\right)-f\left(x+3 \omega_{2}\right)=0
\end{align*}
which is of degree 3 and order 1. Let us define the function as follows: It is equal toxxat a point of the formpomega_(1)+qomega_(2)p \omega_{1}+q \omega_{2}Orp,qp, qare of the same parity, is equal to-x-xat a point of the formpomega_(1)+qomega_(2)p \omega_{1}+q \omega_{2}Orp,qp, qare of different parity and is zero everywhere else.
This function is continuous at the pointx=0x=0, but it does not reduce to a polynomial and is not a solution of the system (10) in our sense. On the contrary, this function verifies the two equations (10) alone.-
This same example shows us that among the values ​​ofmmit is necessary that r appears. In this case in fact we can taker_(1)=r_(2)=2r_{1}=r_{2}=2and we see that the system (8) is a consequence of the two equations (10).
There are special cases where we can greatly reduce the number of values ​​to take formm.
Note thatf(x)f(x)checks the degree equationnnwhere instead ofhhWe haverhr hand "the characteristic equation isG(z)=0\mathrm{G}(z)=0, with
G(z^(r))=F(z)F(alpha z),dots,F(alpha^(r-1)z)\mathrm{G}\left(z^{r}\right)=\mathrm{F}(z) \mathrm{F}(\alpha z), \ldots, \mathrm{F}\left(\alpha^{r-1} z\right)
a being a primitive root of orderrrof unity.
It follows that if all the roots of the characteristic equationF(z)=0\mathrm{F}(z)=0are roots of orderrrof the unit equation (6) is a consequence of the single equation (5).
It follows that if all the roots of the characteristic equationF(z)=0\mathrm{F}(z)=0are roots of orderrrof the unit equation (6) is a consequence of the single equation (5).
We can therefore state the following proposition:
The functionf(x)f(x), verifying the three functional equations
The functionf(x)f(x), verifying the three functional equations
{:sum_(i=0)^(n_(1))a_(i)f(x+iomega_(1))=0,quadsum_(i=0)^(n_(1))a_(i)f(x+iromega_(1))=0,quadsum_(i=0)^(n_(2))b_(i)f(x+iomega_(2)))\left.\sum_{i=0}^{n_{1}} a_{i} f\left(x+i \omega_{1}\right)=0, \quad \sum_{i=0}^{n_{1}} a_{i} f\left(x+i r \omega_{1}\right)=0, \quad \sum_{i=0}^{n_{2}} b_{i} f\left(x+i \omega_{2}\right)\right)
- of orderk(k=k_(1) <= k_(2))k\left(k=k_{1} \leq k_{2}\right), reduces to a polynomial of degreek-1k-1if
- The periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}are independent.
2^(0)2^{0}. All roots of the characteristic equationa_(0)+a_(1)z+dotsa_{0}+a_{1} z+\ldots.dots.+a_(n_(1))z^(n_(1))=0\ldots .+a_{n_{1}} z^{n_{1}}=0are roots of orderrrof unity and all the roots of the equationb_(0)+b_(1)z+dots+b_(n_(2))z^(n_(2))=0b_{0}+b_{1} z+\ldots+b_{n_{2}} z^{n_{2}}=0are roots of a certain order of unity. - The function is continuous inkkpoints.
If we consider the system formed by the first and third equations (withm=1m=1only) the property remains true if
1^(0)1^{0}And2^(0)2^{0}are verified
3^(0)3^{0}The function is continuous innnpoints, n being the degree of the system.
1^(0)1^{0}And2^(0)2^{0}are verified
3^(0)3^{0}The function is continuous innnpoints, n being the degree of the system.