Some generalizations of Levi-Civita functional equation

arXiv:1612.03756v1 [math.CA] 12 Dec 2016

SOME GENERALIZATIONS OF LEVI-CIVITA FUNCTIONAL EQUATION J. M. ALMIRA, E. SHULMAN

Abstract. We study the functional equation n m ÿ ÿ ai pyqvi pxq fi pbi x ` ci yq “ i“1

i“0

with x, y P Rd and bi , ci P GLd pCq, both in the classical context of continuous complex valued functions and in the framework of complex valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets of these equations are, typically, exponential polynomials and, in some particular cases, they reduce to ordinary polynomials. We also apply several results of this paper to the so called characterization problem of the normal distribution in Probability Theory.

1. Introduction Levi-Civita functional equation has the form n ÿ (1) f px ` yq “ ai pyqvi pxq, i“1

where f, ak , vk are complex valued functions defined on a semigroup pΓ, `q. This equation can be restated by claiming that τy pf q P W for all y P Γ, where W “ spantvk unk“1 is a finite dimensional space of functions defined on Γ and τy pf qpxq “ f px ` yq. If Γ “ Rd for some d ě 1, the equation (1) can be formulated also for distributions, since the translation operator τy can be extended, in a natural way, to the space DpRd q1 of Schwartz complex valued distributions. Concretely, we can define τy pf qtφu “ f tτý pφqu for all y P Rd and all test function φ. For these distributions we will also consider, in this paper, the dilation operator 1 σb pf qtφu “ f tσb´1 pφqu, | detpbq| where b P GLd pCq is any invertible matrix, φ P DpRd q is any test function, and σb´1 pφqpxq “ φpb´1 xq for all x P Rd . If Xd denotes either the set of continuous complex valued functions CpRd q or the set of Schwartz complex valued distributions DpRd q1 , and f P Xd , then it is known that τy pf q P W for all y P Γ, where W “ spantvk unk“1 is a finite dimensional 2010 Mathematics Subject Classification. Primary 43B45, 39A70; Secondary 39B52. Key words and phrases. Levi-Civita Functional equation, Polynomials and Exponential Polynomials on abelian Groups, Linear Functional Equations, Montel’s theorem, Fr´ echet’s Theorem, Characterization Problem of the Normal Distribution, Generalized Functions. 1

2

J. M. Almira, E. Shulman

subspace of Xd , if and only if f is equal, in distributional sense, to a continuous exponential polynomial (also named quasi-polynomial). Indeed, if we set M “ τ pf q “ spantf p¨ ` yq : y P Rd u, then M Ď W is finite dimensional and translation invariant, so that Anselone-Korevaar’s Theorem [9] implies that all its elements, including f pxq, are exponential polynomials. This was proved in 1913 by LeviCivita [22] for the case of ordinary continuous functions (see also [21], [23] for other proofs) Furthermore, if twk uN k“1 is a basis of the translation invariant space M , then every f P M satisfies the family of equations τy f “

N ÿ

bi pyqwi py P Rd q.

i“1

Thus, in the context of distributions, it makes sense to say that f P DpRd q1 satisfies the Levi-Civita functional equation if there exist distributions tv1 , ¨ ¨ ¨ , vm u Ď DpRd q1 and ordinary functions ai : Rd Ñ C such that, for every y P Rd τy pf q “

(2)

m ÿ

ai pyqvi .

i“1

Indeed, we can assume that spantv1 , ¨ ¨ ¨ , vm u is translation invariant of dimension m. Then Anselone-Korevaar’s theorem guarantees that v1 , .., vm and f are all of them continuous exponential polynomials. Furthermore, once this is known, we can also demonstrate that a1 , ..., am are also continuous exponential polynomials. This follows from the fact that the translation operator τy pf qpxq “ f px`yq is continuous, which implies that a1 , ¨ ¨ ¨ , am are continuous functions and then a symmetry argument (interchange x and y) will show that they are, indeed, continuous exponential polynomials. Note that when we say that two distributions are equal, this equality is in distributional sense. Hence, if a distribution u is equal to a continuous function f , this means that the distribution is an ordinary function and it is equal almost everywhere, with respect to Lebesgue measure, to f . Thus, when we claim that a distribution is a continuous exponential polynomial, we just state equality almost everywhere in Lebesgue sense. On the other hand if W is a vector subspace of Xd and f, τy pf q P W , then ∆y f “ τy pf q ´ f P W . Thus, for the additive group Γ “ Rd , Levi-Civita functional equation also takes the form ∆y f pxq P W py P Rd q,

(3)

with W a finite dimensional subspace of Xd . In this paper we study the related functional equation (4)

∆m y f “

n ÿ

ai pyqvi , y P Rd

i“1

and its generalization, the functional equation given by (5)

m ÿ

i“0

fi pbi x ` ci yq “

n ÿ

ai pyqvi pxq

i“1

n n for all x, y P Rd , where tfk um k“0 Ytai ui“1 Ytvi ui“1 are unknown functions defined on d R , and bi , ci P GLd pCq are invertible matrices. Note that equation (5) generalizes

Some generalizations of Levi-Civita functional equation

(4) since ∆m y f pxq “ 1 ¨ f pxq `

3

m ˆ ˙ ÿ m p´1qmí f pId x ` piId qyq, i i“1

where Id denotes the identity matrix of size d. We prove a Montel type theorem for the functional equation (4) and we use it to demonstrate that all continuous functions tfk um k“1 that solve (5) are exponential polynomials. We study these equations not only for ordinary functions but also in the distributional setting, where we introduce them in two different frameworks. Concretely, we consider the equation (5) in two different frameworks. Firstly, we consider it as a family of equations on DpRd q1 which depend on the parameter y P Rd . In this case the equation takes the form (6)

m ÿ

i“1

τcri y pfri q P W for all y P Rd

r where cri “ b´1 i ci , fi pxq “ fi pbi xq “ σbi pfi qpxq, and W is an n-dimensional subspace d 1 of DpR q . Moreover, the equation (4) can be formulated directly with distributions since for each h in Rd and every natural number m we have ˜ ¸ m`1 ÿ ˆm ` 1˙ m`1 m`1´k ∆h pf qtφu “ p´1q τkh pf qtφu k k“0 m`1 ÿ ˆm ` 1˙ “ p´1qm`1´k f tτ´kh pφqu k k“0 # + m`1 ÿ ˆm ` 1˙ m`1´k “ f p´1q δ´kh pφq k k“0 “

f t∆m`1 ´h pφqu.

so that we can extend the definition of the operator ∆m h to the distributional setting with the formula m ∆m h pf qtφu “ f t∆´h pφqu. Secondly, we consider it as a unique equation on DpRd ˆ Rd q1 . In this case, the equation takes the form (5) under the imposition that fi , ak , vk P DpRd q1 , bi , ci are invertible matrices, and the distributions fi pbi x ` ci yq, ak pyqvk pxq P DpRd ˆ Rd q1 are defined by * " 1 σ ´1 pϕ1 q ˚ σc´1 pϕ2 q fi pbi x ` ci yqtϕ1 pxqϕ2 pyqu “ fi i | detpbi q|| detpci q| bi and ak pyqvk pxqtϕ1 pxqϕ2 pyqu “ ak pyqtϕ2 pyquvk pxqtϕ1 pxqu for arbitrary ϕ1 , ϕ2 P DpRd q, respectively. Indeed, these last two definitions make sense because the vector subspace of DpRd ˆ Rd q spanned by the functions of the form ϕ1 pxqϕ2 pyq, with ϕ1 , ϕ2 P DpRd q is dense in DpRd ˆ Rd q. Hence, any distribution F P DpRd ˆ Rd q1 is completely determined from its values on these products [30, p. 51]. The case m “ 1 of this equation (which corresponds to our second formulation of Levi-Civita functional equation in distributional setting) has been studied, for example, by Fen¨ yo [12] and Shulman [27].

4


Note that there are important differences between equation (5), where all fi , ai and vi are assumed distributions in Rd and the equality is considered in DpRd ˆRd q1 , and equation (6), which can be written as m ÿ

(7)

i“1

τcri y pfri qpxq “

n ÿ

ai pyqvi pxq

i“1

with tvi uni“1 a basis of W Ď DpRd q1 , since, in this case, we do not assume any particular regularity property on the coefficients ai pyq (but we assume they are ordinary functions!) and for each y P Rd we consider an equation in DpRd q1 . Finally, we demonstrate that ordinary polynomials can be characterized as the solution sets of some particular cases of these equations, and we consider some applications of these results to the so called Characterization Problem of the Normal Distribution in Probability Theory. 2. Generalizations of Anselone-Koreevar’s Theorem Let us state two technical results, which are important for our arguments in this section. These results were, indeed, recently introduced by the first author, and have proved their usefulness for the study of several Montel-type theorems for polynomial and exponential polynomial functions [2, 5, 8]. We include the proofs for the sake of completeness. Definition 1. Let t be a positive integer, E a vector space, L1 , L2 , ¨ ¨ ¨ , Lt : E Ñ E pairwise commuting linear operators. Given a subspace V Ď E, we denote by ˛L1 ,L2 ,¨¨¨ ,Lt pV q the smallest subspace of E containing V which is Li -invariant for i “ 1, 2, . . . , t. Lemma 2. With the notation we have just introduced, if V is an Ln -invariant subspace of E, then the linear space rns

VL

“ V ` LpV q ` ¨ ¨ ¨ ` Ln pV q . rns

rns

is L-invariant. Furthermore VL “ ˛L pV q. In other words, VL L-invariant subspace of E containing V .

is the smallest

rns

Proof. Let v be in VL , then (8)

v “ v0 ` Lv1 ` ¨ ¨ ¨ ` Ln´1 vn´1 ` Ln vn

with some elements v0 , v1 , . . . , vn in V . By the Ln -invariance of V , we have that Ln vn “ u is in V , hence it follows Lv “ Lpv0 ` uq ` L2 v1 ` ¨ ¨ ¨ ` Ln vn´1 , rns

rns

and the right hand side is clearly in VL . This proves that VL is L-invariant. On the other hand, if W is an L-invariant subspace of E, which contains V , then Lk pV q Ď W for k “ 1, 2, . . . , n, hence the right hand side of (8) is in W . Lemma 3. Let t be a positive integer, E a vector space, L1 , L2 , ¨ ¨ ¨ , Lt : E Ñ E pairwise commuting linear operators, and let s1 , ¨ ¨ ¨ , st be natural numbers. Given a subspace V Ď E we form the sequence of subspaces (9)

rs s

V0 “ V, Vi “ pVi´1 qLii , i “ 1, 2, . . . , t .


5

If for i “ 1, 2, . . . , t the subspace V is Lsi i -invariant, then Vt is Li -invariant, and it contains V . Furthermore Vt “ ˛L1 ,L2 ,¨¨¨ ,Lt pV q and dimpV q ă 8 if and only if dimp˛L1 ,L2 ,¨¨¨ ,Lt pV qq ă 8. s

Proof. First we prove by induction on i that Vi is Lj j -invariant and it contains V for each i “ 0, 1, . . . , t and j “ 1, 2, . . . , t. For i “ 0 we have V0 “ V , which is s Lj j -invariant for j “ 1, 2, . . . , t, by assumption. Suppose that i ě 1, and we have proved the statement for Vi´1 . Now we prove it for Vi . If v is in Vi , then we have v “ u0 ` Li u1 ` ¨ ¨ ¨ ` Lsi i usi , where uj is in Vi´1 for j “ 0, 1, . . . , si . It follows for j “ 1, 2, . . . , t s

s

s

s

Lj j v “ pLj j u0 q ` Li pLj j u1 q ` ¨ ¨ ¨ ` Lsi i pLj j usi q . Here we used the commuting property of the given operators, which obviously holds for their powers, too. By the induction hypothesis, the elements in the brackets on s s the right hand side belong to Vi´1 , hence Lj j v is in Vi , that is, Vi is Lj j -invariant. As Vi includes Vi´1 , we also conclude that V is in Vi , and our statement is proved. Now we have Vt “ Vt´1 ` Lt pVt´1 q ` ¨ ¨ ¨ ` Lst t pVt´1 q , and we apply the previous lemma: as Vt´1 is Lst t -invariant, we have that Vt is Lt -invariant. Let us now prove the invariance of Vt under the operators Lj (j ă t). Since V1 is clearly L1 -invariant, by Lemma 12, an induction process gives that Vt´1 is Li -invariant for 1 ď i ď t ´ 1. Thus, if we take 1 ď i ď t ´ 1, then we can use that Li Lt “ Lt Li and Li pVt´1 q is a subset of Vt´1 to conclude that Li pVt q “ Ď

Li pVt´1 q ` Lt pLi pVt´1 qq ` ¨ ¨ ¨ ` Lst t pLi pVt´1 qq Vt´1 ` Lt pVt´1 q ` ¨ ¨ ¨ ` Lst t pVt´1 q “ Vt ,

which completes this part of the proof. Suppose that W is a subspace in E such that V Ď W , and W is Lj -invariant for j “ 1, 2, . . . , t. Then, obviously, all the subspaces Vi for i “ 1, 2, . . . , t are included in W . In particular, Vt is included in W . This proves that Vt is the smallest subspace in E, which includes V , and which is invariant with respect to the family of operators Li . In particular, Vt “ ˛L1 ,L2 ,¨¨¨ ,Lt pV q is uniquely determined by V , and by the family of the operators Li , no matter how we label these operators. The following result generalizes Anselone-Koreevar’s Theorem. Theorem 4. Assume that W is a finite dimensional subspace of Xd , th1 , ¨ ¨ ¨ , ht u Ă k Rd and h1 Z`h2 Z`¨ ¨ ¨`ht Z is dense in Rd . Assume, furthermore, that ∆m hk pW q Ď H, k “ 1, ¨ ¨ ¨ , t, for certain positive integral numbers mk and certain finite dimensional subspace H of Xd satisfying t ď

k“1

k ∆m hk pHq Ď H.

6


Then there exists a finite dimensional subspace Z of Xd which is invariant by translations and contains W Y H. Consequently, all elements of W Y H are continuous exponential polynomials. Proof. We apply Lemma 3 with E “ Xd , Li “ ∆hi , si “ mi , i “ 1, ¨ ¨ ¨ , t, and V “ W ` H, since mi mi mi i ∆m hi pW ` Hq “ ∆hi pW q ` ∆hi pHq Ď H ` ∆hi pHq Ď H Ď W ` H,

so that V Ď Z “ ˛∆h1 ,∆h2 ,¨¨¨ ,∆ht pV q and Z is a finite dimensional subspace of Xd satisfying ∆hi pZq Ď Z , i “ 1, 2, ¨ ¨ ¨ , t. Hence Z is invariant by translations, since h1 Z ` h2 Z ` ¨ ¨ ¨ ` ht Z is dense in Rd . Applying Anselone-Koreevar’s theorem, we conclude that all elements of Z (hence, also all elements of W Y H) are continuous exponential polynomials. Corollary 5. Assume that th1 , ¨ ¨ ¨ , ht u spans a dense subgroup of Rd , f P Xd and k there exist natural numbers tmk utk“1 such that, for every k P t1, ..., tu, ∆m hk f is a continuous exponential polynomial. Then f is a continuous exponential polynomial. k Proof. Let gk “ ∆m hk f be an exponential polynomial for k “ 1, ¨ ¨ ¨ , t. Then we can take, in Lemma 4, W “ spantf u and H “ τ pspantgk utk“1 q, the smallest subspace of Xd which is translation invariant and contains tgk utk“1 , since H is finite dimensional.

3. Characterizations of Exponential Polynomials with Levi-Civita type functional equations In this section we study some distributional generalizations of equation (1) which lead to new characterizations of continuous exponential polynomials. d 1 d Theorem 6. Assume that tfk um k“1 Ă DpR q and, for all y P R , m ÿ

(10)

τci y pfi q P W for all y P Rd

i“1

for an n-dimensional subspace W of DpRd q1 . If all matrices ci and ci ćj (for i ‰ j) are invertible, then fk is a continuous exponential polynomial for k “ 1, ¨ ¨ ¨ , m. Proof. We use induction with respect to the number of summands, m, appearing in the left hand side member of the equation. The case m “ 1 is known. Indeed, it is Levi-Civita functional equation in distributional setting as we already formulated in (2). Suppose that the statement is true whenever this number is strictly smaller than m. Take arbitrary h P Rd and substitute in (10) x ` h for x and y ´ c´1 1 h for y. We will get the condition (11)

m ÿ

τh`ci pyć´1 hq pfi q P τh pW q for all y P Rd . 1

i“1

Subtracting we get (12)

m ÿ

i“2

pτci y`pId ći c´1 qh pfi q ´ τci y pfi qq P W ˚ “ τh pW q ` W for all y P Rd , 1


7

and dim W ˚ ď 2n ă 8. Setting di “ Id ´ ci c´1 1 define, for fixed h, the functions gi by gi pxq “ fi px ` di hq ´ fi pxq. Then (12) will get the form m ÿ

(13)

τci y pgi q P W ˚ for all y P Rd .

i“2

By the induction hypothesis, we obtain that all functions gi are continuous exponential polynomials. The matrices di “ Id ´ ci c´1 1 are invertible by our assumptions, since ker di “ c1 kerpci ´ c1 q “ t0u. Thus, the condition “fi px ` di hq ´ fi pxq is a continuous exponential polynomial for all h” can be written as “fi px ` yq ´ fi pxq is a continuous exponential polynomial for all y” and Corollary 5 implies that fi ř are continuous exponential polynomials for i “ 2, .., m. Consequently, the sums m i“2 τci y pfi q belong to a (fixed!) finite dimensional subspace E of Xd , which does not depend on y. Thus, if we use this information in equation (10), we conclude that f1 satisfies the hypotheses of the theorem for m “ 1. Hence f1 is also a continuous exponential polynomial. This ends the proof. ``m˘ ˘ řm m´k Remark 7. Taking into account that ∆m f , y f pxq “ f ` k“1 τpkId qy k p´1q if f P Xd solves ∆m y f “

(14)

n ÿ

ak pyqvk for all y P Rd .

k“1

defor a certain linearly independent set tvk unk“1 of complex valued distributions ` ˘ m´k fined on Rd and ordinary functions tak unk“1 , then the distributions fk “ m p´1q f, k k “ 1, ¨ ¨ ¨ , m solve (15)

m ÿ

τpkId qy fk “ ´f `

k“1

n ÿ

ak pyqvk for all y P Rd ,

k“1

and iId ´ jId “ pi ´ jqId is an invertible matrix whenever i ‰ j. Hence Theorem 6 implies that all these fk are continuous exponential polynomials and, henceforth, f is a continuous exponential polynomial too. The linear independence of the vk1 s implies, then, that all vk , ak are (in distributional sense) continuous exponential polynomials too. Theorem 8. Assume that m n ÿ ÿ (16) fi px ` ci yq “ ak pyqvk pxq, i“1

k“1

d 1

where fi , ak , vk P DpR q and all ci P GLd pCq are invertible matrices. If all matrices ci ´ cj (for i ‰ j) are invertible, then fk is a continuous exponential polynomial for k “ 1, ¨ ¨ ¨ , m. To prove Theorem 8 we need firstly to demonstrate a technical lemma which should be known to experts in generalized functions, but that we include here for the sake of completeness.

8


Lemma 9. Let f P DpRd q1 , h, k P Rd , and c P GLd pCq be an invertible matrix. Then ∆ph,kq pf px ` cyqq “ p∆h`ck pf qqpx ` cyq Proof. Given ϕ1 , ϕ2 P DpRd q, we have that ∆ph,kq pf px ` cyqqtϕ1 pxqϕ2 pyqu “ f px ` cyqt∆p´h,´kq pϕ1 pxqϕ2 pyqqu “ f px ` cyqtpτ´h ϕ1 qpxqpτ´k ϕ2 pyqq ´ ϕ1 pxqϕ2 pyqu 1 rf tpτ´h ϕ1 q ˚ σc´1 pτ´k ϕ2 qu ´ f tϕ1 ˚ σc´1 pϕ2 qus “ | detpcq| 1 “ rf tpτ´h ϕ1 q ˚ τćk pσc´1 ϕ2 qu ´ f tϕ1 ˚ σc´1 pϕ2 qus | detpcq| “ ‰ 1 “ f tpτ´ph`ckq pϕ1 ˚ σc´1 pϕ2 qu ´ f tϕ1 ˚ σc´1 pϕ2 qu | detpcq| “ ‰ 1 pτph`ckq pf q ´ f qtϕ1 ˚ σc´1 pϕ2 qu “ | detpcq| “ ‰ 1 “ p∆ph`ckq pf qqtϕ1 ˚ σc´1 pϕ2 qu | detpcq| “ p∆ph`ckq pf qqpx ` cyqtϕ1 pxqϕ2 pyqu Proof of Theorem 8. We proceed by induction on m, the number of summands in the left hand side of the equation (16). As we have already noticed, the case m “ 1 of this equation is known (see, e.g., [12] and [27]). Assume the result holds true whenever we have less than m summands. Let fi , ak , vk solve (16). Apply the operator ∆ph,ć´1 hq to both sides of the equation (this is equivalent to substitute 1

´1 x by x ` h and y by y ´ c´1 1 h in the equation). Then Lemma 9 and Id ´ c1 c1 “ 0 imply that ff « m m ÿ ÿ ∆ph,ć´1 hq fi px ` ci yq “ ∆ph,ć´1 hq fi px ` ci yq 1

1

i“1

i“1 m ÿ

“

i“1 m ÿ

“

i“2 m ÿ

“

p∆hći c´1 h pfi qqpx ` ci yq 1

p∆pId ći c´1 qh pfi qqpx ` ci yq 1

gi px ` ci yq,

i“2

with gi “ ∆pId ći c´1 qh pfi q P DpRd q1 for i “ 2, ¨ ¨ ¨ , m. Hence, after applying the 1 operator ∆ph,ć´1 hq to the left hand side of the equation, we reduce in 1 the number 1 of summands in the equation. On the other hand, in the right hand side of the equation we get ¸ ˜ n n n ÿ ÿ ÿ ∆ph,ć´1 hq ak pyqvk pxq “ τć´1 h pak qpyqτh pvk qpxq ´ ak pyqvk pxq, 1

1

k“1

k“1

k“1


9

which is an expression of the form 2n ÿ

Ak pyqVk pxq

k“1

with Ak , Vk P DpRd q1 for k “ 1, ¨ ¨ ¨ , 2n. Hence we can use the induction hypothesis to conclude that gi “ ∆pId ći c´1 qh pfi q P DpRq1 is a continuous exponential polyno1 mial for i “ 2, ¨ ¨ ¨ , m. It follows from Corollary 5 that fi is a continuous exponential polynomial for i “ 2, ¨ ¨ ¨ , m, since the matrices Id ´ ci c´1 1 are invertible. The case f1 is covered by the very same arguments we used at the very end of the proof of Theorem 6. Corollary 10. Assume that (17)

m ÿ

fi pbi x ` ci yq “

i“1

n ÿ

ak pyqvk pxq,

k“1

where fi , ak , vk P DpRd q1 and all bi , ci P GLd pCq are invertible matrices. If all ´1 matrices b´1 i ci ´bj cj (for i ‰ j) are invertible, then fk is a continuous exponential polynomial for k “ 1, ¨ ¨ ¨ , m. Proof. Set fri “ σbi pfi q. A simple computation shows that fri px ` b´1 i ci yq “ fi pbi x ` ci yq,

so that equation (17) can be written as (18)

m ÿ

i“1

fri px ` b´1 i ci yq “

n ÿ

ak pyqvk pxq,

k“1

and we can apply Theorem 8 to this equation since bi , ci invertible implies that b´1 i ci ´1 ´1 is also invertible, and, by hypothesis, we also have that bi ci ´ bj cj is invertible whenever i ‰ j. This proves that fri is an exponential polynomial for every i. It follows that fi is also an exponential polynomial for every i. Remark 11. Note that, if the matrices tb1 , ¨ ¨ ¨ , bm , c1 , ¨ ¨ ¨ , cm u are pairwise commuting, then the statements below are equivalent: ´1 ‚ b´1 i ci ´ bj cj is invertible whenever i ‰ j. ‚ bi cj ´ bj ci is invertible whenever i ‰ j ´1 ´1 ´1 This is so because, if b´1 i ci ´bj cj is invertible, then bj bi pbi ci ´bj cj q “ bj ci ´bi cj is also invertible, and, for the reverse implication we can multiply bj ci ´ bi cj by ´1 ´1 ´1 ´1 b´1 to get b´1 i bj i bj pbj ci ´ bi cj q “ bi ci ´ bj cj .

Remark 12. If f, g P S 1 pRd q are tempered distributions and b, c P Gld pRq are invertible matrices, then it is easy to prove that f pbx ` cyq, f pxqgpyq P S 1 pRd ˆ Rd q, and τch pf q P S 1 pRd q for all h P Rd . Hence, the equations (10) and (16) can be studied for tempered distributions. Now, S 1 pRs q Ă DpRs q1 for all s P N, so that Theorems 6 and 8 and the fact that all continuous exponential polynomials are tempered distributions, imply that all results in this section can be applied also under the additional restriction that all distributions under consideration are tempered distributions, and the same holds for the equations studied in next section.

10


4. Characterizations of Polynomials with Levi-Civita type functional equations There are some particular cases of equation (17) which characterize continuous polynomials in Rd . Moreover, these equations are important because of their connection with the characterization problems of probability distributions by using their characteristic functions and, as we will demonstrate at the very end of this section, they also produce a generalization of Fréchet’s and Kakutani-Nagumo-Walsh’s Theorems for distributional setting. Just to start, we demonstrate the following: Theorem 13. Assume that P, Q, fi P DpRd q1 , i “ 1, ¨ ¨ ¨ , m, and bi , ci P Gld pRq ´1 are such that b´1 i ci ´ bj cj is invertible whenever i ‰ j. If m ÿ

(19)

fi pbi x ` ci yq “ P pxq ¨ 1pyq ` 1pxq ¨ Qpyq,

i“1

where the equality is understood in the sense of DpRd ˆ Rd q1 . Then P pxq and Qpyq are, in distributional sense, continuous polynomials of degree ď m. Furthermore, if Q “ 0 in (19) then the degree of P is m ´ 1 at most. Analogously, if P “ 0, in (19) then the degree of Q is m ´ 1 at most. Proof. There is no loss of generality if we assume bi “ Id for i “ 1, ¨ ¨ ¨ , m. Hence our equation takes the form m ÿ (20) fi px ` ci yq “ P pxq ¨ 1pyq ` 1pxq ¨ Qpyq, i“1

with ci invertible for all i, and ci ´ cj invertible whenever i ‰ j. Take h1 P Rd . and apply the operator ∆ph1 ,ć´1 h1 q to both sides of the equation. Then, in the left 1 hand side we get ff « m m ÿ ÿ (21) ∆ph1 ,ć´1 h1 q fi px ` ci yq “ gi px ` ci yq, 1

i“1

i“2

d 1

with gi “ ∆pId ći c´1 qh1 pfi q P DpR q for i “ 2, ¨ ¨ ¨ , m. On the other hand, in the 1 right hand side of the equation we get

∆ph1 ,ć´1 h1 q pP pxq ¨ 1pyqq “ τć´1 h1 p1qpyq ¨ τh1 pP qpxq ´ 1pyq ¨ P pxq “ ∆h1 P pxq ¨ 1pyq. 1

1

and ∆ph1 ,ć´1 h1 q p1pxq ¨ Qpyqq “ τć´1 h1 pQqpyq¨τh1 p1qpxq´Qpyq¨1pxq “ 1pxq¨∆ć´1 h1 Qpyq. 1

Hence

1

1

m ÿ

gi px ` ci yq “ ∆h1 P pxq ¨ 1pyq ` 1pxq ¨ ∆ć´1 h1 Qpyq 1

i“2

A simple iteration of the argument m times leads to the equation: ¨ ¨ ¨ ∆ć´1 h2 ∆ć´1 h1 Qpyqq, (22) 0 “ p∆hm ¨ ¨ ¨ ∆h2 ∆h1 P pxqq ¨ 1pyq ` 1pxq ¨ p∆ć´1 m hm 2

1

¨ ¨ ¨ ∆ć´1 h2 ∆ć´1 h1 Qpyq are which implies that ∆hm ¨ ¨ ¨ ∆h2 ∆h1 P pxq and ∆ć´1 m hm 1 2 both constant functions. In particular, taking hm`1 P Rd and applying the operator ∆phm`1 ,0q to both sides of the equation, we get ∆hm`1 ∆hm ¨ ¨ ¨ ∆h2 ∆h1 P pxq ¨ 1pyq “ 0


11

and, analogously, if we apply ∆p0,hm`1 q to both sides of the equation, we get ¨ ¨ ¨ ∆ć´1 h2 ∆ć´1 h1 Qpyq “ 0 1pxq ¨ ∆hm`1 ∆ć´1 m hm 1

2

Thus ¨ ¨ ¨ ∆ć´1 h2 ∆ć´1 h1 Qpyq “ 0 ∆hm`1 ∆hm ¨ ¨ ¨ ∆h2 ∆h1 P pxq “ ∆hm`1 ∆ć´1 m hm 1

2

for all h1 , ¨ ¨ ¨ , hm`1 in Rd (with equality in the sense of DpRd q1 ), and the result follows from the corresponding Fréchet’s type theorem for distributions, which is known (see, e.g., [1], [2], [5] or [7]). Note that, if we assume Q “ 0 in the hypotheses of the theorem, then equation (22) takes the form 0 “ p∆hm ¨ ¨ ¨ ∆h2 ∆h1 P pxqq ¨ 1pyq, which leads to ∆hm ¨ ¨ ¨ ∆h2 ∆h1 P pxq “ 0 and henceforth, in that case, P is a polynomial of degree ď m ´ 1. Of course, an analogous argument proves that, if P “ 0 in (19), then Q is a polynomial of degree ď m ´ 1. řm řm Remark 14. Note that, if we impose P pxq “ i“1 fi pbi xq and Qpyq “ i“1 fi pci yq, the equation (19) is transformed into m m m ÿ ÿ ÿ (23) fi pbi x ` ci yq “ fi pbi xq ¨ 1pyq ` 1pxq ¨ fi pci yq, i“1

i“1

i“1

which is a distributional version of the equation you get taking logarithms at both sides of Skitovich-Darmois functional equation: m m m ź ź ź (24) µpi pbi x ` ci yq “ µpi pbi xq µpi pci yq. i“1

i“1

i“1

Here, µpi represents the characteristic function of a probability distribution µi . If fi “ ´ log µpi for all i, then we get the equation (23) for ordinary functions. This equation is connected to the characterization of normal distributions problem. Concretely, its study leads to a proof of the following result: Theorem 15 (Ghurye-Olkin [14], [19]). Assume that Xi , i “ 1, ¨ ¨ ¨ , m are independent d-dimensional random vectors such that the linear forms L1 “ bt1 X1 ` ... ` btm Xm and L2 “ ct1 X1 ` ... ` ctm Xm are independent, with bi , ci P Gld pRq for i “ 1, ¨ ¨ ¨ , m. Then Xi is Gaussian for all i. We include here, just to introduce some extra motivation, the arguments which justify that equation (19) is connected to Ghurye-Olkin’s Theorem. If µi is the distribution function of the random vector Xi and µpi pxq “ Epeixx,Xi y q is its characteristic function, the independence of L1 and L2 implies that Epeipxx,L1 y`xy,L2 yq q “ Epeixx,L1 y eixy,L2 y q “ Epeixx,L1 y qEpeixy,L2 y q

Now, Epeipxx,L1 y`xy,L2 yq q

t

t

t

t

“ Epeipxx,b1 X1 `...`bm Xm y`xy,c1 X1 `...`cm Xm yq q “ Epeipxb1 x`c1 y,X1 y`...`xbm x`cm y,Xm yq q “ Epeixb1 x`c1 y,X1 y ¨ ¨ ¨ eixbm x`cm y,Xm y q “ Epeixb1 x`c1 y,X1 y q ¨ ¨ ¨ Epeixbm x`cm y,Xm y q m ź µpi pbi x ` ci yq, “ i“1

12


since the Xi ’s are independent. With analogous computations, we get m ź Epeixx,L1 y q “ µpi pbi xq i“1

and

Epeixx,L2 y q “

m ź

i“1

µpi pci yq.

Hence the characteristic functions µpi satisfy the Skitovich-Darmois functional equation (23). Now, the assumption that they are characteristic functions, and that matrices bi , ci are invertible, imply that, for each i, µpi doesn’t vanish anywhere (this was proved, e.g., in [14, Lemma 1]). Hence we can take logarithms at both sides of the equation to get the Levi-Civita type functional equation m m m ÿ ÿ ÿ (25) fi pbi x ` ci yq “ fi pbi xq ` fi pci yq, i“1

i“1

i“1

where fi “ ´ log µpi . Now, Theorem 13 implies that, under the additional řm assumption that b´1 ci ´ b´1 i ř j cj is invertible for all i ‰ j, the functions P pxq “ i“1 fi pbi xq m and Qpyq “ i“1 fi pci yq are polynomials. Hence the characteristic function of, for example, L1 , is of the form eP pxq with P pxq a polynomial. Then Marcinkiewicz’s theorem [19, Lemma 1.4.2] implies that P pxq is a quadratic polynomial and L1 is Gaussian. Finally, Cramér’s theorem for Rd claims that, in this case, all the random vectors Xi must be Gaussian (see [10, p. 112]). This proves Theorem 15 under the ´1 additional assumption that detpb´1 i ci ´ bj cj q ‰ 0 for all i ‰ j. The equation (25) and its applications to the so called “characterization problems” of probability distributions, has been studied with great detail, by Feld’man [13], for ordinary functions defined on quite general commutative groups. May be our contribution here is that we formulate the equation, for the very first time, in distributional setting. The equation (17) also generalizes an equation which was studied by Ghurye and Olkin in the 1960’s. Concretely, in [14, Lemma 3], they introduced the equation m ÿ (26) fi px ` ci yq “ Apx, yq ` Bpy, xq, i“1

d

where fi is defined on R and takes complex values for all i, and functions A, B are such that, for each y P Rd , Apx, yq and Bpx, yq are polynomials in the variable x with degrees ď r, s, respectively (here r, s do not depend on y). This equation has also proven to be a useful tool in the characterization problem of normal distributions (see, for example, [24, Chapter 7]). ř Now, the hypotheses just mentioned on A, B imply that, Apx, yq “ |α|ďr aα pyqxα ř and Bpy, xq “ |β|ďs bβ pxqy β for certain functions aα pyq , bβ pxq . This motivates us to study the equation (26) in distributional setting: Theorem 16. Assume that fi , aα , bβ P DpRd q1 for 1 ď i ďřm, 0 ď |α| ď r and 0 ď |β| ď s, and equation (26) is satisfied with Apx, yq “ |α|ďr xα ¨ aα pyq and ř Bpy, xq “ |β|ďs bβ pxq ¨ y β . Assume, furthermore, that all matrices ci (for all i) and ci ćj (for i ‰ j) are invertible. Then fi is (in distributional sense) an ordinary polynomial, for i “ 1, ¨ ¨ ¨ , m.


13

Let us first study the case m “ 1 of the equation: Lemma 17. Assume that c P Gld pRq and ÿ ÿ f px ` cyq “ xα ¨ aα pyq ` bβ pxq ¨ y β |α|ďr

|β|ďs

for some complex valued distributions f , aα , bβ P DpRd q1 . Then f is a polynomial. Proof. Obviously, this equation is a particular case of (17). Hence f is an exponenřm tial polynomial, which means that f “ k“1 pk pxqexλk ,xy for certain m and certain polynomials pk and complex vectors λk P Cd . Hence m ÿ ÿ ÿ (27) pk px ` cyqexλk ,x`cyy “ xα ¨ cα pyq ` dβ pxq ¨ y β “: Φpx, yq k“1

|α|ďr

|β|ďs

r`1 Given h1 , h2 P Rd , we have that ∆ph ∆s`1 p0,h2 q Φpx, yq “ 0. On the other hand, 1 ,0q

∆ph1 ,0q ppk px ` cyqexλk ,x`cyyq “ “ “ and

pk px ` cy ` h1 qexλk ,x`cy`h1 y ´ pk px ` cyqexλk ,x`cyy ´ ¯ pk px ` cy ` h1 qexλk ,h1 y ´ pk px ` cyq exλk ,x`cyy ´ ¯ exλk ,h1 y τh1 ´ 1d ppk qpx ` cyqexλk ,x`cyy ∆p0,h2 q ppk px ` cyqexλk ,x`cyyq

“ “ “ Hence 0 “ “ “

pk px ` cy ` ch2 qexλk ,x`cy`ch2y ´ pk px ` cyqexλk ,x`cyy ´ ¯ pk px ` cy ` ch2 qexλk ,ch2 y ´ pk px ` cyq exλk ,x`cyy ´ ¯ exλk ,ch2 y τch2 ´ 1d ppk qpx ` cyqexλk ,x`cyy .

r`1 ∆p0,h ∆s`1 p0,h2 q Φpx, yq 2q m ´ ¯ ÿ r`1 s`1 xλk ,x`cyy ∆p0,h ∆ p px ` cyqe k q p0,h q 2 2

k“1 m ´ ÿ

exλk ,h1 y τh1 ´ 1d

k“1

Hence

m ´ ÿ

k“1

exλk ,h1 y τh1 ´ 1d

¯s`1 ´

¯s`1 ´

exλk ,ch2 y τch2 ´ 1d

exλk ,ch2 y τch2 ´ 1d

¯r`1

¯r`1

ppk qpx ` cyqexλk ,x`cyy

ppk qpxqexλk ,xy “ 0,

since, for any distribution u and any invertible matrix c P Gld pRq, we have that upx ` cyq “ 0 as an element of DpRd ˆ Rd q1 if and only if u “ 0 as an element of DpRd q1 . We can assume that vectors λk , k “ 1, ¨ ¨ ¨ , m are pairwise distinct and all pk are different from 0. Then [28, Lemma 4.3] (see also [29, Theorem 5.10]) implies that ´ ¯s`1 ´ ¯r`1 exλk ,h1 y τh1 ´ 1d ppk qpxq “ 0, k “ 1, ¨ ¨ ¨ , m. exλk ,c1 h2 y τch2 ´ 1d

14


for all h1 , h2 P Rd . In particular, taking h2 “ c´1 h1 , we have that ´ ¯s`r`2 ppk qpxq “ 0, k “ 1, ¨ ¨ ¨ , m, exλk ,h1 y τh1 ´ 1d

for all h1 P Rd . Consider the polynomial pk and the variety it generates: Vk “ τ ppk q “ spantpk px ` αq : α P Rd u, which is a translation invariant space of finite dimension. Then for any operator L : Vk Ñ Vk which commutes with translations we have that L “ 0 if and only if Lppk q “ 0. Moreover, τh : Vk Ñ Vk defines an automorphism for every h P Rd , and there exists an integral number M ą 0 such d M M that ∆M h pk “ 0 for all h P R , since pk is a polynomial. But ∆h “ pτh ´ 1d q M implies that pz ´ 1q is a multiple of the minimal polynomial associated to the operator τh : Vk Ñ Vk . This implies that the only eigenvalue of τh is 1 and the minimal polynomial of τh is of the form pz ´ 1qmphq for some mphq ą 1, for all ` ˘s`r`2 h ‰ 0 (since τh is different from the identity). Hence, exλk ,h1 y z ´ 1 must xλk ,h1 y be a multiple of pz ´ 1q for every h1 . In particular, e “ 1, for all h1 , which implies that λk “ 0. Hence k “ 1, λ1 “ 0, and f is a polynomial. Proof of Theorem 16. Lemma above proves the case m “ 1. Assume m ą 1 and apply the operator ∆ph1 ,c´1 h1 q to both sides of the equation. This transforms it 1 into a similar one, with m ´ 1 summands, m ÿ ÿ ÿ gi px ` ci yq “ xα ¨ a˚α pyq ` b˚β pxq ¨ y β (28) i“2

|α|ďr

|β|ďs

d 1

with gi “ ∆pId ći c´1 qh1 pfi q P DpR q for i “ 2, ¨ ¨ ¨ , m. Thus, we can apply the 1 operator ∆ph2 ,c´1 h2 q to both sides of the new equation to reduce again by one the 2 number of summands, m ÿ ÿ ÿ β ∆pId ći c´1 qh2 ∆pId ći c´1 qh1 pfi qpx`ci yq “ (29) xα ä˚˚ b˚˚ α pyq` β pxqÿ , 1

2

i“3

|α|ďr

|β|ďs

and, iterating the process, we get an equation with just one summand in the left hand side member: ∆pId ćm c´1 qhm´1 ¨ ¨ ¨ ∆pId ćm c´1 qh2 ∆pId ćm c´1 qh1 pfm qpx ` cm yq 1 2 m´1 ÿ ÿ α ˚˚˚ ˚˚˚ β “ x ¨ aα pyq ` bβ pxq ¨ y . |α|ďr

|β|ďs

Applying case m “ 1 we conclude that ∆pId ćm c´1

m´1 qhm´1

¨ ¨ ¨ ∆pId ćm c´1 qh2 ∆pId ćm c´1 qh1 pfm q 1

2

is a polynomial for all h1 , ¨ ¨ ¨ , hm´1 P Rd . In particular, fm satisfies the distribu`1 tional version of Fréchet’s functional equation ∆N fm “ 0, for all h and for certain h N . Hence fm is a polynomial (see again [1], [2], [5] or [7]). A simple permutation in the summation process, before applying the very same arguments above, produces the very same result for fi for all i. Remark 18. Note that the equation m ÿ ÿ ÿ fi px ` ci yq “ xα ¨ aα pyq ` bβ pxq ¨ y β i“1

|α|ďr

|β|ďs


15

generalizes (19) (It coincides, when r “ s “ 0). Hence Theorem 13 follows, with a completely different proof, as a direct corollary of Theorem 16, since, if Φpx, yq “ P pxq ¨ 1pyq ` 1pxq ¨ Qpxq is a polynomial in Rd ˆ Rd , then both P and Q are polynomials in Rd . Remark 19. A similar result to Theorem 16 was already proved by Ghurye and Olkin for continuous functions. Indeed, for d ą 1, Theorem 16 generalizes to distributional setting, with a simpler proof, the particular case we get from [14, Lemma 3] when we impose the additional condition detpci ´ cj q ‰ 0 for i ‰ j. For d “ 1, see [14, Lemma 4]. d d Given f P DpRd q1 , we can think of F “ ∆m y f pxq as a distribution in R ˆ R . To m distinguish this from the standard interpretation of the operator ∆y , we introduce a new notation: m ˆ ˙ ÿ m r m f pxq “ f pxq ¨ 1pyq ` ∆ p´1qmí f pId x ` piId qyq, y i i“1

and then we can prove the following generalization of Fréchet’s theorem: Theorem 20 (Fréchet’s type theorem). Let f P DpRd q1 be such that m ˆ ˙ ÿ m m r (30) ∆y f pxq “ f pxq ¨ 1pyq ` p´1qmí f pId x ` piId qyq “ 0, i i“1

where the equality is understood in the sense of DpRd ˆRdq1 . Then f is equal almost everywhere to a continuous polynomial function in Rd of total degree ď m ´ 1. ` ˘ Proof. Use Theorem 13 with P “ f , Q “ 0, fi “ mi p´1qm`1í f , bi “ Id , and ´1 ci “ iId , i “ 1, ¨ ¨ ¨ , m, which obviously satisfy that b´1 i ci ´ bj cj “ pi ´ jqId is invertible whenever i ‰ j. We end this section with a commentary about harmonic polynomials of degree at most N (i.e, real and imaginary parts of complex polynomials ppzq “ a0 ` a1 z ¨ ¨ ¨ ` aN z N ). S. Kakutani and M. Nagumo [20] and J. L. Walsh [32] introduced, in the 1930’s, the functional equation (31)

N ´1 1 ÿ f pz ` wk hq “ 0, for all z, h P C, N k“0

where w is any primitive N -th root of 1. This equation was extensively studied by S. Haruki [16, 17, 18] in the 1970’s and 1980’s. Its continuous solutions f : C Ñ R are harmonic polynomials of degree at most N and, if we restrict our attention to solutions f : C Ñ C which are holomorphic, the equation characterizes the complex polynomials of degree at most N . Now we can demonstrate the following Theorem 21. Assume that f P DpR2 q1 is a solution of the equation (31), which we formulate as an identity in DpR2 ˆ R2 q1 . Then f is equal, in distributional sense, to an harmonic polynomial of degree at most N . Proof. Under these hypotheses, Theorem 13 can be applied to f , since the linear operator La : C Ñ C given by La pzq “ az is invertible as soon as a P Czt0u and, if w is a primitive N -th root of 1 and 0 ď k, l ď N ´ 1, k ‰ l, then wk ´ wl ‰ 0. Hence f is a polynomial in R2 which satisfies the Kakutani-Nagumo-Walsh equation (31) and, henceforth, is an harmonic polynomial.

16


5. Another characterization of polynomials In this section we introduce another variation of the distributional version of Fréchet’s Theorem, different in spirit of Theorem 20 above, which represents, up to our knowledge, a new characterization of ordinary polynomials which is, furthermore, stronger than the original Fréchet’s theorem. Let f P DpRd q1 be a complex valued distribution defined on Rd , and let qpzq “ a0 ` a1 z ` ¨ ¨ ¨ ` an z n be a polynomial of one variable. We define, for each y P Rd , the new distribution n ÿ qpτy qpf q “ ak pτy qk pf q k“0

which, in case of f being an ordinary function, can be written as qpτy qpf qpxq “

n ÿ

ak f px ` kyq.

k“0

The following result is an improvement of [4, Theorem 2.2.]: Theorem 22. Assume d ą 1. Let f be a complex valued distribution defined on Rd . Assume that qpτy qpf q “ 0 for all y P Rd with }y} “ δ, for certain δ ą 0 and certain polynomial qpzq “ a0 ` a1 z ` ¨ ¨ ¨ an z n , with an ‰ 0. Then f is, in distributional sense, an ordinary polynomial. In particular, if f is a continuous function, then it is an ordinary polynomial. Let us first prove two technical Lemmas: Lemma 23. Consider on Rd the Euclidean norm, the sphere Sd pδq “ tx P Rd : }x} “ δu, and the ball Bd pδq “ tx P Rd : }x} ď δu. If d ą 1, then Bd p2δq “ Sd pδq ´ Sd pδq Proof. The inclusion Sd pδq ´ Sd pδq Ď Bd p2δq follows directly from triangle inequality, for any norm we consider on Rd . Let us demonstrate the other inclusion. Take x P Bd p2δq. If }x} “ 2δ then x “ x2 ´ ´x 2 and we are done. If }x} ă 2δ, take Γ any plane passing through the origin of coordinates and containing x as an element (this can be done because d ą 1). The inequality }x} ă δ ` δ, and the fact that we are considering the Euclidean norm, imply that in this plane there exists a triangle one of whose sides is x and the other two sides have both length δ. If Ľ Ľ ´ OP Ľ and O, P, Q are the vertices of such triangle, with x “ P Q, then x “ OQ Ľ } “ }OQ} Ľ “δ }OP

Lemma 24. Assume d ą 1. Then for any δ ą 0 there exist points th1 , ¨ ¨ ¨ , hs u Ă Sd pδq in the sphere of Rd of radius δ, which span a dense additive subgroup of Rd .

Proof. It follows directly from a well known theorem by Kronecker [15, Theorem 442, page 382] (see also [3]) that every open neighborhood of 0 “ p0, 0, ¨ ¨ ¨ , 0q P Rd contains a finite set of vectors ty1 , ¨ ¨ ¨ , yt u which span a dense subgroup of Rd . In particular, there exist ty1 , ¨ ¨ ¨ , yt u Ď Bd p2δq such that y1 Z ` ¨ ¨ ¨ ` yt Z is dense in Rd . On the other hand, d ą 1 implies that Bd p2δq “ Sd pδq ´ Sd pδq, so that we can take h1 , h2 , ¨ ¨ ¨ , h2t P Sd pδq such that yi “ h2i ´ h2i´1 , i “ 1, ¨ ¨ ¨ , t. Then h1 Z ` ¨ ¨ ¨ ` h2t Z is dense in Rd .


17

Proof of Theorem 22. Let th1 , ¨ ¨ ¨ , hs u Ă Sd pδq be such that they span a dense subgroup of Rd (these vectors exist thanks to Lemma 24). Then qpτhi qpf q “ 0 implies that dim spantf, τhi pf q, ¨ ¨ ¨ , pτhi qn pf qu ď n, for i “ 1, ¨ ¨ ¨ s, and we can apply Theorem 2.1 from [3] to claim that f is, in distributional sense, an exponentialř polynomial. Thus, f “ ri“1 pi pxqexλi ,xy (with equality in distributional sense) for certain complex vectors λi P Cd and certain polynomials pi (we assume λi ‰ λj for i ‰ j). Let us now demonstrate that r “ 1 and λ1 “ 0. By hypothesis, qpτy qpf q “ 0 for all y with }y} “ δ. This means that, for all y P Sd pδq, 0

“ qpτy qp r ÿ

“

i“1

˜

r ÿ

i“1 n ÿ

k“0 ˜ r n ÿ ÿ

“

i“1 r ÿ

“

pi pxqexλi ,xy q “

r ÿ

i“1

qpτy qppi pxqexλi ,xy q ¸

ak pτy qk ppi pxqexλi ,xy q ak pe

xλi ,yy k

¸

q pτy q ppi qpxq exλi ,xy k

k“0

Qi,y pτy qppi qpxqexλi ,xy ,

i“1

with

Qi,y pzq “

n ÿ

ak pexλi ,yy qk z k , i “ 1, ¨ ¨ ¨ , r.

k“0

It follows that, for all y P Sd pδq, Qi,y pτy qppi q vanishes identically (see [28, Lemma 4.3 ]). Let us assume that λi ‰ 0 and consider the polynomial pi and the variety it generates: Vi “ τ ppi q “ spantpi px ` αq : α P Rd u, which is a translation invariant space of finite dimension. The arguments used in the proof of Lemma 17 for the study of the translation operator τy : Vi Ñ Vi tell us that the minimal polynomial of this operator is of the form pz ´ 1qmpyq for some mpyq ą 1. Now, Qi,y pτy qppi q “ 0 means that Qi,y ppτy q|Vi q “ 0, which implies that Qi,y pzq is necessarily a multiple of pz ´ 1qmpyq . In particular, (32)

n ÿ

ak pexλi ,yy qk “ Qi,y p1q “ 0 for all y P Sd pδq.

k“0

Choose y0 , ¨ ¨ ¨ , yn P Sd pδq such that ρt ‰ ρl for all t ‰ l, where ρt “ exλi ,yt y (this can be done because d ą 1 and λi ‰ 0). Then (32), used with these values, leads to the linear system of equations fi fi » fi» » 0 a0 1 ρ0 ¨ ¨ ¨ pρ0 qn — 1 ρ1 ¨ ¨ ¨ pρ1 qn ffi — a1 ffi — 0 ffi ffi ffi — ffi— — (33) ffi — .. ffi “ — .. ffi , — .. .. .. .. – fl – fl – . . . fl . . . 1 ρn

¨¨¨

pρn qn

an

0

18


which imply a0 “ a1 “ ¨ ¨ ¨ “ an “ 0 since Vandermonde’s determinant is different from zero because all ρi ’s are different. This of course contradicts an ‰ 0. Thus λi “ 0 necessarily, which ends the proof. Remark 25. Note that d ą 1 is necessary in Theorem 22, since S1 pδq “ t´δ, δu and, for example, if we set qpzq “ z ´ 1 and f pxq “ cosp 2πix δ q, then qpτ˘δ qpf qpxq “ p∆˘δ f qpxq “ 0, but f is not a polynomial. On the other hand, for d “ 1, if f P DpRq1 and qpzq “ a0 ` a1 z ` ¨ ¨ ¨ ` an z n is a polynomial with an ‰ 0 such that qpτy qpf q “ 0 for all y P U , for a certain open set U Ď R. Then U contains two elements h1 , h2 such that h1 {h2 R Q, and the very same arguments used for the proof of Theorem 22 guarantee that f is (in distributional sense) an ordinary polynomial, since h1 {h2 R Q guarantees that th1 , h2 u span a dense subgroup of R, and Theorem 2.1 from [3] can be used also with d “ 1. Furthermore, for the last part of the proof, we will just need to find, given n P N and λi any non-zero complex number, a set of points ty1 , ¨ ¨ ¨ , yn u Ă U such that eλi yt ‰ eλi yl for all t ‰ l, and this can (obviously) be done for any open set U Ď R. 6. Representations of Abelian groups The standard representation of pRd , `q in the vector space CpRd q is given by T : Rd Ñ AutpCpRd qq, T phq “ τh . In this case, the T -invariant subspaces V of CpRd q are those satisfying T phqpV q “ τh pV q Ď V for all h P Rd (i.e., they are just the translation invariant subspaces ) and, for the case dimpV q ă 8, AnseloneKorevaar’s theorem states that all their elements are exponential polynomials. Let G be an arbitrary topological group and let us consider any continuous representation T : G Ñ AutpCpGqq (e.g., we can think of the one defined by T pgqpf qpxq “ f pxgq). Obviously the vector space V Ď CpGq is T -invariant if and only if it is ∆T -invariant,where ∆T : G Ñ AutpCpGqq is given by p∆T qphq “ T phq ´ I, where I represents the identity operator Ipf q “ f . This is so because V is a vector subspace, so that T phqpf q, f P V imply T phqpf q ´ f “ pT phq ´ Iqpf q P V and, reciprocally, if f, pT phq ´ Iqpf q P V , then T phqpf q “ f ` pT phq ´ Iqpf q P V . We can introduce the map p∆T qm : G Ñ AutpCpGqq, defined inductively by p∆T qm`1 phq “ p∆T qphqp∆T qm phq, and say that V Ă CpGq is p∆T qm -invariant if p∆T qm phqpV q Ď V for all g P G. It is easy to check that, if th1 , ¨ ¨ ¨ , ht u generates a dense subgroup of G, then V Ď CpGq is T -invariant (or, what is the same, ∆T -invariant) if and only if T phi qpV q Ď V , i “ 1, ..., t (analogously, it is enough that p∆T qphi qpV q Ď V , i “ 1, ..., t). On the other hand, if G is Abelian, then T phgq “ T pghq and this implies that p∆T qpgqp∆T qphq

“ “

pT pgq ´ IqpT phq ´ Iq “ T pgqT phq ´ T pgq ´ T phq ` I T pghq ´ T pgq ´ T phq ` I “ T phgq ´ T phq ´ T pgq ` I

“ “

T phqT pgq ´ T phq ´ T pgq ` I “ pT phq ´ IqpT pgq ´ Iq p∆T qphqp∆T qpgq.

Hence we can use Lemmas 2, 3 to demonstrate, in this context, the following Proposition 26. Let G be a topological Abelian group and let T : G Ñ AutpCpGqq be a continuous representation. Assume that W is a finite dimensional subspace of CpGq and the elements th1 , ¨ ¨ ¨ , ht u Ă G generate a dense subgroup of G. Assume,


19

furthermore, that p∆T qm phk qpW q Ď H, k “ 1, ¨ ¨ ¨ , t, for certain finite dimensional space H of CpGq satisfying t ď

p∆T qm phk qpHq Ď H.

k“1

Then there exists a finite dimensional subspace Z of CpGq which is T -invariant and contains W Y H. Proof. We apply Lemma 3 with E “ CpGq, Li “ p∆T qphi q, i “ 1, ¨ ¨ ¨ , t, and V “ W ` H, since p∆T qm phi qpW ` Hq “ Ď

p∆T qm phi qpW q ` p∆T qm phi qHq H ` p∆T qm phi qpHq Ď H Ď W ` H,

so that V Ď Z “ ˛m p∆T qph1 q,p∆T qph2 q,¨¨¨ ,p∆T qpht q pV q and Z is a finite dimensional subspace of CpGq satisfying p∆T qphi qpZq Ď Z , i “ 1, 2, ¨ ¨ ¨ , t. Hence Z is T invariant, since th1 , ¨ ¨ ¨ , ht u generates a dense subgroup of G. Theorem 27. Let G be a topological Abelian group with representation T : G Ñ AutpCpGqq. Assume that Γ “ h1 Z ` ¨ ¨ ¨ ` ht Z is a dense subgroup of G and f P CpGq satisfies p∆T qm phk qpf q P H, k “ 1, ¨ ¨ ¨ , t; for a certain finite dimensional T -invariant subspace H of CpGq. Then f belongs to some finite dimensional T -invariant subspace of CpGq. Proof. Just take, in Proposition 26, W “ spantf u.

In [26, Lemma 2] the following result was proved: Proposition 28. Let T be a continuous representation of a topologically finitely generated semigroup G on a topological linear space X. Suppose that a vector x P X and a finite-dimensional subspace L Ă X have the property that for any g P G, there is a finite-dimensional invariant subspace Rpgq Ă X with T pgqpxq P L`Rpgq . Then x belongs to a finite dimensional T -invariant subspace of X. We use Proposition 28 to demonstrate the following result, which, for the case of continuous functions on Rd , is stronger than Theorem 6 above. d Theorem 29. Assume that r P N, tfk um k“1 Ă CpR q, W is a finite dimensional d d subspace of CpR q and, for each y P R , there is a finite dimensional translation invariant space Rpyq Ă CpRd q such that

(34)

m ÿ

τci y pfi q P W ` Rpyq,

i“1

where the matrices ci and ci ćj (for i ‰ j) are invertible. Then fi is an exponential polynomial for i “ 1, ¨ ¨ ¨ , m. Proof. We proceed by induction on m. The case m “ 1 is solved by Proposition 28, since c1 is invertible. Take m ą 1 and assume that W “ span tvi uni“1 is a finite dimensional subspace of CpRd q, and for each y P Rd there exists a finite dimensional

20


translation invariant subspace Rpyq of CpRd q, such that (34) is satisfied. Then, if tvk unk“1 is a basis of W , for each y P Rd , we have that m ÿ

(35)

τci y pfi q “

i“1

n ÿ

ak pyqvk ` ry

k“1

for certain coefficients ak pyq, and some function ry P Rpyq. Hence, given h P Rd , if we substitute y by y ´ c´1 1 h in (35), we get (36)

m ÿ

τci pyć´1 hq pfi q “ 1

i“1

n ÿ

ak py ´ c´1 1 hqvk ` ryć´1 h 1

k“1

Apply the translation operator τh to both sides of (36) (which is equivalent to substituting x by x ` h in the equation), to obtain řm pfi q i“1 τh`ci pyć´1 1 hq řn (37) ´1 “ k“1 ak py ´ c1 hqτh pvk q ` τh pryć´1 h q P τh pW q ` Rpy ´ c´1 1 hq, 1

c´1 1 hq

since τa`b “ τa τb and Rpy ´ is translation invariant. Subtracting (35) from (37) we get (38)

m ÿ

pτci y`pId ći c´1 qh pfi q ´ τci y pfi qq P W1 ` R1 pyq, 1

i“2

where W1 “ τh pW q ` W has finite dimension ď 2n and R1 pyq “ Rpyq ` Rpy ´ c´1 1 hq is translation invariant with finite dimension d dim R1 pyq ď dim Rpyq ` dim Rpy ´ c´1 1 hq ă 8 for all y P R .

Setting di “ Id ći c´1 1 define the functions gi by gi pxq “ fi px`di hq´fi pxq “ ∆di h fi . Then (38) will get the form (39)

m ÿ

τci y pgi q P W1 ` R1 pyq, for all y P Rd .

i“2

Thus, the induction step confirms us that all functions gi “ ∆di h fi are exponential polynomials. Furthermore, this holds for arbitrary h P Rd , and the matrices di are all of them invertible, since ci , ci ´ cj are invertible matrices for all i ‰ j. Hence Proposition 26 implies řm that fi is an exponential polynomial for i “ 2, ¨ ¨ ¨ , m. Consequently, the sums i“2 τci y pfi q belong to a finite dimensional subspace E of CpRd q, which does not depend on y. Thus, if we use this information in equation (34), we conclude that f1 satisfies the hypotheses of the theorem for m “ 1. Hence f1 is also an exponential polynomial. This ends the proof. Corollary 30. Let f P CpRd q be such that, for all h P Rd , ∆m h f P L ` Rphq with L, Rphq vector subspaces of CpRd q, dim L ă 8 and Rphq translation invariant with dim Rphq ă 8 for all h. Then f is an exponential polynomial. Corollary 31. Let N P N be a natural number, n ě 3. Let w be a primitive N -th řN ´1 root of 1 and let f P CpCq be such that, for all h P C, N1 k“0 f pz `wk hq P L`Rphq with L, Rphq vector subspaces of CpCq, dim L ă 8 and Rphq translation invariant with dim Rphq ă 8 for all h. Then f is an exponential polynomial.


21

7. Anselone-Koreevar’s type theorem for closed subgroups of Rd In this section we demonstrate that density of G “ h1 Z ` h2 Z ` ¨ ¨ ¨ ` ht Z in Rd is a necessary hypothesis in Corollary 5. Moreover, under the hypothesis that this group G is not a dense subgroup of Rd , we characterize the continuous functions f satisfying that, for a certain finite dimensional space H Ď CpRd q and certain natural Ťt nk k numbers nk , mk , k “ 1, ¨ ¨ ¨ , t, the relations k“1 ∆m hk pHq Ď H and ∆hk f P H, k “ 1, ¨ ¨ ¨ , t, hold. For d “ 1, the condition that th1 , ...., ht u spans a dense subgroup of Rd can be resumed to: t ě 2 and hi {hj R Q for some i ‰ j. We prove that, if h1 {h2 P Q and m ě 1 then there are finite dimensional subspaces V, H of CpRq such that m m ∆m h1 pV q Y ∆h2 pV q Ď H, H is ∆hk -invariant (indeed, it can be chosen translation invariant), and no finite dimensional translation invariant subspace W of CpRq satisfies V Ď W . To construct these spaces we need to use the following technical results: Lemma 32. Assume that f : R Ñ C satisfies ∆m h f “ 0, and let p P Z. Then ∆m f “ 0. ph Proof. For m “ 1 the result is trivial, since the periods of the function f form an additive subgroup of R. For m ě 2 the result follows from commutativity of the composition of the operators ∆h (i.e., we use that ∆h ∆k f “ ∆k ∆h f ). Concretely, m´1 ∆m f q “ 0 implies that h is a period of ∆m´1 f . Hence ph is also a h f “ ∆h p∆h h period of this function and ∆ph p∆m´1 f q “ 0. Now we use that ∆ph p∆m´1 fq “ h h m ∆h p∆ph ∆m´2 f q and iterate the argument several times to conclude that ∆ ph f “ h 0. Lemma 33. Let h ą 0 and assume that g P CpRq satisfies gphZq “ t0u. Then there exists f P CpRq such that f phZq “ t0u and ∆h f “ g. Consequently, for each m ě 1 there exists Fm P CpRq such that Fm phZq “ t0u and ∆m h Fm “ g. Proof. Given g P CpRq satisfying $ ř k´1 ’ & j“0 gpx ` jhq ř f pzq “ ´ kj“1 gpx ´ jhq ’ % 0

gphZq “ t0u, it is easy to check that the function if z “ x ` kh, k P Nzt0u, and x P r0, hq if z “ x ´ kh, k P Nzt0u, and x P r0, hq if z P r0, hq

is continuous and satisfies f phZq “ t0u and ∆h f “ g. The second claim of the lemma follows by iteration of this argument. Example 34. Let us now assume that h1 , h2 ą 0, h1 {h2 P Q , and m P N (m ě 1). Obviously, there exists h ą 0 and p, q P N such that h1 “ ph and h2 “ qh. Consider the function φ P CpRq defined by φpxq “ |x| for |x| ď h{2 and φpx ` hq “ φpxq for all x P R and use Lemma 33 with g “ φ to construct, for each m ě 2, a function fm P CpRq such that ∆m´1 fm “ φ. Take f1 “ φ. Then ∆m h fm “ ∆h φ “ 0 for all h m. This, in conjunction with Lemma 32, implies that the space Vm “ spantfm u m satisfies ∆m h1 pVm q Y ∆h2 pVm q “ t0u. Obviously, H “ t0u is translation invariant. On the other hand, Vm is not a subset of any finite dimensional translation invariant subspace of CpRq, since fm P Vm is not an exponential polynomial (indeed, it is not a differentiable function). Let d be a positive integer. If G denotes the additive subgroup of Rd generated by the elements th1 , ¨ ¨ ¨ , ht u, then it is well-known [31, Theorem 3.1] that G, the

22


topological closure of G with the Euclidean topology, satisfies G “ V ‘ Λ, where V is a vector subspace of Rd and Λ is a discrete additive subgroup of Rd . Furthermore, the case when G is dense in Rd , or, what is the same, the case whenever V “ Rd , has been characterized in several different ways (see e.g., [15, Theorem 442, page 382], [31, Proposition 4.3]). In the next proposition we prove that, under the notation we have just introduced, if V is a proper vector subspace of Rd , there are k continuous functions f satisfying that ∆m hk f is a continuous exponential polynomial for k “ 1, 2, ¨ ¨ ¨ , t but f is not an exponential polynomial. Proposition 35. Let t be a positive integer, let n1 , n2 , . . . , nt be natural numbers, and assume that the subgroup G of Rd generated by th1 , h2 , . . . , ht u satisfies G “ V ‘ Λ, where V is a proper vector subspace of Rd , and Λ is a discrete additive subgroup of Rd . Then there exist H linear finite dimensional subspace of CpRd q satisfying t ď ∆hk pHq Ď H, k“1

and ϕ : Rd Ñ R, continuous function satisfying ∆nhkk ϕ P H

for k “ 1, ¨ ¨ ¨ , t, such that ϕ is not an exponential polynomial on Rd . Proof. Obviously, we can take Vr a vector subspace of Rd of dimension d ´ 1 that contains V , such that Vr ` Λ is not dense in Rd . Observe that for every h in Rd , Vr ` h is an affine subspace of Rd which is parallel to Vr . Hence, if w0 is a unitary normal vector to Vr , then for every h in Rd , Vr ` h “ Vr ` sphqw0 for a certain real number sphq which depends on h. Indeed, if PVr : Rd Ñ Rd denotes the orthogonal projection of Rd onto Vr , then every vector h P Rd admits a unique decomposition h “ PVr phq ` sphqw0 and, obviously, Vr ` h “ Vr ` pPVr phq ` sphqw0 q “ Vr ` sphqw0

Furthermore, as a direct consequence of the linearity of PVr and the uniqueness of the decompositions h “ PVr phq ` sphqw0 , we have that, for all h1 , h2 P Rd , sph ` h2 q “ sph1 q ` sph2 q. Hence Vr ` Λ can be decomposed as a union of sets Ť 1 r sPΓ V ` sw0 with Γ “ tsphq : h P Λu a subgroup of R which is finitely generated since Λ is finitely generated. Moreover, Γ is not dense in R because Vr ` Λ is not dense in Rd . It follows that Γ “ rZ for certain positive real number r. Hence Vr ` Λ “ Vr ` rw0 Z

Take h “ rw0 , m “ mintnk utk“1 and consider the function fm constructed in Example 34 with h1 “ 1 and h2 “ 2. If we take epxq any exponential polynomial r “ τ pspanteuq the smallest translation invariant subspace of CpRd q which on Rd , H contains teu, and we define ϕpx ` shq “ epxq ` fm psq for all x P Vr and all s P R. Let us consider the linear space r H “ tf pPVr pzqq : f P Hu.

r and let v P Vr be fixed. Then Take F pzq “ f pPVr pzqq with f P H

F pz ` vq “ f pPVr pz ` vqq “ f pPVr pzq ` vq “ pτv f qpPVr pzqq P H


r and since τv f P H,

23

F pz ` hq “ f pPVr pz ` hqq “ f pPVr pzqq P H

It follows that ∆h pHq Ď H and ∆v pHq Ď H for all v P Vr . Consequently, if we define pk P Z by pk r “ sphk q, for k “ 1, ¨ ¨ ¨ , t, then t ď

∆hk pHq Ď H,

k“1

since ∆hk pHq “ “ “

∆PVĂ phk q`sphk qw0 pHq ∆PVĂ phk q p∆pk h pHqq

∆PVĂ phk q p∆phk pHqq Ď H, k “ 1, 2, ¨ ¨ ¨ , t.

Furthermore, ϕ satisfies that, if z “ PVr pzq ` sh, then m ˆ ˙ ÿ m m ∆hk ϕpzq “ p´1qmí ϕpz ` ihk q i i“0 m ˆ ˙ ÿ m “ p´1qmí ϕpPVr pzq ` sh ` ipPVr phk q ` pk hqq i i“0 m ˆ ˙ m ˆ ˙ ÿ ÿ m m mí “ p´1q epPVr pzq ` iPVr phk qq ` p´1qmí fm ps ` ipk q i i i“0 i“0 “

“ “

m ∆m P Ă phk q epPVr pzqq ` ∆pk fm psq V

m pk ∆m P Ă phk q epPVr pzqq ` p∆1 q fm psq V

∆m P Ă phk q epPVr pzqq P H for k “ 1, 2, ¨ ¨ ¨ , t, V

since PVr phk q P Vr , epPVr pzqq P H and ∆v pHq Ď H for all v P Vr . Thus, for every k we have that nk ě m, ∆hk pHq Ď H and ∆m hk ϕ P H, which obviously implies that ∆nhkk ϕ “ ∆hnkk ´m p∆m hk ϕq P H.

On the other hand, ϕ is not an exponential polynomial on Rd , since it is not an analytic function because fm is non-differentiable. We now give a description, for the case when V is a proper subspace of Rd (arbitrary d), of the sets of continuous functions f satisfying that ∆nhkk f P H for k k “ 1, ¨ ¨ ¨ , t, for a certain finite dimensional subspace H of CpRd q which is ∆m hk invariant for some mk , k “ 1, ¨ ¨ ¨ , t. Theorem 36. Let t be a positive integer, let n1 , n2 , . . . , nt , m1 , m2 , ¨ ¨ ¨ , mt be natural numbers, let H Ď CpRd q be a linear finite dimensional subspace satisfying t ď

k ∆m hk pHq Ď H.

k“1 d

Further let f : R Ñ R be a continuous function satisfying ∆nhkk f P H

24


for k “ 1, ¨ ¨ ¨ , t. If the subgroup G in Rd generated by th1 , h2 , . . . , ht u satisfies G “ V ‘ Λ, where V is a vector subspace of Rd , and Λ is a discrete additive subgroup of Rd , then for each λ in Λ there exist a continuous exponential polynomial eλ : Rd Ñ R such that f px ` λq “ eλ pxq for all x P V. For the proof of this result, we need firstly to introduce the following technical result: Lemma 37. Let pG, `q be a commutative topological group, h1 , ¨ ¨ ¨ , ht P G and G1 “ h1 Z ` ¨ ¨ ¨ ` ht Z. Let H be a vector subspace of CG such that τh pHq Ď H for all h P G1 , and assume that f : G Ñ C satisfies ∆nhii f P H for certain natural numbers ni and i “ 1, ¨ ¨ ¨ , t. Take N “ n1 ` ¨ ¨ ¨ ` nt . Then ∆N h f P H for all h P G1 . Moreover, if H is a closed subspace of CpGq, then ∆N f P H for all h P G1 . h Proof. The proof follows similar arguments to those used in [7, Theorem 2] and, in fact, this lemma is a generalization of that result, which follows as a Corollary just imposing H “ t0u. Take N “ n1 ` ¨ ¨ ¨ ` nt and let h P G1 . Then there exist m1 , ¨ ¨ ¨ , mt P Z such that h “ m1 h1 ` ¨ ¨ ¨ ` mt ht and ∆N h f

“

N ∆N m1 h1 `¨¨¨`mt ht f “ pτm1 h1 `¨¨¨`mt ht ´ 1d q pf q

“

pτhm1 τhm2 . . . τhmt t ´ 1d qN pf q “ 1m1 2m2 pτh1 τh2 . . . τhmt t ´ τhm22 . . . τhmt t q ` pτhm22 . . . τhmt t ´ τhm33 . . . τhmt t q

“

`pτhm33 τhm44 . . . τhmt t ´ τhm44 . . . τhmt t q ` pτhm44 . . . τhmt t ´ τhm55 . . . τhmt t q

“

¨¨¨ ‰N mt´1 mt mt´1 mt mt´2 mt´1 mt τht ´ τhmt t q ` pτhmt t ´ 1d q pf q τht q ` pτht´1 τht´1 τht ´ τht´1 `pτht´2 “ m1 pτh1 ´ 1d qτhm22 . . . τhmt t ` pτhm22 ´ 1d qτhm33 . . . τhmt t ‰N mt´1 ´ 1d qτhmt t ` pτhmt t ´ 1d q pf q . ` ¨ ¨ ¨ ` pτht´1

Expanding the N -th power, by the Multinomial Theorem, we obtain a sum of the form t ź ÿ N! mi`1 . . . τhmt t qαi pτhmi i ´ 1d qαi pf q , pτhi`1 (40) α ! . . . α ! 1 t i“1 0ďα ,...,α ďN 1

t

where the sum is also restricted by α1 ` α2 ` ¨ ¨ ¨ ` αt “ N , which implies that, for some k P t1, ¨ ¨ ¨ , tu, αk ě nk . In particular, for this concrete k, if mk ą 0, we have that pτhmkk ´ 1d qαk pf q “ pτhmkk ´ 1d qαk ńk pτhmkk ´ 1d qnk pf q P pτhmkk ´ 1d qαk ńk pHq Ď H, which follows from the assumption that H is G1 -invariant and the equation pτhmkk ´ 1d qnk “ pτhmkk ´1 ` τhmkk ´2 ` ¨ ¨ ¨ ` τhk ` 1d qnk pτhk ´ 1d qnk . If mk ă 0, we can apply a similar argument, since we have that τh´1 ´ 1d “ τ´h ´ 1d “ ´τ´h pτh ´ 1d q . Consequently, all summands in (40) belong to the vector space H, which proves the first claim of the lemma. If H is a closed subspace of CpGq and h P G1 then there


25

exist a sequence tgn u Ă G1 converging to h and the sequence given by fn “ ∆N gn f converges to ∆N f , which belongs to H since H is closed and g P H for all n. n h Proof of Theorem 36. Lemma 3 allow us to substitute H satisfying t ď

k ∆m hk pHq Ď H.

k“1

r of CpRd q which contains H and satisfies by a finite dimensional subspace H t ď

k“1

r Ď H. r ∆hk pHq

If we take W “ spantf u, we can apply again Lemma 3 with E “ CpRd q, Li “ ∆hi , r since i “ 1, ¨ ¨ ¨ , t, to the vector space M “ W ` H, r ∆nhii pW ` Hq “

Ď Ď

r ∆nhii pW q ` ∆nhii pHq

r H `H r Ď W ` H, r H

Hence M Ď Z “ ˛∆h1 ,∆h2 ,¨¨¨ ,∆ht pM q and Z is a finite dimensional subspace of CpRd q satisfying ∆hi pZq Ď Z , i “ 1, 2, ¨ ¨ ¨ , t. Let V K denote the orthogonal complement of V in Rd with respect to the standard scalar product and let PV : Rd Ñ Rd denote the orthogonal projection onto V with respect to the standard scalar product of Rd . We define the function F : Rd Ñ R by F pxq “ f pPV pxqq. r admits a basis Obviously, F is a continuous extension of f|V . Furthermore, if H Ă m Ă tgk um k“1 , we introduce the new vector space H “ spantGk uk“1 , where Gk pxq “ d d gk pPV pxqq for all x P R . Then, if x “ v`w P R with v P V , w P V K , k P t1, ¨ ¨ ¨ , tu, and j P t1, ¨ ¨ ¨ , mu, we have that ∆hk Gj pxq

“ Gj pv ` w ` hk q ´ Gj pv ` wq “ gj pv ` hk q ´ gj pvq m ÿ r ĎH r “ αi,j gi pvq, since ∆hk pHq “

“

i“1 m ÿ

i“1 m ÿ

αi,j Gi pv ` wq αi,j Gi pxq.

i“1

Ă Ă Ă for k “ 1, ¨ ¨ ¨ , t. Ă ĎH Thus, ∆hk pHq ˚ Take th1 , ¨ ¨ ¨ , h˚s u Ď V K such that th1 , ¨ ¨ ¨ , ht , h˚1 , ¨ ¨ ¨ , h˚s u spans a dense subgroup of Rd . Then if x “ v ` w P Rd with v P V , w P V K , k P t1, ¨ ¨ ¨ , su, and j P t1, ¨ ¨ ¨ , mu, we have that ∆h˚ Gj pxq k

“

Gj pv ` w ` h˚k q ´ Gj pv ` wq

“

Gj pvq ´ Gj pvq “ 0.

26


Ă Ă Ă for k “ 1, ¨ ¨ ¨ , s. Anselone-Korevaar’s Ă Ď H Hence we also have that ∆h˚ pHq k Ă Ă are continuous exponential polynomials on theorem implies that all elements of H d R . Let us now do the computations for F . Take N “ n1 ` ¨ ¨ ¨ ` nt . Then N ˆ ˙ ÿ N F pxq “ ∆N p´1qN ´k F px ` khi q hi k k“0 ˆ ˙ N ÿ N “ p´1qN ´k F pPV pxq ` kPV phi q ` rpx ´ PV pxqq ` kphi ´ PV phi qqsq k k“0 N ˆ ˙ ÿ N “ p´1qN ´k f pPV pxq ` kPV phi qq k k“0 “

“ “

∆N PV phi q f pPV pxqq m ÿ

j“1 m ÿ

j“1

r ai,j gj pPV pxqq since ∆N PV phi q f P H

Ă Ă ai,j Gj pxq P H.

r is G-invariant and Lemma 37 to conclude that ∆N Here we have used that H PV phi q f P d r r H since PV phi q P V Ď G and H is a closed subspace of CpR q, since it is finite Ă Ă dimensional. On the other hand, ∆h˚ F “ 0 P H for j “ 1, ¨ ¨ ¨ , s. Hence we j can apply Corollary 5 to F to prove that F is an exponential polynomial whose restriction to V is f|V . Thus, if we set e0 “ F , we have that e0 is a continuous exponential polynomial on Rd and f pxq “ e0 pxq for all x in V . Now let λ be arbitrary in Λ and we consider the function gλ : V Ñ R defined by gλ pxq “ f px ` λq for x in V and λ in Λ. Then gλ “ τλ pf q, so that r ĎH r ∆nhii gλ “ ∆nhii τλ pf q “ τλ p∆nhii pf qq P τλ pHq Ď τλ pHq

Thus we can repeat all arguments above with gλ instead of f to get that, for some continuous exponential polynomial eλ defined on Rd we have that gλ pxq “ eλ pxq for all x P V . Hence, if x P V and λ P Λ, f px ` λq “ gλ pxq “ eλ pxq. This ends the proof. References [1] A. Aksoy, J. M. Almira, On Montel and Montel-Popoviciu Theorems in several variables, Aequationes Mathematicae, (2014) [2] J. M. Almira, Montel’s theorem and subspaces of distributions which are ∆m -invariant, Numer. Functional Anal. Optimiz. 35 (4) (2014) 389–403. [3] J. M. Almira, On Popoviciu-Ionescu functional equation, Annales Mathematicae Silesianae, to appear, 2016. [4] J. M. Almira, On Loewner’s characterization of polynomials, Ja´ en Journal on Approximation, to appear, 2016.


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[5] J. M. Almira, K. F. Abu-Helaiel, On Montel’s theorem in several variables, Carpathian Journal of Mathematics, 31 (2015), 1–10. [6] J. M. Almira, K. F. Abu-Helaiel, A note on invariant subspaces and the solution of some classical functional equations, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity 11 (2013) 3-17. [7] J. M. Almira, L. Sz´ ekelyhidi, Local polynomials and the Montel theorem, Aequationes Mathematicae, 89 (2015) 329-338. [8] J. M. Almira, L. Sz´ ekelyhidi, Montel–type theorems for exponential polynomials, Demonstratio Mathematica, 49 (2) (2016) 197-212. [9] P. M. Anselone, J. Korevaar, Translation invariant subspaces of finite dimension, Proc. Amer. Math. Soc. 15 (1964), 747-752. er, Random variables and probability distributions, Cambridge Tracts in Maths. [10] H. Cram´ and Math. Physics 36 Cambridge University Press, London, 1937. [11] M. Engert, Finite dimensional translation invariant subspaces, Pacific J. Maths. 32 (2) (1970) 333-343. ¨ , Sur les ´ [12] I. Fenyo equations distributionnelles, in Functional Equations and Inequalities, Ed. B. Forte, C.I.M.E. summer schools, 54, Springer, 1971 (Reprinted in 2010), 45-109. [13] G. Feldman, Functional equations and characterization problems on locally compact Abelian groups, EMS, 2008. [14] S. G. Ghurye, I. Olkin, A characterization of the multivariate normal distribution, Ann. Math. Statist. 33 (2) (1962) 533–541. [15] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. [16] S. Haruki, On the mean value property of harmonic and complex polynomials, Proc. Japan Acad. Ser. A, 57 (1981) 216-218. [17] S. Haruki, On the theorem of S. Kakutani-M. Nagumo and J.L. Walsh for the mean value property of harmonic and complex polynomials, Pacific J. Math. 94 (1) (1981) 113-123. [18] S. Haruki, On two functional equations connected with a mean-value property of polynomials, Aequationes Math. 6 (1971) 275-277. [19] A.M. Kagan, Yu. V. Linnik, C. R. Rao, Characterization problems in Mathematical Statistics, John Wiley & Sons, 1973. ř qi p 2vπ 2 q “ nf pzq, [20] S. Kakutani, M. Nagumo, About the functional equation n´1 v“0 f pz ` e Zenkoku Shijˆ o Danwakai, 66 (1935) 10-12 (in Japanese). [21] K. O. Leland, Finite dimensional translation invariant spaces, Amer. Math. Monthly 75 (1968) 757-758. ` , Sulle funzioni che ammettono una formula d’addizione del tipo f px ` yq “ [22]řT. Levi-Civita n X pxqY pyq, Atti Accad. Naz. Lincei Rend. 22 (5) (1913) 181-183. i i i“1 [23] C. Loewner, On some transformation semigroups invariant under Euclidean and nonEuclidean isometries, J. Math. Mech. 8 (1959) 393-409. [24] A. M. Mathai, G. Pederzoli, Characterizations of the Normal Probability Law, Wiley Eastern Limited, 1977. [25] E. Shulman, Subadditive set-functions on semigroups, applications to group representations and functional equations, J. Funct. Anal. 263 (2012) 1468-1484. [26] E. Shulman, Decomposable functions and representations of topological semigroups, Aequat. Math. 79 (2010) 13-21. [27] E. Shulman, Functional equations of homological type, PhD Thesis, Moscow State Pedagogical University (1994). [28] L. Szekelyhidi Convolution type functional equations on Topological Abelian Groups, World Scientific, 1991. [29] L. Szekelyhidi Discrete spectral synthesis and its applications, Springer, 2006. [30] V. S. Vladimirov, Generalized Functions in Mathematical Physics, 1979. [31] M. Waldschmidt, Topologie des Points Rationnels, Cours de Troisi` eme Cycle 1994/95 Universit´ e P. et M. Curie (Paris VI), 1995. [32] J. L. Walsh, A mean value theorem for polynomials and harmonic polynomials, Bull. Amer. Math. Soc. 42 (1936) 923-930.

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´ ticas, Universidad de Ja´ Departamento de Matema en, E.P.S. Linares, Campus Cient´ıfico ´ gico de Linares, 23700 Linares, Spain Tecnolo E-mail address: [email protected] ´ Instytut Matematyki, Uniwersytet Slaski, Ul. Bankowa 14, 40-007 Katowice E-mail address: [email protected], [email protected]