Integrable mean periodic functions on locally compact abelian groups

Rana, Inder K.; Navada, K. Gowri

doi:10.1090/S0002-9939-1993-1111221-3

Integrable mean periodic functions on locally compact abelian groups
HTML articles powered by AMS MathViewer

by Inder K. Rana and K. Gowri Navada PDF

Proc. Amer. Math. Soc. 117 (1993), 405-410 Request permission

Abstract:

Let $G$ be a locally compact abelian group with a Haar measure ${\lambda _G}$. A function $f$ on $G$ is said to be mean-periodic if there exists a nonzero finite regular measure $\mu$ of compact support on $G$ such that $f{\ast }\mu = 0$. It is known that there exist no nontrivial integrable mean periodic functions on ${{\mathbf {R}}^n}$. We show that there exist nontrivial integrable mean periodic functions on $G$ provided $G$ has nontrivial proper compact subgroups. Let $f \in {L_1}(G)$ be mean periodic with respect to a nonzero finite measure $\mu$ of compact support. If $\mu (G) \ne 0$ and ${\lambda _G}(\operatorname {supp} (\mu )) > 0$, then there exists a compact subgroup $K$ of $G$ such that $f{\ast }{\lambda _K} = 0$, i.e., $f$ is mean periodic with respect to ${\lambda _K}$, where ${\lambda _K}$ denotes the normalized Haar measure of $K$. When $G$ is compact, abelian and meterizable, we show that there exists continuous (hence integrable and almost periodic) functions on $G$ that are not mean periodic.

References

Carlos A. Berenstein and B. A. Taylor, Mean-periodic functions, Internat. J. Math. Math. Sci. 3 (1980), no. 2, 199–235. MR 570178, DOI 10.1155/S0161171280000154

Les fonctions moyenne-périodiques

14

J. Delsarte, Théorie des fonctions moyenne-périodiques de deux variables, Ann. of Math. (2) 72 (1960), 121–178 (French). MR 130568, DOI 10.2307/1970149
Leon Ehrenpreis, Mean periodic functions. I. Varieties whose annihilator ideals are principal, Amer. J. Math. 77 (1955), 293–328. MR 70047, DOI 10.2307/2372533
L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier transform on semi-simple Lie groups. II, Trans. Amer. Math. Soc. 84 (1957), 1–55. MR 83683, DOI 10.1090/S0002-9947-1957-0083683-7

Abstract harmonic analysis

Lectures on mean periodic functions

Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990
Inder K. Rana, A uniqueness problem for probability measures on locally-compact abelian groups, Sankhyā Ser. A 39 (1977), no. 4, 309–316. MR 492188
Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834

Théorie générale des fonctions moyenne-périodiques

48

Alladi Sitaram, Some remarks on measures on noncompact semisimple Lie groups, Pacific J. Math. 110 (1984), no. 2, 429–434. MR 726500