RANDOM TEMPERED DISTRIBUTIONS ON

RANDOM TEMPERED DISTRIBUTIONS ON (LOCALLY)
COMPACT ABELIAN GROUPS
MLE
Abstract. We try to state the problem in form of questions, some of which,
to be answered need only known results.
1. Some General Comments
It is well known [7][p. 268, 269] that, for a compact abelian group G, the Fourier
inversion formula for f ∈ L2 (G) takes the form:
f (x) =
< χ, x > fˆ(χ) ,
χ∈Ĝ
where fˆ(χ) is non zero only on a countable set of elements χ ∈ Ĝ.
Consider now, a family (aχ )χ∈Ĝ which is square summable, that is, such that
| aχ |2 < +∞ .
χ∈Ĝ
In that case, aχ = 0 on a countable set of elements χ ∈ Ĝ and so, t, defined on G
by:
t(x) =
aχ < χ, x >
χ∈Ĝ
is a L2 function, as the characters form a complete orthonormal system in L2 (G).
This harmonic analysis analogy, between the torus T and a general compact abelian
group, allows us to define, at least formally, both test functions on the group and,
by duality, tempered distributions on the group in the following way.
Definition 1.1. A test function φ on the group G is a function defined by:
φ(x) =
aχ < χ, x > ,
χ∈Ĝ
where the family (aχ )χ∈Ĝ is rapidly decreasing in a generalized sense associated
with the notion of summability.
Date: Informal report, August 20, 2003.
1991 Mathematics Subject Classification. Primary —–, —–; Secondary —–, ——.
Key words and phrases. Harmonic Analysis on (Locally) Compact Abelian Groups, Random
Tempered Distributions.
1
2
MLE
Definition 1.2. T is a tempered distribution on the group G if there is a family
(bχ )χ∈Ĝ , which is slowly increasing in a generalized sense associated with the notion
of summability, such that:
T =
bχ χ
χ∈Ĝ
meaning that, for a test function φ defined by φ(x) =
< T, φ >=
aχ bχ .
χ∈Ĝ
aχ < χ, x >:
χ∈Ĝ
2. Questions
Question 2.1. To define precisely the notions of rapidly decreasing and slowly
increasing in a generalized sense associated with the notion of summability.
By analogy with the classical notion one can propose the following definition for
a family of complex numbers to be rapidly decreasing.
Let A = (ai )i∈I be a family of complex numbers indexed by an arbitrary set
I. Denote by S = S(A, J) the set of bijective maps from J ⊂ I, a finite set, to
{1, . . . , (J)} such that for σ ∈ S:
| aσ−1 (1) |≤| aσ−1 (2) |≤ · · · ≤| aσ−1 ((J)) | .
Definition 2.2. The family A = (ai )i∈I of complex numbers is rapidly decreasing
in a generalized sense associated with the notion of summability if


(J)
∀k ∈ N∗
sup
sup 
| aσ−1 (n) | nk  < +∞
J⊂I,(J)<+∞ σ∈S
n=1
Then, exactly in the same way:
Definition 2.3. The family A = (ai )i∈I of complex numbers is slowly increasing
in a generalized sense associated with the notion of summability if


(J)
| aσ−1 (n) |
 < +∞
∃k ∈ N∗
sup
sup 
nk
J⊂I,(J)<+∞ σ∈S
n=1
Question 2.4. Is a test function analytic in the sense of Mackey [10]? Would that
have as a consequence that a test function is derivable in the sense of Bruhat [3]?
Question 2.5. Can one show that a tempered distribution is derivable in some
sense? Is there an analog of the harmonic analysis on distributions on the torus
for distributions on compact abelian groups?
Question 2.6. Can the result on the mean of random tempered distributions on
the torus be stated and proved for random distributions on a compact group?
Question 2.7. What can be said for general locally compact abelian groups?
Question 2.8. Is there relevant applications for all this? Or, is there any particular case, besides the torus, that deserves particular study?
RANDOM TEMPERED DISTRIBUTIONS ON (LOCALLY) COMPACT ABELIAN GROUPS 3
References
1. C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer Verlag
1975.
2. D. Bernard, Techniques d’Analyse Mathématique, Masson et Cie. Éditeurs, Paris 1968.
3. Bruhat, Distributions sur un Groupe Localement Compact et Applications à l’Étude des
Répresentations des Groupes p-Adiques, Bull. Soc. Math. de France 89 (1961), 43–75.
4. A. Guichardet, Analyse Harmonique Commutative, Dunod, Paris 1968.
5. E. Hewitt, K. A. Ross, Abstract harmonic Analysis I, second edition, Springer Verlag 1979.
6. E. Hewitt, K. A. Ross, Abstract harmonic Analysis II, second edition, Springer Verlag 1970.
7. V. P. Khavin, N. K. Nikol’skij (Eds.), Commutative Harmonic Analysis I, Springer Verlag,
1991.
8. V. P. Khavin, N. K. Nikol’skij (Eds.), Commutative Harmonic Analysis II, Springer Verlag,
1998.
9. V. P. Khavin, N. K. Nikol’skij (Eds.), Commutative Harmonic Analysis III, Springer Verlag,
1995.
10. Mackey, Laplace Transforms for locally compact abelian groups, Proceedings of the National
Academy of Sciences of USA 34 (1948), 156–.
11. M. B. Marcus, G. Pisier, Random Fourier Series with Applications to Harmonic Analysis,
Princeton University Press, Princeton 1981.
12. Suzana Matello de Nápoles, Sur les fonctions analytiques dans les groupes localement compacts, C. R. Acad. Sc. Paris, 298 I, 5 (19).
13. Suzana Matello de Nápoles, Théorème des Noyaux pour les Fonctions Généralisées, Bull Sc.
Math., 110 II (1994), 85–112.
14. Suzana Matello de Nápoles, Sur certsains espaces de fonctions dans les groupes localement
compacts, Portugaliae Mathematica, 44 2 (1987), 221–226.
15. Suzana Matello de Nápoles, Silva Distributions for Certain Locally Compact Groups, Portugaliae Mathematica, 51 3 (1994), 351–362.
16. H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Mathematical
Monographs, 1968.
17. Riss, Éléments de Calcul differentiel et Théorie des Distributions sur les Groupes Abéliens
Localement Compacts, Acta Mathematica 89 (1953), 45–150.
18. W. Rudin, Fourier Analysis on Groups, John Wiley & Sons, 1962.
Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade
Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, PORTUGAL and CMAF-UL
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