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On the convolution of Roumieu ultradistributions through the \(\epsilon \) tensor product

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Abstract

The equivalence of several definitions of convolution of two Roumieu ultradistributions is proved. For that purpose, the \(\varepsilon \) tensor product of \(\dot{\tilde{\mathcal{B }}}^{\{M_p\}}\) and a locally convex space \(E\) is considered.

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References

  1. Dierolf, P., Dierolf, S.: Topological Properties of the Dual Pair \((\dot{B}(\Omega )^{\prime }, \dot{B}(\Omega )^{\prime \prime })\). Pac. J. Math 108, 51–82 (1983)

    Google Scholar 

  2. Dierolf, P., Voigt, J.: Convolution and S-convolution of distributions. Collectanea Math 29, 185–196 (1978)

    MathSciNet  Google Scholar 

  3. Horvath, J.: Sur la convolution des distributions. Bull. Sci. Math 98(2), 183–192 (1974)

    MathSciNet  Google Scholar 

  4. Kaminski, A.: Convolution, product and Fourier transform of distributions. Studia Math 74, 83–96 (1982)

    MATH  MathSciNet  Google Scholar 

  5. Kamiński, A., Kovačević, D., Pilipović, S.: The equivalence of various defnitions of the convolution of ultradistributions. Trudy Mat. Inst. Steklov 203, 307–322 (1994)

    Google Scholar 

  6. Komatsu, H.: Ultradistributions, I: Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo, Sect. IA Math 20(1), 25–105 (1973)

    Google Scholar 

  7. Komatsu, H.: Ultradistributions, II: The kernel theorem and ultradistributions with support in submanifold. J. Fac. Sci. Univ. Tokyo, Sect. IA Math 24(3), 607–628 (1977)

    Google Scholar 

  8. Komatsu, H.: Ultradistributions, III: Vector valued ultradistributions and the theory of kernels. J. Fac. Sci. Univ. Tokyo, Sect. IA Math 29(3), 653–717 (1982)

    Google Scholar 

  9. Ortner, N.: On convolvability conditions for distributions. Monatsh. Math 160, 313–335 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ortner, N., Wagner, P.: Applications of weighted DLp-spaces to the convolution of distributions. Bull. Polish Acad. Sci. Math 37, 579–595 (1990)

    MathSciNet  Google Scholar 

  11. Ortner, N., Wagner, P.: Distribution-Valued Analytic Functions—Theory and Applications. Max-Plank-Institut für Mathematik in den Naturwissenschaften, Leipzig (2008)

  12. Pilipović, S.: On the convolution in the space of Beurling ultradistributions. Comment. Math. Univ. St. Paul 40, 15–27 (1991)

    MATH  MathSciNet  Google Scholar 

  13. Pilipović, S.: Characterizations of bounded sets in spaces of ultradistributions. Proc. Am. Math. Soc. 120(4), 1191–1206 (1994)

    MATH  Google Scholar 

  14. Schwartz, L.: Théorie des distributions á valeurs vectorielles. I, Ann. Inst. Fourier 7, 1–141 (1957)

    Google Scholar 

  15. Schwartz, L.: Théorie des distributions. I, II. 2nd edn. Hermann, Paris (1966)

  16. Shiraishi, R.: On the definition of convolution for distributions. J. Sci. Hiroshima Univ. Ser. A 23, 19–32 (1959)

    MATH  MathSciNet  Google Scholar 

  17. Shiraishi, R., Itano, M.: On the multiplicative product of distributions. J. Sei. Hiroshima Univ. A -I 28, 223–235 (1964)

    Google Scholar 

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Acknowledgments

The research of this project is supported by the Serbian Ministry of Education, Science and Technological Development 17424 as well as of the DAAD project Center of Excellence for Applications of Mathematics.

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Correspondence to Stevan Pilipović.

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Communicated by A. Constantin.

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Pilipović, S., Prangoski, B. On the convolution of Roumieu ultradistributions through the \(\epsilon \) tensor product. Monatsh Math 173, 83–105 (2014). https://doi.org/10.1007/s00605-013-0503-4

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