Abstract
By means of a certain kind of ‘atomic’ representation a new Segal algebraS 0(G) of continuous functions on an arbitrary locally compact abelian groupG is defined. From various characterizations ofS 0(G), e. g. as smallest element within the family of all strongly character invariant Segal algebras, functorial properties of the symbolS 0 are derived, which are similar to those of the spacel (G) of Schwartz-Bruhat functions, e. g. invariance under the Fourier transform, or compatibility with restrictions to closed subgroups. The corresponding properties of its Banach dualS' 0(G) as well as some of their applications are to be given in a subsequent paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Argabright, L., andGil De Lamadrid, J.: Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups. Mem. Amer. Math. Soc. 145. Providence, R. I.: Amer. Math. Soc. 1974.
Bertrandias, J. P., Datry, C., andDupuis, C.: Unions et intersections d'espacesL p invariantes par translation ou convolution. Ann. Inst. Fourier28/2, 53–84 (1978).
Bruhat, H.: Distributions sur un groupe localement compact et applications à l'étude des représentations des groupesp-adiques. Bull. Soc. Math. France89, 43–75 (1961).
Bürger, R.: Functions of translation type and functorial properties of Segal algebras. Mh. Math.90, 101–115 (1980); Part II. Mh. Math.92, 253–268 (1981).
Coifman, R., andWeiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc.83, 569–645 (1977).
Cowling, M.: Some applications of Grothendieck's theory of topological tensor products in harmonic analysis. Math. Ann.232, 273–285 (1978).
Dixmier, J., andMalliavin, P.: Factorisation de fonctions et de vecteurs indéfinement differentiable. Bull. Sci. Math.102, 305–330 (1978).
Edwards, R. E.: Approximation by convolutions. Pac. J. Math.15, 85–95 (1965).
Eymard, P., andTerp, M.: La transformation de Fourier et son inverse sur le groupe desax+b d'un corps local. In: Analyse Harmonique sur les Groupes de Lie II. pp. 207–248. Lecture Notes Math. 739. Berlin-Heidelberg-New York: Springer. 1979.
Feichtinger, H. G.: A characterization of Wiener's algebra on locally compact groups. Arch. Math.29, 136–140 (1977).
Feichtinger, H. G.: Multipliers fromL 1(G) to a homogeneous Banach space. J. Math. Anal. Appl.61, 341–356 (1977).
Feichtinger, H. G.: The minimal strongly character invariant Segal algebra, I and II. Preprint. Univ. Wien. 1979.
Feichtinger, H. G.: Un espace de Banach de distributions tempérées sur les groupes localement compacts abéliens. C. R. Acad. Sci. Paris, Sér. A290, 791–794 (1980).
Feichtinger, H. G.: A characterization of minimal homogeneous Banach spaces. Proc. Amer. Math. Soc.81, 55–61 (1981).
Feichtinger, H. G.: Banach convolution algebras of Wiener's type. In: Proc. Conf. Functions, Series, Operators, Budapest 1980. (To appear).
Feichtinger, H. G.: Banach spaces of Wiener's type and interpolation. In: Proc. Funct. Anal. ApproxO Oberwolfach, 1980, pp. 153–165. Intern. Ser. Numer. Math. 60, Basel-Stuttgart: Birkhäuser. 1981.
Feichtinger, H. G.: Good vectors and Besov spaces of order zero. Announcement in Abstracts. Amer. Math. Soc. 1981.
Feichtinger, H. G., Graham, C. C., andLakien, E. H.: Factorization in commutative weakly self-adjoint Banach algebras. Pac. J. Math.80, 117–125 (1979).
Gaudry, G. I.: Quasimeasures and operators commuting with convolution. Pac. J. Math.18, 461–476 (1966).
Holland, F.: On the representation of functions as Fourier transforms of unbounded measures. Proc. London Math. Soc. (3)30, 347–365 (1975).
Kahane, J. P.: Séries de Fourier Absolument Convergentes. Ergeb. d. Math. 50. Berlin-Heidelberg-New York: Springer. 1970.
Krogstad, H. E.: Multipliers of Segal algebras. Math. Scand.38, 285–303 (1976).
Krogstad, H. E.: Segal algebras on quotient groups. Preprint. Univ. Trondheim.
Larsen, R.: Introduction to the Theory of Multipliers. Berlin-Heidelberg-New York: Springer, 1971.
Leptin, H.: On one-sided harmonic analysis in non commutative locally compact groups. J. reine angew. Math.306, 122–153 (1979).
Losert, V.: A characterization of the minimal strongly character invariant Segal algebra. Ann. Inst. Fourier30/3, 129–139 (1980).
Osborne, M. S.: On the Schwartz-Bruhat space and the Paley-Wiener theorem for locally compact abelian groups. J. Funct. Anal.19, 40–49 (1975).
Petzeltová, H., andVrbova, P.: Factorization in the algebra of rapidly decreasing functions on ℝ n . Comm. Univ. Car.19/3, 489–499 (1978).
Poguntke, D.: Gewisse Segalsche Algebren auf lokalkompakten Gruppen. Arch. Math.33, 454–460 (1980).
Reiter, H.: Classical Harmonic Analysis and Locally Compact Groups. Oxford: University Press, 1968.
Reiter, H.: Über den Satz von Weil-Cartier. Mh. Math.86, 13–62 (1978).
Rieffel, M. A.: Induced Banach representations of Banach algebras and locally compact groups. J. Funct. Anal.1, 443–491 (1967).
Rieffel, M. A.: Multipliers and tensor products ofL p-spaces of locally compact groups. Stud. Math.33, 71–82 (1969).
Stewart, J.: Fourier transforms of unbounded measures. Can. J. Math.31, 1281–1292 (1979).
Tewari, U. B.: Isomorphisms of some convolution algebras and their multiplier algebras. Bull. Austral. Math. Soc.7, 321–335 (1972).
Tewari, U. B.: Multipliers of Segal algebras. Proc. Amer. Math. Soc.54, 157–161 (1976).
Wiener, N.: The Fourier Integral and Certain of Its Applications. Cambridge: University Press. 1933.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Feichtinger, H.G. On a new Segal algebra. Monatshefte für Mathematik 92, 269–289 (1981). https://doi.org/10.1007/BF01320058
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01320058