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Almost periodic ultradistributions of Beurling and of Roumieu type. (English) Zbl 0984.46023

To extend results of I. Cioranescu [Proc. Am. Math. Soc. 116, No. 1, 127-134 (1992; Zbl 0765.46021)] on almost periodic ultradistributions on the real line, the author introduces for a given weight function \(\omega\) the bounded ultradistributions of Beurling (resp. Roumieu) type as \({\mathcal D}_{L^1,(\omega)}'(\mathbb{R}^N)\) (resp. \({\mathcal D}_{L^1,\{\omega\}}'(\mathbb{R}^N)\)), where \[ \begin{aligned} {\mathcal D}_{L^1,(\omega)}(\mathbb{R}^N) & =\{f\in{\mathcal D}(\mathbb{R}^N): \forall\lambda> 0:\|f\|_\lambda< \infty\},\\ {\mathcal D}_{L^1,\{\omega\}}(\mathbb{R}^N) &= \{f\in{\mathcal D}(\mathbb{R}^N): \exists\lambda> 0:\|f\|_\lambda< \infty\},\\ \|f\|_\lambda &:= \sup_{\alpha\in \mathbb{N}^N_0}\|f^{(\alpha)}\|_{L^1(\mathbb{R}^N)} \exp\Biggl(-\lambda\varphi^*\Biggl({|\alpha|\over \lambda}\Biggr)\Biggr),\end{aligned} \] and where \(\varphi^*\) is the Young conjugate of the convex function \(\varphi(t):= \omega(e^t)\).
A bounded ultradistribution \(T\) is called almost periodic, if \(T\) is the \(\beta({\mathcal D}_{L^1,*}'(\mathbb{R}^N)\), \({\mathcal D}_{L^1,*}(\mathbb{R}^N))\)-limit of a sequence of trigonometric polynomials. Various characterizations of almost periodic ultradistributions are derived and it is investigated how an almost periodic ultradistribution in one variable can be obtained from its spectrum and its Fourier coefficients.

MSC:

46F05 Topological linear spaces of test functions, distributions and ultradistributions

Citations:

Zbl 0765.46021
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References:

[1] Luigi Amerio and Giovanni Prouse, Almost-periodic functions and functional equations, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne, 1971.
[2] Göran Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351 – 407 (1966). · Zbl 0166.36501 · doi:10.1007/BF02590963
[3] Rüdiger W. Braun, An extension of Komatsu’s second structure theorem for ultradistributions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1993), no. 2, 411 – 417. · Zbl 0811.46031
[4] R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), no. 3-4, 206 – 237. · Zbl 0735.46022 · doi:10.1007/BF03322459
[5] Ioana Cioranescu, The characterization of the almost periodic ultradistributions of Beurling type, Proc. Amer. Math. Soc. 116 (1992), no. 1, 127 – 134. · Zbl 0765.46021
[6] Ioana Cioranescu, Asymptotically almost periodic distributions, Appl. Anal. 34 (1989), no. 3-4, 251 – 259. · Zbl 0661.46031 · doi:10.1080/00036818908839898
[7] C. Corduneau, Almost periodic functions, Tracts in Pure Appl. Math. 22, Interscience, New York (1968).
[8] Hikosaburo Komatsu, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25 – 105. · Zbl 0258.46039
[9] Michael Langenbruch, Surjective partial differential operators on spaces of ultradifferentiable functions of Roumieu type, Results Math. 29 (1996), no. 3-4, 254 – 275. · Zbl 0859.35019 · doi:10.1007/BF03322223
[10] Reinhold Meise and B. Alan Taylor, Whitney’s extension theorem for ultradifferentiable functions of Beurling type, Ark. Mat. 26 (1988), no. 2, 265 – 287. · Zbl 0683.46020 · doi:10.1007/BF02386123
[11] Raghavan Narasimhan, Analysis on real and complex manifolds, 2nd ed., Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Advanced Studies in Pure Mathematics, Vol. 1. · Zbl 0188.25803
[12] Stevan Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1191 – 1206. · Zbl 0816.46026
[13] Stevan Pilipović, Structural theorems for ultradistributions, Dissertationes Math. (Rozprawy Mat.) 340 (1995), 223 – 235. Different aspects of differentiability (Warsaw, 1993). · Zbl 0837.46031
[14] Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966 (French). · Zbl 0962.46025
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