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Tytuł książki
Tytuł rozdziału
On convolution of ultradistributions
Treść / Zawartość
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![C:\Users\Tomek\Desktop\DOZROB\rm340\pdf\rm34018.pdf [zdalny]](images/mime-icons/pdf.gif "article_8")
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Abstrakty
EN
Abstract: Several definitions of convolution of ultradistributions of Beurling type as well as of tempered ultradistributions of Beurling type are presented. The definitions are analogous to the known definitions of the convolution of distributions and tempered distributions introduced by C. Chevalley, L. Schwartz, R. Shiraishi, V. S. Vladimirov and others. Similarly to the case of distributions, all the considered definitions of the convolution of ultradistributions are equivalent and the same is true in the case of tempered ultradistributions. Proofs are presented in separate papers.
Słowa kluczowe
Kategorie tematyczne
Twórcy
autor
- Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8, 40-013 Katowice, Poland
- Institute of Mathematics, Silesian Technical University, Kaszubska 23, 44-100 Gliwice, Poland
autor
- Institute of Mathematics, University of Novi Sad, Trg Obradovića 4, 21000 Novi Sad, Yugoslavia
autor
- Institute of Mathematics, University of Novi Sad, Trg Obradovića 4, 21000 Novi Sad, Yugoslavia
Strony
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] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 (1990), 206-237.
[4] G. Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351-407.
[5] C. Chevalley, Theory of Distributions, Columbia University, 1950-51.
[6] I. Ciorănescu and L. Zsidó, ω-ultradistributions and their applications to operator theory, in: Spectral Theory, Banach Center Publ. 8, PWN, Warszawa, 1982, 77-220.
[7] P. Dierolf and J. Voigt, Convolution and S'-convolution of distributions, Collect. Math. 29 (1978), 185-196.
[8] Y. Hirata, On the convolutions in the theory of distributions, J. Sci. Hiroshima Univ. Ser. A 22 (1958), 89-98.
[9] Y. Hirata and H. Ogata, On the exchange formula for distributions, ibid., 147-152.
[10] J. Horvath, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, 1966.
[11] A. Kamiński, Remarks on delta- and unit-sequences, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom Phys. 26 (1978), 25-30.
[12] A. Kamiński, Convolution, product and Fourier transform of distributions, Studia Math. 74 (1982), 83-86.
[13] 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.
[14] A. Kamiński, D. Perišić and S. Pilipović, Existence theorems for convolution of ultradistributions, this volume, 93-114.
[15] A. Kamiński, D. Perišić and S. Pilipović, The convolution in the space of tempered ultradistributions, preprint.
[16] H. Komatsu, Ultradistributions I-III, J. Fac. Sci. Univ. Tokyo Sect. IA Mat. 20 (1973), 25-105; 24 (1977), 607-628; 29 (1982), 653-718.
[17] H. Komatsu, Microlocal analysis in Gevrey classes and in complex domains, Lecture Notes in Math. 1726, Springer, Berlin, 1989, 426-493.
[18] 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.
[19] D. Kovačević and S. Pilipović, Structural properties of the spaces of tempered ultradistributions, Proc. of the Conf. on Complex Analysis and Generalized Functions, Varna, 1991, Publ. House of the Bulgar. Acad. Sci., Sofia, 1993, 169-184.
[20] R. Meise, Sequence space representations for zero-solutions of convolution equations on ultradifferentiable functions of Roumieu type, Studia Math. 85 (1987), 203-227.
[21] S. Pilipović, On the convolution in the space $D^{(M_p)}_{L²}$, Rend. Sem. Mat. Univ. Padova 79 (1988), 25-36.
[22] S. Pilipović, Tempered ultradistributions, Boll. Un. Mat. Ital. (7) 2-B (1988), 235-251.
[23] S. Pilipović, On the convolution in the space of Beurling ultradistributions, Comm. Math. Univ. St. Pauli 40 (1991), 15-27.
[24] S. Pilipović, Multipliers, convolutors and hypoeliptic convolutors for tempered ultradistributions, in: Proc. of the Conference on Generalized Functions and their Applications, Banaras Hindu University, 1991.
[25] S. Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191-1206.
[26] R. Shiraishi, On the definition of convolution for distributions, J. Sci. Hiroshima Univ. Ser. A 23 (1959), 19-32.
[27] L. Schwartz, Produits tensoriels topologiques et d'espaces vectoriels topologiques. Espaces vectoriels topologiques nucléaires, Séminaire Schwartz, Institut Henri Poincaré, 1953-54, Paris, 1954.
[28] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.
[29] J. Uryga, On tensor product and convolution of generalized functions of Gelfand-Shilov type, in: Generalized Functions and Convergence, World Scientific, Singapore, 1990, 251-264.
[30] V. S. Vladimirov, Equations of Mathematical Physics, Nauka, Moscow, 1968 (in Russian); English edition: Marcel Dekker, New York, 1971.
[31] P. Wagner, Zur Faltung von Distributionen, Math. Ann. 276 (1987), 467-485.
[32] R. Wawak, Improper integrals of distributions, Studia Math. 86 (1987), 205-220.
[33] K. Yoshinaga and H. Ogata, On convolutions, J. Sci. Hiroshima Univ. 22 (1958), 15-24.
[2] A. Beurling, Quasi-analyticity and general distributions, Lectures 4 and 5, AMS Summer Institute, Stanford, 1961.
[3] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 (1990), 206-237.
[4] G. Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351-407.
[5] C. Chevalley, Theory of Distributions, Columbia University, 1950-51.
[6] I. Ciorănescu and L. Zsidó, ω-ultradistributions and their applications to operator theory, in: Spectral Theory, Banach Center Publ. 8, PWN, Warszawa, 1982, 77-220.
[7] P. Dierolf and J. Voigt, Convolution and S'-convolution of distributions, Collect. Math. 29 (1978), 185-196.
[8] Y. Hirata, On the convolutions in the theory of distributions, J. Sci. Hiroshima Univ. Ser. A 22 (1958), 89-98.
[9] Y. Hirata and H. Ogata, On the exchange formula for distributions, ibid., 147-152.
[10] J. Horvath, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, 1966.
[11] A. Kamiński, Remarks on delta- and unit-sequences, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom Phys. 26 (1978), 25-30.
[12] A. Kamiński, Convolution, product and Fourier transform of distributions, Studia Math. 74 (1982), 83-86.
[13] 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.
[14] A. Kamiński, D. Perišić and S. Pilipović, Existence theorems for convolution of ultradistributions, this volume, 93-114.
[15] A. Kamiński, D. Perišić and S. Pilipović, The convolution in the space of tempered ultradistributions, preprint.
[16] H. Komatsu, Ultradistributions I-III, J. Fac. Sci. Univ. Tokyo Sect. IA Mat. 20 (1973), 25-105; 24 (1977), 607-628; 29 (1982), 653-718.
[17] H. Komatsu, Microlocal analysis in Gevrey classes and in complex domains, Lecture Notes in Math. 1726, Springer, Berlin, 1989, 426-493.
[18] 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.
[19] D. Kovačević and S. Pilipović, Structural properties of the spaces of tempered ultradistributions, Proc. of the Conf. on Complex Analysis and Generalized Functions, Varna, 1991, Publ. House of the Bulgar. Acad. Sci., Sofia, 1993, 169-184.
[20] R. Meise, Sequence space representations for zero-solutions of convolution equations on ultradifferentiable functions of Roumieu type, Studia Math. 85 (1987), 203-227.
[21] S. Pilipović, On the convolution in the space $D^{(M_p)}_{L²}$, Rend. Sem. Mat. Univ. Padova 79 (1988), 25-36.
[22] S. Pilipović, Tempered ultradistributions, Boll. Un. Mat. Ital. (7) 2-B (1988), 235-251.
[23] S. Pilipović, On the convolution in the space of Beurling ultradistributions, Comm. Math. Univ. St. Pauli 40 (1991), 15-27.
[24] S. Pilipović, Multipliers, convolutors and hypoeliptic convolutors for tempered ultradistributions, in: Proc. of the Conference on Generalized Functions and their Applications, Banaras Hindu University, 1991.
[25] S. Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191-1206.
[26] R. Shiraishi, On the definition of convolution for distributions, J. Sci. Hiroshima Univ. Ser. A 23 (1959), 19-32.
[27] L. Schwartz, Produits tensoriels topologiques et d'espaces vectoriels topologiques. Espaces vectoriels topologiques nucléaires, Séminaire Schwartz, Institut Henri Poincaré, 1953-54, Paris, 1954.
[28] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.
[29] J. Uryga, On tensor product and convolution of generalized functions of Gelfand-Shilov type, in: Generalized Functions and Convergence, World Scientific, Singapore, 1990, 251-264.
[30] V. S. Vladimirov, Equations of Mathematical Physics, Nauka, Moscow, 1968 (in Russian); English edition: Marcel Dekker, New York, 1971.
[31] P. Wagner, Zur Faltung von Distributionen, Math. Ann. 276 (1987), 467-485.
[32] R. Wawak, Improper integrals of distributions, Studia Math. 86 (1987), 205-220.
[33] K. Yoshinaga and H. Ogata, On convolutions, J. Sci. Hiroshima Univ. 22 (1958), 15-24.
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