Revisión de la teoría de Boehmians
Palabras clave:
Calculo operacional, operaciones de Mikusinski, funciones generalizadas, distriubuciones de Schwartz, Boehmian, espacio de BeohmianResumen
Este artículo está organizado en dos secciones seguidas de una lista de artículos seleccionados de los autores. Probablemente este es el primer intento de escribir con todo detalle acerca de los operadores de Boehmians y J. Mikusinski junto con las contribuciones de varios matemáticos respecto a Bohemians. La teoría de distribuciones de Schwartz fue desarrollada para dar un soporte y fundamentos matemáticos comprensibles en la generalización de las propiedades de la función delta de Dirac. A partir del trabajo de Sobolev y Schwartz, se hicieron intentos para generalizar el concepto de distribuciones. Colombeau construyó una nueva algebra diferenciable de funciones generalizadas conteniendo el espacio de distribución, en el cual el producto puede ser definido. El concepto de Boehmians, la mas reciente generalización de la teoría de distribuciones de Schwartz, esta motivada por los operadores regulares introducidos y por Boehme. Boehme no adoptó el nombre de Teoría de Bohemians, sino J. Mikusinski y P. Miku- siński fueron inspirados para desarrollar la Teoría de Boehme, que posiblemente adopto el nombre de Boehmians.
Descargas
Referencias
Plesner, A. I. On inclusion of Heaviside’s operational calculus in the spectral theory of maximal operators, Dokaldy-Akad. Nauk USSR, Vol. 26, N° 1, (1940), 10-12.
Detkin, V. A. Operational Calculus, Uspehi Mat. Nauk, Vol. 26, N° 22, (1947), 75-158.
Mikusiński, J. On the work of Polish mathematicians in the theory of generalized functions and operational calculus, Uspehi Mat. Nauk, Vol. 11, N° 6, (1956), 172.
Shtokalo, I. Z. Operational Calculus (Original Russian Edition) Hindustan Publ. Coorp. Delhi (1976), Ed. Sneddon, I. N. (English Edition), pp. 30-31.
Mikusiński, J. G. Rachunek Operatów (Polskie Towarzystwo, Warsaw), Engl. Trans.: “The Calculus of Operators”, Pergamon Press, Oxford 1959 (work published in 1953).
Sneddon, I. N. The Use of Integral Transforms, Tata McGraw-Hill Publ. Co. Ltd., New Dehi, India (1974). Original Edition : McGraw-Hill, Inc. (1972). Art. 9-1-4, pp. 488-89.
Buschman, R. G. Integral Transformations, Operational Calculus and Generalized Functions, Kluwer Academic Publishers, Boston, London (1996).
Bochner, S.. Vorlesungen über Fourierische Integrale, Akademie-Verlag, Berlin (1932)
Sobolev, S. L. Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaire hyperboliques normales, Math. Sbornik, Vol. 1, (1936), 39-72.
Schwartz, L. Théorie des Distributions, 2 Vols., Hermann, Paris (1950, 1951), Vol. I and II are republished by Actualitées Scientifiques et Industrilles, Herman & Cie, Paris, (1957, 1959).
Zemanian, A. H. Distribution Theory and Transform Analysis : An Introduction to Generalized Functions with Applications, McGraw-Hill Book Co., New York (1965). Republished by Dover Publications, Inc., New York (1987).
Zemanian, A. H. Generalized Integral Transformations, Interscience Publishers, New York (1968). Republished by Dover Publications, Inc., New York (1987).
Dirac, P. A. M. The physical interpretation of the quantum dynamics, Proc. Roy. Soc. London Ser. A., Vol.133, (1927), 621-641.
Mikusiński, J. G. Sur la méthode de généralization de M. Laurent Schwartz et sur la convergence faible, Fund. Math. Vol. 35, (1948), 235-239.
Temple, G. Theories and applications of generalized functions, J. London Math. Soc. Vol. 28, (1953), 134.
Temple, G. The theory of generalized functions, Proc. Royal Soc., London, Vol. 228, (1955), 175.
Beltrami, E. J. and Wohlers, M. R. Distributions and the Boundary Values of Analytic Functions, Academic Press, New York (1966).
Bremermann, H. Distributions, Complex Variables and Fourier Transforms, Addison- Wesley, Reading, Mass. (1965).
Carmichael, R. D. and Pilipovic, S. The Cauchy and Poisson kernels as ultradifferentiable functions, Complex variables, Vol. 35 (1998), 171-186.
Debnath, L. Integral Transforms and their Applications, CRC Press, Inc., New York, London (1995).
Debnath, L. and Mikusiński, P. Introduction to Hilbert Spaces with Applications, Second ed. Academic Press, New York (1999).
Erdélyi, A. The Stieltjes transform of generalized functions, Proc. Roy. Soc. Edinburgh, Vol. 76A (1977), 231-249.
Friedman, A. Generalized Functions and Partial Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey (1963).
Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 1, Properties and Operations, Academic Press, New York, London (1964).
Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 3, Theory of Differential Equations, Academic Press, New York, London (1967)
Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 2, Spaces of Fundamental and Generalized Functions, Academic Press, New York, London (1968).
Gel’fand, I. M. and Vilenkin, N. Ya. Generalized Functions, Vol. 4, Applications of Harmonic Analysis, Academic Press, New York, London (1964).
Hoskins, R. F. Generalized Functions, Ellis Horwood Ltd., Chichester, New York (1979).
Korevaar, J. Pansions and the theory of Fourier transforms, Amer. Math. Soc. Vol. 91, (1959), 53-101.
Mikusiński, J. and Sikorski, R. The Elementary Theory of Distributions I, Rozprawy Mat. 12, Warszwa (1957).
Pandey, J. N. On the Stieltjes transform of generalized functions, Proc. Cambridge Phil. Soc. Vol. 71, (1972), 85-96
Schwartz, L. Généralisation de la notion de fonction de dérivation, de transformation de fourier, et applications mathematques et physiques, Annales Univ. Grenoble Vol. 21, (1945), 57-64.
Mikusiński, J. On the value of distribution at a point, Bull. Acad. Pol. Sci. Ser. Sci. Math., Vol. 8, No. 10, (1960), 681-683.
Mikusiński, J. Irregular operations on distributions, Studia Math., Vol. 20, (1961), 163-169.
Antosik, P., Mikusi n ski, J. and Sikorski, R. Theory of Distributions : The Sequential Approach, Elsevier Scientific Publishing Co. Amsterdam; PWN-Polish Scientific Publishers, Warsaw (1973).
Mikusiński, J. Criteria of the existence and of associativity of the product of distributions, Studia Math. Vol. 21, (1962), 253-259.
Kaminski, A. On convolutions, products and Fourier transforms of distributions, Bull. Acad. Pol. Sci. Ser. Sci. Math. Vol. 25, (1977), 369-374.
Antosik, P. and Ligez a, J. Product of measures and functions of finite variation, Proc. Conference of Generalized Functions and Operational Calculi, Varna, Bulgaria (1975).
Lojasiewicz, S. Sur la valeur et le limite d’une distribution dans un point, Bull. Pol. Acad. Sci. Cl. Vol. III, N° 4, (1956), 239-242.
Zielezny, Z. Sur la défintion de Lojasiewicz de la valeur d’une distributions dans un point, Bull. Pol. Acad. Sci. Cl. Vol. III, N° 3, (1955), 519-520.
Loonker, Deshna. Distribution Spaces and Transform Analysis, Ph. D. Thesis, J. N. V. University, Jodhpur, India (2001).
Shornik, Krystyna. Prof. Jan Mikusiński -- life and work, in Notices from the ISMS (Internat. Soc. for Math. Scis.), July 2007, pp. 1-20, Sci. Math. Japonica (e-print); source : http://www.jams.or.jp.
Mikusiński, J. Une definition de distribution, Bull. Acad. Pol. Sci. Cl. III 3 (1955), 589-591.
Mikusiński, J. and Sikorski, R. The Elementary Theory of Distributions II, Rozprawy Mat. 25,
Warszwa (1961) .
Mikusiński, J. Bochner Integral, Birkhäuser, Basel and Stuttgart (1978).
Boehme, T. K. Operational Calculus and the Finite Part of Divergent Integrals, Ph. D. Dissertation, California, Institute of Technology , Pasadena, California (1960). Source : http:/etd.caltech.edu/etd/ available/etd_02222006-154540/.
Lighthill, M. Introduction to Fourier Analysis and Generalized Functions, Cambridge (1959).
Butzer, P. Singular integral equations of Volterra type and the finite part of divergent integrals, Arch. Rat. Mech. Analysis 3 (1959).
Hadamard, J. Lectures on Cauchy’s problem in Linear Hyperbolic Differential Equations, Dover (1952).
Bureau, F. Divergent integrals and partial differential equations, Comm. Pure Appl. Math. Vol. 8, (1955).
Boehme, T. K. The support of Mikusinski operators, Trans. Amer. Math. Soc. Vol.176, (1973), 319-334.
Mikusiński, J. and Mikusiński, P. Quotients de suites et lures applications dans l’analyse fonctionnlle, C. R. Acad. Sci. Paris, Vol. 293, Ser. I (1981), 463-464.
Mikusiński, P. Convergence of Boehmians, Japan J. Math. Vol. 9, N° 1, (1983), 159-179.
Mikusiński, P. Fourier transform for integrable Boehmians, Rocky Mountain J. Math. Vol. 17, (1987), 577-582.
Mikusiński, P. The Fourier transform of tempered Boehmians pp.303-309, In Fourier Analysis, Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York (1994).
Mikusiński, P. Tempered Boehmians and ultradistributions, Proc. Amer. Math. Soc. Vol. 123, N° 3, (1995), 813-817.
Debnath, L. Wavelet transform and their applications, PINSA-A, Vol. 64 (A), N° 6, (1998), 685-713.
Koornwinder, T. H. Wavelets : An Elementary Treatment of the Theory and Applications, World Scientific, Singapaore (1993).
Nemzer, D. Lacunary Boehmians, Integral Transforms and Special Functions, Vol. 16, (2005), 451- 459.
Samko, S. G., Kilbas, A. A. and Marichev, O. I. Fractional Integrals and derivatives, Gordon and Breach Science Publishers, Switzerland, India, Japan, (1993).
Podlubny, I. Fractional Differential Equations, Academic Press, New York, (1999).
Banerji, P. K. and Loonker, Deshna. An introduction to Boehmians and their transforms, Proc. 7th Internat. Conf. SSFA, Vol.7, (2006), 49-56.
Banerji, P. K. Unfolding the concepts of Boehmians and their applications, The Math. Student, Vol. 75, N° 1-4, (2006), 87-121.
Loonker, Deshna and Banerji, P. K. Plancherel theorem for wavelet transform for vector valued functions and Boehmians, J. Indian Math. Soc. Vol. 73, N° 1-2, (2006), 31-39.
Loonker, Deshna and Banerji, P.K. An approach to Boehmians : Discussing mathematical concepts without mathematical riddles, News Bull. Calcutta Math. Soc. Vol. 30, N° 4-6, (2007), 17-20.
Loonker, Deshna and Banerji, P. K. On the Laplace and the Fourier transformations of fractional integrals for integrable Boehmians, Bull. Cal. Math. Soc. Vol. 99, N° 4, (2007), 345-354.
Loonker, Deshna and Banerji, P. K.Mehler-Fock transform for integrable Boehmians, J. Indian Acad. Math. Vol. 29, N° 2, (2007), 475-481.
Banerji, P. K. and Loonker, Deshna. Laplace transform for integrable Boehmians, Bull. Cal. Math. Soc. Vol. 98, N° 5, (2006), 465-470.
Singh, A., Loonker, Deshna and Banerji, P. K. Fourier – Bessel transform for temepered Boehmians, Int. J. Math. Analysis, Vol. 4, N° 45 (2010), 2199- 2210.
Loonker, Deshna and Banerji, P. K. Kontorovich – Lebedev transform for integrable Boehmians, J. Indian Acad. Math. Vol. 32, N° 2, (2010), 683 -689.
Singh, A. and Banerji, P. K. Dunkl transform of integrable Boehmians, J. Raj. Acad. Phy. Sci. Vol. 10, N° 2, (2011), 169 -176.
Singh, A. and Banerji, P. K. Dunkl transform for temepered Boehmians, J. Indian Acad. Math. Vol. 34, N° 1, (2012), 9- 18.
Loonker, Deshna and Banerji, P. K. Natural transform for distribution and Boehmian spaces, Math. in Engg., Sci. and Aerospace (MESA), Vol. 4, N° 1, (2013), 69 – 76.
Singh, A. , Banerji, P. K. and Kalla, S. L. A uniquness theorem for Mellin transform for quotient spaces, SCIENTIA, Ser. A., Vol. 23 (2012), 25 – 30.
Descargas
Publicado
Número
Sección
Licencia
Derechos de autor 2014 @copyright
Esta obra está bajo una licencia Creative Commons Reconocimiento 3.0 Unported.
Avenida 2 “El Milagro”, entrada autónoma de la 