Slow Growth for Universal Harmonic Functions
© M. Carmen G´omez-Collado et al. 2010
Received: 8 April 2010
Accepted: 17 June 2010
Published: 7 July 2010
Given any continuous increasing function such that , we show that there are harmonic functions on satisfying the inequality for every , which are universal with respect to translations. This answers positively a problem of D. H. Armitage (2005). The proof combines techniques of Dynamical Systems and Operator Theory, and it does not need any result from Harmonic Analysis.
A classic result of Birkhoff from 1929  says that there are entire functions whose translates approximate any other entire function as accurately as we want on an arbitrary compact set of the complex plane. More precisely, given any entire function , there exists an increasing sequence of integers such that , uniformly on compact sets of . Then is called a universal entire function (with respect to translation). In terms of Dynamical Systems, the (continuous and linear) operator , admits elements whose orbit is dense in . Within the framework of Operator Theory, is called a hypercyclic operator, and is a hypercyclic vector for . The same phenomenon happens for any translation operator with respect to a nonzero .
then there is a universal harmonic function on that satisfies for all . Armitage  asked whether the condition (1.1) can be relaxed: is it true that, for every , one can find universal harmonic functions on with arbitrarily slow transcendental growth? In other words, can the exponent of be reduced to ? We answer positively this question. It is easy to notice that is equivalent to saying that for all , and no universal harmonic function can satisfy that for any , since this would force to be a polynomial. This is why we can only consider transcendental growths for universal harmonic functions. Let us recall that the case can be obtained as a consequence of the result in [4, 5] (as was noticed in ) since, for example, the real part of a universal entire function is a universal harmonic function on .
We will recall some notions. For the basic theory of harmonic functions we refer the reader to the books [7, 8]. For a good source on the theory of hypercyclic operators and related properties, we refer the reader to [9–12].
A continuous function on a metric space is topologically transitive (resp., mixing) if, for any pair of nonempty open sets, there is (resp., ) such that (resp., for every ). We will work with (continuous and linear) operators on separable, metric, and complete topological vector spaces . In our framework it is well known that topological transitivity is equivalent to hypercyclicity.
2. Universal Harmonic Functions of Slow Growth and Strong Approximation of Polynomials
Given a transcendental growth determined by , to find universal harmonic functions whose growth is bounded by (i.e., for all ), it suffices to find a Banach space consisting of harmonic functions, whose topology is finer than the topology of uniform convergence on compact sets of , containing the harmonic polynomials, and such that, on the one hand, the translation operator , , is (well-defined, continuous and) hypercyclic for every , and on the other hand every harmonic function in the unit ball of has a growth bounded by . Indeed, in this case, there are functions in the unit ball of whose orbit under is dense in , therefore every harmonic polynomial can be approximated in the topology of by a (multiple of a fixed ) translation of , and we obtain that the harmonic functions can be approximated, uniformly on compact sets of , by translations of .
Then there is a Banach space of harmonic functions, whose topology is finer than the topology of uniform convergence on compact sets of , containing the harmonic polynomials, such that the translation operator is a (well-defined and bounded) mixing operator, for any , and every with satisfies for all .
In particular, there are universal harmonic functions of arbitrarily slow transcendental growth.
The case is even simpler since the basis of harmonic polynomials can be taken as such that and for every , . The corresponding space is the direct sum of two weighted -spaces, and a simplification of the above argument does the job.
What we actually showed in the proof of Theorem 2.1 is that there are with for all such that, for any harmonic polynomial , there is an increasing sequence of integers such that in the topology of , which is finer than the topology of uniform convergence on compact sets of . Moreover, the function is universal with respect to translations in the direction of , a result also stronger than Armitage's in the sense that in  all translations were allowed for a universal function.
3. Final Comments
It is possible to obtain the main result of the paper by using tensor product techniques developed in . However the proofs seem to be technically much more complicated since one has to represent the space of harmonic functions on as a quotient of a direct sum of the complete tensor product of spaces of harmonic functions with fewer variables, while we produced here a, somehow, direct proof. We want to thank Antonio Bonilla for pointing out to us the possibility of a tensor product approach.
When we fix a nonzero , one can consider the parametric family of translation operators on our space . This is a -semigroup of operators, which we showed to be hypercyclic on . We refer to [12, 13] for the basic theory of hypercyclic -semigroups. In  we proved that every hypercyclic vector of a -semigroup is shared by all operators of the semigroup (except, obviously, ). In particular, if is a universal harmonic function for , then it is also universal (or a hypercyclic vector) for for all . Let us notice that the inheritance of hypercyclicity by discrete subsemigroups depends strongly on the fact that the index set is or : there are hypercyclic -semigroups of operators whose index set is a sector of the complex plane and such that no discrete subsemigroup is hypercyclic .
The operator can be considered as a (infinite type) differential operator since . In  we develop techniques based on a kind of generalized backward shifts which allow us to extend the main results in this paper and  to other differential operators, and then we show the existence of harmonic and entire functions of arbitrarily slow transcendental growth which are universal with respect to some differential operators.
This work was partially supported by the MEC and FEDER Projects MTM2007-64222, MTM2010-14909, and MTM2007-62643. The third author was also supported by Generalitat Valenciana, Project PROMETEO/2008/101. The authors would like to thank the referees, whose reports produced a great improvement of the paper's presentation.
- Birkhoff GD: Démonstration d'un théorème élémentaire sur les fonctions entières. Comptes Rendus de l'Académie des sciences 1929, 189: 473–475.MATHGoogle Scholar
- Dzagnidze OP: On universal double series. Sakharthvelos SSR Mecnierebatha Akademiis Moambe 1964, 34: 525–528.MathSciNetMATHGoogle Scholar
- Blasco O, Bonilla A, Grosse-Erdmann K-G: Rate of growth of frequently hypercyclic functions. Proceedings of the Edinburgh Mathematical Society. Series II 2010, 53(1):39–59. 10.1017/S0013091508000564MathSciNetView ArticleMATHGoogle Scholar
- Duĭos Ruis SM: Universal functions and the structure of the space of entire functions. Doklady Akademii Nauk SSSR 1984, 279(4):792–795. English translation Soviet Mathematics—Doklady, vol. 30, pp. 713–716, 1984 English translation Soviet Mathematics—Doklady, vol. 30, pp. 713-716, 1984MathSciNetGoogle Scholar
- Chan KC, Shapiro JH: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana University Mathematics Journal 1991, 40(4):1421–1449. 10.1512/iumj.1991.40.40064MathSciNetView ArticleMATHGoogle Scholar
- Armitage DH: Permissible growth rates for Birkhoff type universal harmonic functions. Journal of Approximation Theory 2005, 136(2):230–243. 10.1016/j.jat.2005.07.006MathSciNetView ArticleMATHGoogle Scholar
- Axler S, Bourdon P, Ramey W: Harmonic Function Theory, Graduate Texts in Mathematics. Volume 137. Second edition. Springer, New York, NY, USA; 2001:xii+259.View ArticleGoogle Scholar
- Gardiner SJ: Harmonic Approximation, London Mathematical Society Lecture Note Series. Volume 221. Cambridge University Press, Cambridge, UK; 1995:xiv+132.Google Scholar
- Grosse-Erdmann K-G: Universal families and hypercyclic operators. Bulletin of the American Mathematical Society 1999, 36(3):345–381. 10.1090/S0273-0979-99-00788-0MathSciNetView ArticleMATHGoogle Scholar
- Grosse-Erdmann K-G: Recent developments in hypercyclicity. RACSAM. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2003, 97(2):273–286.MathSciNetMATHGoogle Scholar
- Bayart F, Matheron É: Dynamics of Linear Operators, Cambridge Tracts in Mathematics. Volume 179. Cambridge University Press, Cambridge; 2009:xiv+337.View ArticleGoogle Scholar
- Grosse-Erdmann KG, Peris A: Linear Chaos, Universitext. Springer, New York, NY, USA; 2010.Google Scholar
- Desch W, Schappacher W, Webb GF: Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory and Dynamical Systems 1997, 17(4):793–819. 10.1017/S0143385797084976MathSciNetView ArticleMATHGoogle Scholar
- Martínez-Giménez F, Peris A: Universality and chaos for tensor products of operators. Journal of Approximation Theory 2003, 124(1):7–24. 10.1016/S0021-9045(03)00118-7MathSciNetView ArticleMATHGoogle Scholar
- Conejero JA, Müller V, Peris A: Hypercyclic behaviour of operators in a hypercyclic -semigroup. Journal of Functional Analysis 2007, 244(1):342–348. 10.1016/j.jfa.2006.12.008MathSciNetView ArticleMATHGoogle Scholar
- Conejero JA, Peris A: Hypercyclic translation -semigroups on complex sectors. Discrete and Continuous Dynamical Systems. Series A 2009, 25(4):1195–1208.MathSciNetView ArticleMATHGoogle Scholar
- Martínez-Giménez F, Peris A: Hypercyclic differential operators on spaces of entire and harmonic functions. , Preprint , PreprintGoogle Scholar
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