Abstract
Let G be a locally compact abelian group. The concern of the present note is to extend (for exponents p>2) the saturation theorem on G stated as Theorem 4 in [5]. The extension will be established for approximation processes (It)t>0 acting on the submodule CP(G), p∈]1,+∞[, of the convolutionM 1(G)-module LP(G) which consists of all functions f∈LP(G) admitting as their Fourier transformsF Gf (in the sense of the theory of quasimeasures) complex Radon measures not necessarily absolutely continuous with respect to any Haar measure on the dual group Ĝ. Moreover, the relationship of the complex vector spaces CP(G) to some other function spaces, in particular to the vector spaces BP(G) introduced in [5], will be investigated.
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Bibliography
BOURBAKI, N.: Topologie générale, Chap. X, 2e éd. Actualités Sci. et Ind.1084. Paris: Hermann 1967.
BOURBAKI, N.: Intégration, Chap. I–IV, 2e éd. Actualités Sci. et Ind.1175. Paris: Hermann 1965.
BOURBAKI, N.: Théories spectrales, Chap. I–II. Actualités Sci. et Ind.1332. Paris: Hermann 1967
BUCHWALTER, H.: Saturation sur un groupe abélien localement compact. C. R. Acad. Sci. Paris250, 808–810 (1960).
DRESELER, B., SCHEMPP, W.: Saturation on locally compact abelian groups. Manuscripta Math.7, 141–174 (1972).
DRESELER, B., SCHEMPP, W.: On the Fourier transformation on spaces of (p,q)-multipliers (to appear).
DRESELER, B., SCHEMPP, W.: A general saturation theorem on differentiable manifolds (to appear).
EDWARDS, R. E., PRICE, J. F.: A naively constructive approach to boundedness principles, with applications to harmonic analysis. Enseignement Math.16, 255–296 (1970).
GAUDRY, G. I.: Quasimeasures and operators commuting with convolution. Pacific J. Math.18, 461–476 (1966).
GAUDRY, G. I.: Multipliers of type (p,q). Pacific J. Math.18, 477–488 (1966).
GAUDRY, G. I.: Topics in harmonic analysis. Lecture notes. Department of Mathematics, Yale University, New Haven 1969.
GAUDRY, G. I.: Bad behavior and inclusion results for multipliers of type (p,q). Pacific J. Math.35, 83–94 (1970).
HEWITT, E., ROSS, K. A.: Abstract harmonic analysis. Vol. I. Die Grundlehren der math. Wissenschaften, Band115. Berlin-Göttingen-Heidelberg: Springer-Verlag 1963.
KAHANE, J.-P., SALEM, R.: Ensembles parfaits et séries trigonométriques. Actualités Sci. et Ind.1301. Paris: Hermann 1963.
LARSEN, R.: An introduction to the theory of multipliers. Die Grundlehren der math. Wissenschaften, Band175. Berlin-Heidelberg-New York: Springer-Verlag 1971.
SALEM, R.: On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat.1, 353–365 (1950). Also in “Oeuvres mathématiques”, B. 34, pp. 481–493. Paris: Hermann 1967.
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Dreseler, B., Schempp, W. Saturation on locally compact abelian groups: An extended theorem. Manuscripta Math 8, 271–286 (1973). https://doi.org/10.1007/BF01297692
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DOI: https://doi.org/10.1007/BF01297692