On spectral synthesis in the space of tempered functions on finitely generated Abelian groups

S. S. Platonov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2016), pp. 42-59.

Résumé

Let $G$ be an arbitrary locally compact Abelian group (LCA-group) and let ${\mathscr F}$ be a topological vector space (TVS) consisting of complex-valued functions on $G$. The space ${\mathscr F}$ is said to be translation invariant if ${\mathscr F}$ is invariant with respect to the transformations $\tau_y: f(x)\mapsto f(x-y), \quad f(x)\in{\mathscr F}, y\in G$, and all operators $\tau_y$ are continuous on ${\mathscr F}$. A closed linear subspace ${\mathscr H}\subseteq{\mathscr F}$ is referred to as an invariant subspace if $\tau_y({\mathscr H})\subseteq {\mathscr H}$ for every $y\in G$. By an exponential function or generalized character we mean an arbitrary continuous homomorphism from a group $G$ to the multiplicative group ${\mathbb C}_*:={\mathbb C}\setminus \{0\}$ of nonzero complex numbers. Continuous homomorphisms of $G$ to the additive group of complex numbers are referred to as additive functions. A function $x\mapsto P(a_1(x), \dots, a_m(x))$ on $G$ is said to be polynomial function if $P(z_1,\dots, z_m)$ is a complex polynomial in $m$ variables and $a_1,\dots, a_m$ are additive functions. A product of a polynomial and an exponential function is referred to as an exponential monomial, and a sum of exponential monomials is referred to as an exponential polynomial on $G$. Let ${\mathscr F}$ be a translation-invariant function space on the group $G$ and let be ${\mathscr H}$ an invariant subspace of ${\mathscr F}$. An invariant subspace ${\mathscr H}$ admits spectral synthesis if it coincides with the closure in ${\mathscr F}$ of the linear span of all exponential monomials belonging to ${\mathscr H}$. We say that spectral synthesis holds in ${\mathscr F}$ if every invariant subspace ${\mathscr H}\subseteq{\mathscr F}$ admits spectral synthesis. One of the natural function space is the space ${\mathscr S}'(G)$ of all tempered distributions on a LCA-group $G$. In the present paper we study spectral synthesis in the space ${\mathscr S}'(G)$ for the case when $G$ is a discrete Abelian group. In this case the distributions from ${\mathscr S}'(G)$ coincide with usual functions, thus we will refer to ${\mathscr S}'(G)$ as the space of tempered functions. Let us consider a convenient definition of the space ${\mathscr S}'(G)$ on a discrete finite generated Abelian group $G$. Let $G$ be a finitely generated Abelian group, $v_1, \dots, v_n$ be a system of generators of $G$. Any element $x\in G$ can be representd in the form $x=t_1 v_1+\dots +t_n v_n$, where $t_j\in{\mathbb Z}$ (this representation can be not unique). For $x\in G$, we define the number $|x|\in{\mathbb Z}_+=\{0,1,2,\dots\}$ by $|x|:=\min\{ |t_1|+\dots+|t_n| : x=t_1 v_1+\dots +t_n v_n, \,\, t_j\in{\mathbb Z}, \, j=1,\dots,n\}$. The function $|x|$ is a special example of a quasinorm on $G$. For every $k>0$, we denote by ${\mathscr S}'_k(G)$ the set of all compex-valued functions $f(x)$ on $G$ that satisfy $|f(x)| (1+|x|)^{-k} \to 0 \quad\text{ as }\quad |x|\to \infty$. The set ${\mathscr S}'(G)$ is a Banach space with respect to the norm $\|f\|_{G,k}=\|f\|_k:=\sup_{x\in G} |f(x)| (1+|x|)^{-k}$. Clearly, ${\mathscr S}'_{k_1}(G)\subseteq {\mathscr S}'_{k_2}(G)$ if $k_1$, and this embedding is continuous. We equip the space ${\mathscr S}'(G):=\bigcup_{k>0} {\mathscr S}'_k(G)$ with the topology of the inductive limit of the Banach spaces ${\mathscr S}'_k(G)$. Thus ${\mathscr S}'(G)$ is a translation invariant locally convex space. The main results of the paper is the theorem, that spectral synthesis holds in the space ${\mathscr S}'(G)$ for any finitely generated Abelian group $G$.

Détail

Keywords: spectral synthesis, locally compact Abelian groups, finitely generated Abelian groups, tempered functions, Bruhat–Schwartz functions.

Bibliographie

[1] Y. Berg, Y. Lefstrem, Interpolation Spaces. An Introduction, Mir Publ., M., 1980, 264 pp. | MR

[2] D.\;I. Gurevich, “Counterexamples to a Problem of L. Schwartz”, Funct. Anal. Appl., 9:2 (1975), 29–35 | Zbl

[3] B. Malgranzh, Ideals of Differentiable Functions, Mir Publ., M., 1968, 131 pp.

[4] V.\;V. Napalkov, Convolution Equations in Multidimensional Spaces, Nauka Publ., M., 1982, 240 pp.

[5] N.\;K. Nikolskiy, “Invariant Subspaces in Operator Theory and Function Theory”, Itogi Nauki i Tekhniki. Mat. Anal., 12, 1974, 199–412 | Zbl

[6] S.\;S. Platonov, “On Spectral Synthesis on Zero-Dimensional Abelian Groups”, Sbornik: Mathematics, 204:9 (2013), 99–114 | DOI | Zbl

[7] S.\;S. Platonov, “On Spectral Synthesis on Element-Wise Abelian Groups”, Sbornik: Mathematics, 206:8 (2015), 127–152 | DOI | Zbl

[8] S.\;S. Platonov, “On the Structure of Exponential Monomials on Some Locally Compact Abelian Groups”, Issues of Analysis, 2012, no. 19 (1), 3–14

[9] S.\;S. Platonov, “Spectral Synthesis in Some Topological Vector Spaces of Functions”, St. Petersburg Mathematical Journal, 22:5 (2010), 154–185

[10] A. Robertson, B. Robertson, Topological Vector Spaces, Mir Publ., M., 1967, 261 pp. | MR

[11] E. Hewitt, K. Ross, Abstract Harmonic Analysis, v. 1, Mir Publ., M., 1978, 654 pp.

[12] F. Bruhat, “Distributions sur un groupe localement compact et applications à létudedes représentations des groupes $p$-adiques”, Bull. Soc. math. France, 89 (1961), 43–75 | DOI | MR | Zbl

[13] M. Laczkovich, L. Székelyhidi, “Spectral synthesis on discrete groups”, Math. Proc. Camb. Phil. Soc., 143 (2007), 103–120 | DOI | MR | Zbl

[14] M.\;S. Osborne, “On the Schwartz–Bruhat space and Paley–Wiener theorem for locally compact Abelian groups”, J. of Funct. Anal., 19 (1975), 40–49 | DOI | MR | Zbl

[15] L. Schwartz, “Analyse et synthése harmonique dans les espaces de distributions”, Can. J. Math., 3:2 (1951), 503–512 | DOI | MR | Zbl

[16] L. Schwartz, “Théorie générale des fonctions moynne-périodiques”, Ann. of Math., 48:2 (1947), 875–929 | MR

[17] L. Székelyhidi, Discrete spectral synthesis and its applications, Springer, Berlin, 2006, 135 pp. | MR | Zbl