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Existence theorems for convolution of ultradistributions

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 Abstract: In the paper we give sufficient conditions for the existence of convolution of two ultradistributions of Beurling type. We distinguish two types of sufficient conditions: a) conditions in terms of the supports of ultradistributions; b) conditions in terms of subspaces of ultradistributions on which convolution can be defined as a bilinear mapping.
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Twórcy
  • Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8, 40-013 Katowice, Poland
  • Institute of Mathematics, University of Novi Sad, Trg Obradovića 4, 21000 Novi Sad, Yugoslavia
  • Institute of Mathematics, University of Novi Sad, Trg Obradovića 4, 21000 Novi Sad, Yugoslavia
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Bibliografia
[1] P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier-PWN, Amsterdam-Warszawa, 1973.
[2] A. Beurling, Quasi-analyticity and general distributions, Lectures 4 and 5, AMS Summer Institute, Stanford, 1961.
[3] K. Floret and J. Wloka, Einführung in die Theorie der lokalkonvexen Räume, Lecture Notes in Math. 71, Springer, Berlin, 1968.
[4] J. Horvath, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, 1966.
[5] A. Kamiński, Integration and irregular operations, Ph.D. Thesis, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1975.
[6] A. Kamiński, On convolutions, products and Fourier transforms of distributions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 369-374.
[7] A. Kamiński, D. Kovačević and S. Pilipović, The equivalence of various definitions of the convolution of ultradistributions, Proc. Steklov Inst. Math., 203 (1994), 307-322.
[8] A. Kamiński, D. Perišić and S. Pilipović, On the convolution of ultradistributions, this volume, 79-91.
[9] A. Kamiński and J. Uryga, Convolution in $K'{M_p}$-spaces, in: Generalized Functions, Convergence Structures and their Applications, B. Stanković, E. Pap, S. Pilipović and V. S. Vladimirov (eds.), Plenum Press, New York, 1988, 187-196.
[10] H. Komatsu, Ultradistributions, I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Mat. 20 (1973), 25-105.
[11] D. Kovačević, Some operations on the space $S'^{(M_p)}$ of tempered ultradistributions, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., to appear.
[12] D. Kovačević, The spaces of weighted and tempered ultradistributions, ibid., to appear.
[13] D. Kovačević and S. Pilipović, Structural properties of the spaces of tempered ultradistributions, in: Proc. of the Conf. on Complex Analysis and Generalized Functions, Varna, 1991, Publ. House of the Bulgar. Acad. Sci., Sofia, 1993, 169-184.
[14] D. Kovačević and S. Pilipović, Integral transforms on the spaces of tempered ultradistributions, preprint.
[15] S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris, 1952.
[16] N. Ortner and P. Wagner, Applications of weighted $D'_{L^p}$ spaces to the convolution of distributions, Bull. Polish Acad. Sci. 37 (1989), 579-595.
[17] N. Ortner and P. Wagner, Sur quelques propriétés des espaces $D'_{L^p}$ de Laurent Schwartz, Boll. Un. Mat. Ital. (6) 2-B (1983), 353-357.
[18] S. Pilipović, On the convolution in the space of Beurling ultradistributions, Comm. Math. Univ. St. Pauli 40 (1991), 15-27.
[19] S. Pilipović, On the convolution in the space $D^(M_p)_{L^2}$, Rend. Sem. Mat. Univ. Padova 79 (1988), 25-36.
[20] A. P. Robertson and W. Robertson, Topological Vector Spaces, Cambridge University Press, Cambridge, 1964.
[21] J. Uryga, On a criterion of existence of convolution of generalized functions in spaces of the type $K{M_p}'$, Preprint 15, Ser. B, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1986 (in Polish).
[22] J. Uryga, On compatibility of supports of generalized functions of Gelfand-Shilov type, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 36 (1988), 143-150.
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DML-PL
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