J. M. Almira A MONTEL-TYPE THEOREM FOR MIXED DIFFERENCES

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1 Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 75, 2 (2017), 5 10 J. M. Almira A MONTEL-TYPE THEOREM FOR MIXED DIFFERENCES Abstract. We prove a generalization o classical Montel s theorem or the mixed dierences case, or polynomials and exponential polynomial unctions, in commutative setting. 1. Introduction. Given G a group, the unction : G C is called a polynomial o degree n i satisies Fréchet s mixed dierences equation (1) hn 1 hn h1 0 or all h 1,,h n 1 G, n i satisies Fréchet s unmixed dier- and it is called a semipolynomial o degree ences equation (2) n 1 h 0 or all h G. Here, h : C G C G is the orward dierences operator given by h x xh x, hn 1 hn h1 hn 1 hn h1 denotes the composition o the operators h1,, hn 1 and n 1 h h n h. I G is commutative, Djoković s Theorem [7] guarantees that equations (1) and (2) are equivalent (see also [17] or a dierent proo o this result) so that both concepts represent the very same class o unctions. Finally, we say that : G C is an exponential polynomial i the space τ span τ h : h G is inite dimensional. Here τ h : C G C G is the translation operator deined by τ h x xh. Obviously, a unction is an exponential polynomial i and only i it belongs to some inite-dimensional translation invariant vector subspace V o C G. A main motivation or this deinition is that, or G R d,, a continuous unction : R d C is an exponential polynomial i and only i it is a inite sum o unctions o the orm p x e x,λ or certain ordinary polynomials p x and vectors λ C d. Moreover, i is a complex valued Schwartz distribution which belongs to a inite-dimensional translation invariant space o distributions, then is equal, in distributional sense, to a continuous complex valued exponential polynomial on R d (see, e.g., [6], [8], [10], [11] or [16] or the proos o these claims). Just to ix the terminology, we recall that a set S G generates the group G i the smallest subgroup o G which contains S is G. Moreover, i G is a topological group, the set S G topologically generates G i the smallest subgroup o G which contains S is a dense subgroup o G. The ollowing result, which was proved in [4] (see also [1], [3]), generalizes a well known theorem o Montel [12], [13]: THEOREM 1. Let G be a commutative (topological) group and : G C be a (continuous) unction. Assume that E h 1,,h s (topologically) generates G and 5

2 6 J. M. Almira let n k 1 h k 0, or k 1,,s. Then is a polynomial on G o degree at most n 1 n s. Moreover, i G R d,, h 1,,h s topologically generates R d, and is a complex valued Schwartz distribution on R d such that n k 1 h k 0, or k 1,,s, then is, in distributional sense, a continuous polynomial on R d o degree at most n 1 n s. In particular, is equal almost everywhere to an ordinary polynomial with total degree at most n 1 n s. In 1948 Montel [14], [15] demonstrated, or mixed dierences, a version o his theorem, or complex unctions o one and two real variables. We generalize his result, with an easier proo, to complex unctions deined on any initely generated (topologically initely generated) group (topological group) G, and to complex valued Schwartz distributions deined on R d. The result also applies to complex valued unctions depending on any inite number o real variables. In particular, Theorem 2 characterizes polynomials and exponential polynomials as solutions o certain inite sets o mixeddierences unctional equations. 2. Main results THEOREM 2. Let G be a commutative group and let : G C be a unction. I there exist exponential polynomials P k : G C, k 1,,s and elements h i, j 1 i n;1 j s o G such that, or every i 1,,i s 1,2,,n s, the set is a generating system o G, and h i1,1,,h is,s (3) h1,k,h 2,k,,h n,k x P k x, or all k 1,2,,s, and all x G, then is an exponential polynomial. Proo. We proceed by induction on n. I we assume that h1,k P k, k 1,,s, then we have that h1,k V V or k 1,,s, with V span τ P 1 τ P 2 τ P s, since h1,k τ P i τ P i V or all k,i and h1,k span span P k V or all k. Hence V is a inite dimensional translation invariant space, which implies that all its elements are exponential polynomials, and V. This proves the result or n 1. Assume the result holds true or n 1 and let satisy (3). Take i i 1,,i s 1,2,,n s and deine the unction F i hi1,1,h i2,2,. Then, i we denote by h ik,k the act that the step h ik,k is hidden (i.e. it does not appear) in a ormula, we have that h1,k,h 2,k,,h ik,k,,h n,k F i h1,k,h 2,k,,h ik,k,,h n,k hi1,1,h i2,2, hi1,1,h i2,2,,h ik,k, h1,k,h 2,k,,h ik,k,,h n,k hi1,1,h i2,2,,h ik,k, P k Q k

3 Montel theorem or mixed dierences 7 with Q k being an exponential polynomial or k 1,2,s. Hence the induction hypothesis tell us that F i is an exponential polynomial or every i. We want to reduce the size o the operator hi1,1,h i2,2, used or the deinition o F i. So, consider, or each k 1,,s, the new unction G i,k hi1,1,,h ik,k, (which results rom deleting the step h ik,k in the deinition o F i ). Then we have that h1,k,h 2,k,,h n 1,k G i,k h1,k,h 2,k,,h n 1,k hi1,1,,h ik,k, h1,k,,h k,k,,h n 1,k hi1,1,,h ik 1,k 1,h k,k,h ik 1,k 1, is an exponential polynomial, since hi1,1,,h ik 1,k 1,h k,k,h ik 1,k 1, is an exponential polynomial. Furthermore, or j k we have that h1, j,h 2, j,,h i j, j,,h n, j G i,k h1, j,h 2, j,,h i j, j,,h n,k hi1,1,,h ik,k, hi1,1,,h i j, j,,h ik,k, h1, j,h 2, j,,h i j 1, j,h i j, j,h i j 1, j,h n, j hi1,1,,h i j, j,,h ik,k, P j which is also an exponential polynomial. Hence the unction G i,k satisies a set o equations like (3) with n 1 steps, and the induction hypothesis implies that G i,k is also an exponential polynomial. In other words, every unction o the orm h j1,a 1,h j2,a 2,,h js 1,a s 1 with 0 j k n and a 1,,a s 1 an arbitrary subset o cardinality s 1 o 1,,s, is an exponential polynomial. Let us still do a step more: Take t 1,,s, t k, and consider the unction G i,k,t hi1,1,,h ik,k,,h i t,t,. Then h1,k,h 2,k,,h n 1,k G i,k,t h1,k,h 2,k,,h n 1,k hi1,1,,h ik,k,,h i t,t, h1,k, h k,k,,h n 1,k hi1,1,,h ik 1,k 1,h k,k,h ik 1,k 1,,h i t,t, is an exponential polynomial, since hi1,1,,h ik 1,k 1,h k,k,h ik 1,k 1,,h i t,t, is an exponential polynomial. Furthermore, h1,t,h 2,t,,h n 1,t G i,k,t h1,t,h 2,t,,h n 1,t hi1,1,,h ik,k,,h i t,t, h1,t,,h t,t,,h n 1,t hi1,1,,h ik,k,,h it 1,t 1,h t,t,h it 1,t 1, is an exponential polynomial, since hi1,1,,h ik,k, h it 1,t 1,h t,t,h it 1,t 1, is an ex-

4 8 J. M. Almira ponential polynomial. Finally, or j 1,,s k,t we have that h1, G j,h 2, j,,h i j, j,,h n, j i,k,t h1, j,h 2, j,,h i j, j,,h n,k hi1,1,,h ik,k,,h i t,t, hi1,1,,h i j, j,,h ik,k,,h i t,t,,h h1, is,s j,h 2, j,,h i j 1, j,h i j, j,h i j 1, j,h n, j hi1 P,1,,h i j, j,,h ik,k,,h i t,t, j which is also an exponential polynomial. Hence the unction G i,k,t satisies a set o equations like (3) with n 1 steps, and the induction hypothesis implies that G i,k,t is also an exponential polynomial. Thus, every unction o the orm h j1,a 1,h j2,a 2,,h js 2,a s 2 with 0 j k n and a 1,,a s 2 an arbitrary subset o cardinality s 2 o 1,,s, is an exponential polynomial. We can repeat the argument to delete another step in the dierence operators used or the deinition o each one o the unctions G i,k,t above, and maintain the property that the new unctions are still exponential polynomials. Iterating the argument as many times as necessary we lead to the act that all unctions hi, j are exponential polynomials. Then we apply the case n 1 to conclude that is an exponential polynomial. This ends the proo. The ollowing are easy corollaries o Theorem 2: COROLLARY 1. Let G be a topological commutative group and let : G C be a continuous unction. I there exist exponential polynomials P k : G C, k 1,,s and elements h i, j 1 i n;1 j s o G such that, or every i 1,,i s 1,2,,n s, the set h i1,1,,h is,s topologically generates G, and satisies (3), then is an exponential polynomial. COROLLARY 2. I h i1,1,,h is,s topologically generates R d, or every element i 1,,i s 1,,n s, and D R d is a complex valued Schwartz distribution on R d which satisies the equations (3) or certain continuous exponential polynomials P k, then is, in distributional sense, a continuous exponential polynomial. In particular, there exists a continuous exponential polynomial p such that p almost everywhere. Proo. It is enough to ollow the same steps o the demonstration o Theorem 2, just taking into account Anselone-Korevaar s theorem and that the operators τ h and h are deined or Schwartz distributions D R d by the expressions τ h,ϕ,τ h ϕ and h,ϕ, h ϕ, respectively. These inherit all properties rom their corresponding versions, originally deined or ordinary unctions. REMARK 1. I we impose P k 0 or all k in Theorem 2 or Corollaries 1, 2, then the unction will be a polynomial. Thus, these results generalize Theorem 1 to the mixed dierences case.

5 Montel theorem or mixed dierences 9 REMARK 2. The condition that, or every i 1,,i s 1,2,,n s, the set h i1,1,,h is,s either generates or (in the case that G is a topological group) topologically generates G, is a very natural necessary condition. It can be shown, or G R d, that i exists i 1,,i s 1,2,,n s such that h i1,1z h i2,2z h is,sz is not dense in R d, then there exists a non-dierentiable continuous unction : R d R such that hik 0 or k 1,,s and, henceorth, solves the unctional equations,k (3) with P k 0 or all k (see [2] or a proo o this claim) but is not a polynomial nor an exponential polynomial, since every continuous polynomial on R d is an ordinary polynomial in d variables [9] and, henceorth, is a dierentiable unction, and continuous exponential polynomials on R d are also dierentiable unctions. The merit o Theorem 2 and Corollaries 1, 2 is, thus, to prove that their hypotheses are suicient to guarantee that is an exponential polynomial. 3. Acknowledgement The author wants to express his gratitude to the anonymous reeree or his/her deep reading o the manuscript and his/her advice, who helped to make the paper more transparent. Reerences [1] A. AKSOY, J.M.ALMIRA, On Montel and Montel-Popoviciu Theorems in several variables, Aequationes Mathematicae, 89 (5) (2015) [2] J. M. ALMIRA, Characterization o distributions whose orward dierences are exponential polynomials, Commentationes Mathematicae Universitatis Carolinae, 58 (4) (2017) [3] J. M. ALMIRA, K. F. ABU-HELAIEL, On Montel s theorem in several variables, Carpathian Journal o Mathematics, 31 (2015) [4] J. M. ALMIRA, L.SZÉKELYHIDI, Local polynomials and the Montel theorem, Aequationes Mathematicae 89 (2) (2015) [5] J. M. ALMIRA, L.SZÉKELYHIDI, Montel type theorems or exponential polynomials, Demonstratio Mathematica 49 (2) (2016) [6] P. M. ANSELONE, J. KOREVAAR, Translation invariant subspaces o inite dimension, Proc. Amer. Math. Soc. 15 (1964) [7] D. Z. DJOKOVIĆ, A representation theorem or X 1 1 X 2 1 X n 1 and its applications, Ann. Polon. Math. 22 (1969/1970) [8] M. ENGERT, Finite dimensional translation invariant subspaces, Paciic J. Maths. 32 (2) (1970)

6 10 J. M. Almira [9] M. FRÉCHET, Une deinition onctionelle des polynomes, Nouv. Ann. 9 (1909) [10] P.G. LAIRD, On characterizations o exponential polynomials, Paciic J. Math. 80 (1979) [11] C. LOEWNER, On some transormation semigroups invariant under Euclidean and non-euclidean isometries, J. Math. Mech. 8 (1959) [12] P. MONTEL, Sur un théoreme du Jacobi, Comptes Rend. Acad. Sci. París, 201 (1935) [13] P. MONTEL, Sur quelques extensions d un théorème de Jacobi, Prace Matematyczno-Fizyczne 44 (1) (1937) [14] P. MONTEL, Sur quelques équations aux dierences mêlées. Ann. Sci. École Norm. Sup. 65 (3) (1948) [15] P. MONTEL, Sur un système d équations onctionnelles. Ann. Soc. Polon. Math. 21 (1948) [16] H. STETKAER, Functional equations on groups, World Scientiic, 2013 [17] L. SZÉKELYHIDI, On Fréchet s unctional equation, Monatsch. Für Math., 175 (4) (2014) AMS Subject Classiication: 39A05, 39A70 Jose Maria Almira Departamento de Ingeniería y Tecnología de Computadores, Facultad de Inormática Universidad de Murcia Campus de Espinardo Murcia (Spain) and Departamento de Matemáticas, Universidad de Jaén E.P.S. Linares, C/Alonso X el Sabio, Linares (Jaén) Spain jmalmira@um.es; jmalmira@ujaen.es Lavoro pervenuto in redazione il