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On the continuation of the ideas of Heaviside and Mikusiński in operational calculus

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 1. Introduction. In this survey we consider operational calculus as a branch of linear functional analysis. The purpose of this survey is to give some ideas what the modern theory of operational calculus is about. We shall concentrate mainly on the basic problems of the theory, occasionally touching upon analytic and algebraic aspects. In several cases they lead to work of Heaviside and Mikusiński.
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  • Computer Center, Russian Academy of Sciences, Vavilova St. 40, Moscow 117333, Russia
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[Am] L. Amerio, Sulla trasformata doppia di Laplace, Atti Acad. Italia Mem. Cl. Sci. Fis. Mat. Natur. 12 (1941), 707-780.
[An1] P. Antosik, The isomorphism of certain rings of operators, Studia Math. 25 (1965), 227-230.
[An2] P. Antosik, On a topology of convergence, Colloq. Math. 21 (1970), 205-209.
[An3] P. Antosik, The commutativity of the limit with regular operations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 325-327.
[An4] P. Antosik, On the Mikusiński diagonal theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1971), 305-310.
[An5] P. Antosik and J. Mikusiński, On the Hermite expansions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 787-791.
[An6] P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier and PWN, Amsterdam-Warszawa, 1973.
[An7] P. Antosik and C. Swartz, Matrix Methods in Analysis, Lecture Notes in Math. 1113, Springer, Berlin, 1985.
[B1] S. Bellert, On foundations of operational calculus, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 855-858.
[B2] S. Bellert, Całka Fouriera i badanie czwórników elektrycznych metodą impulsową, Zeszyty Nauk. Politechn. Warsz. 42 (1959), 79-109 (in Polish).
[B3] S. Bellert, Metoda operatorów liczbowych, Rozpr. Elektrotechn. 5 (1959), 515-586 (in Polish).
[B4] S. Bellert, Rachunek operatorowy w przestrzeniach liniowych, Rozpr. Elektrotechn. 6 (1960), 169-212 (in Polish).
[B5] S. Bellert, On the continuation of the idea of Heaviside in the operational calculus, J. Franklin Inst. 276 (1963), 411-440.
[Be1] L. Berg, Moderne Operatorenrechnung, Z. Angew. Math. Mech. 39 (1959), 342-346.
[Be2] L. Berg, Asymptotische Auffassung der Operatorenrechnung, Studia Math. 21 (1962), 215-229.
[Be3] L. Berg, Einführung in die Operatorenrechnung, Deutsche Verlag Wiss., Berlin, 1962.
[Be4] L. Berg, Konstruktion von Operatorenkörper, Z. Angew. Math. Mech. 44 (1964), 3-4.
[Ber] D. L. Bernstein, The double Laplace integral, Duke Math. J. 8 (1941), 460-496.
[Bi1] R. Bittner, On certain axiomatics for the operational calculus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 7 (1959), 1-9.
[Bi2] R. Bittner, Operational calculus in linear spaces, Studia Math. 20 (1961), 1-18.
[Bi3] R. Bittner, Algebraic and analytic properties of solutions of abstract differential equations, Rozprawy Mat. 41 (1964).
[Bi4] R. Bittner, Rachunek operatorów w przestrzeniach liniowych, PWN, Warszawa, 1974 (in Polish).
[Bi5] R. Bittner, Rachunek operatorów w grupach, PWN, Warszawa, 1992 (in Polish).
[Boc] S. Bochner, L. Schwartz, Théorie des distributions, t. I et II, Bull. Amer. Math. Soc. 58 (1952), 78-85.
[Bo1] T. K. Boehme, On Mikusiński operators, Studia Math. 33 (1969), 127-140.
[Bo2] T. K. Boehme, The Mikusiński operators as a topological space, Amer. J. Math. 98 (1976), 55-66.
[Bro] M. S. Brodskiĭ, A problem suggested by I. M. Gel'fand, Uspekhi Mat. Nauk 12 (2) (1957), 129-132.
[Br1] Yu. A. Brychkov, A. P. Prudnikov and V. S. Shishov, Operational calculus, in: Itogi Nauki, Mat. Analiz, 16, VINITI, Akad. Nauk SSSR, Moscow, 1979, 99-148 (in Russian).
[Br2] Yu. A. Brychkov, H.-J. Glaeske, A. P. Prudnikov and Vu Kim Tuan, Multidimensional Integral Transformations, Gordon and Breach, Reading, Mass., 1992.
[Bu1] J. Burzyk, On convergence in the Mikusiński operational calculus, Studia Math. 75 (1983), 313-333.
[Bu2] J. Burzyk, On type II convergence in the Mikusiński operational calculus, Studia Math. 77 (1983), 17-27.
[Bu3] J. Burzyk, Nonharmonic solutions of the Laplace equation, in: Generalized Functions, Convergence Structures and their Applications, Plenum Press, New York, 1988, 3-11.
[Bu4] J. Burzyk, A Paley-Wiener type theorem for regular operators of bounded support, Studia Math. 93 (1989), 187-200.
[Car] J. R. Carson, Heaviside operational calculus, Bell System Techn. J. 1 (1922), 43; 4 (1923), 685; 5 (1926), 50 and 5 (1926), 336.
[Col] J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984.
[Coo] J. L. Cooper, Heaviside and the operational calculus, Math. Gazette 36 (315) (1952), 5-19.
[Cr] M. M. Crum, On the resultant of two functions, Quart. J. Math. Oxford 12 (1941), 108-111.
[Cur] H. B. Curry, The Heaviside operational calculus, Amer. Math. Monthly 50 (1943), 365-379.
[DAd] R. D'Adhemar, Exercises et leçons d'analyse, Paris, 1908.
[Dal] J. P. Dalton, Symbolic operators, Witwatersrand Univ. Press, Johannesburg, 1954.
[Del] J. Delsarte, Le calcul linéaire, Bull. Soc. Math. France, 1937, 42-53, and in: Oeuvres de Jean Delsarte, I, Paris, 1971, 411-422.
[Di1] I. H. Dimovski, The uniqueness of the Mikusiński field, Fiz.-Mat. Spis. 5 B"lgar. Akad. Nauk 5 (38) (1) (1962), 56-59.
[Di2] I. H. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulgare Sci. 19 (1966), 1111-1114.
[Di3] I. H. Dimovski, On an operational calculus for a differential operator, C. R. Acad. Bulgare Sci. 21 (1968), 513-516.
[Di4] I. H. Dimovski, The contemporary operational calculus, Fiz.-Mat. Spis. B"lgar. Akad. Nauk 19 (1976), 54-71 (in Bulgarian).
[Di5] I. H. Dimovski, Convolutional Calculus, Publ. House of the Bulgar. Acad. Sci., Sofia, 1982.
[Di6] I. H. Dimovski, Convolutional Calculus, 2nd ed., Kluwer Academic Publishers, Dordrecht, 1990.
[DV1] V. A. Ditkin, Operational calculus, Uspekhi Mat. Nauk 2 (6) (1947), 72-158 (in Russian).
[DV2] V. A. Ditkin, Operational calculus for functions defined on the whole real line, Dokl. Akad. Nauk SSSR 112 (1957), 191-194 (in Russian).
[DV3] V. A. Ditkin, Operational calculus theory, Dokl. Akad. Nauk SSSR 116 (1957), 15-17 (in Russian).
[DV4] V. A. Ditkin, Operational calculus theory, Dokl. Akad. Nauk SSSR 123 (1958), 395-396 (in Russian).
[DV5] V. A. Ditkin, The generalized Laplace transforms, Inzh.-Fiz. Zh. 1 (11) (1958), 98-104 (in Russian).
[DV6] V. A. Ditkin, The convolutional Hilbert spaces, Dokl. Akad. Nauk SSSR 286 (1986), 528-530 (in Russian).
[DV7] V. A. Ditkin, Integral representations in convolutional spaces, Dokl. Akad. Nauk SSSR 295 (1987), 791-795 (in Russian).
[DV8] V. A. Ditkin and P. I. Kuznetsov, Operational Calculus Manual, Gostekhizdat, Moscow, 1951 (in Russian).
[DV9] V. A. Ditkin and A. P. Prudnikov, Operational Calculus for Two Variables and its Applications, Fizmatgiz, Moscow, 1958 (in Russian).
[DV10] V. A. Ditkin and A. P. Prudnikov, Integral Transformation and Operational Calculus, Fizmatgiz, Moscow, 1961 (in Russian).
[DV11] V. A. Ditkin and A. P. Prudnikov, The operational calculus of Bessel operators, Zh. Vychisl. Mat. i Mat. Fiz. 2 (1962), 997-1018 (in Russian).
[DV12] V. A. Ditkin and A. P. Prudnikov, The operational calculus arising from the Bessel equations, Zh. Vychisl. Mat. i Mat. Fiz. 3 (1963), 223-238 (in Russian).
[DV13] V. A. Ditkin and A. P. Prudnikov, Operational calculus for functions with integral arguments and its applications in discrete analysis, Inzh.-Fiz. Zh. 7 (7) (1964), 101-115 (in Russian).
[DV14] V. A. Ditkin and A. P. Prudnikov, Operational calculus, in: Itogi Nauki, Mat. Analiz, 1964, VINITI, Akad. Nauk SSSR, Moscow, 1966, 7-75 (in Russian).
[DV15] V. A. Ditkin and A. P. Prudnikov, Operational Calculus, Izd. Vysshaya Shkola, Moscow, 1965 (in Russian).
[DV16] V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.
[DV17] V. A. Ditkin and A. P. Prudnikov, Operational Calculus Manual, Izd. Vysshaya Shkola, Moscow, 1965 (in Russian).
[DV18] V. A. Ditkin and A. P. Prudnikov, A method of justification for operational calculus, Inzh.-Fiz. Zh. 24 (1973), 1114-1117 (in Russian).
[Do] G. Doetsch, Handbuch der Laplace-transformation, bd. I-IV, Birkhäuser, Basel, 1950-1966.
[Dr] M. Drobot and J. Mikusiński, The translation operator and its applications to the statics of beams, Uspekhi Mat. Nauk 13 (2) (1958), 73-92 (in Russian).
[Du1] J. Dufresnoy, Sur le produit de composition de deux fonctions, C. R. Acad. Sci. Paris 225 (1947), 857-859.
[Du2] J. Dufresnoy, Autour du théorème de Phragmén-Lindelöf, Bull. Sci. Math. 72 (1948), 17-22.
[Éf1] A. M. Éfros, Some applications of operational calculus in analysis, Mat. Sb. 42 (1935), 699-706 (in Russian).
[Éf2] A. M. Éfros and A. M. Danilevskiĭ, Operational Calculus and Contour Integrals, Khar'kov, 1937 (in Russian).
[Er1] A. Erdélyi, On a generalisation of the Laplace transformation, Proc. Edinburgh Math. Soc. 10 (1954), 53-55.
[Er2] A. Erdélyi, Operational Calculus, Pasadena, 1955.
[Er3] A. Erdélyi, Lectures on Mikusiński's Theory of Operational Calculus and Generalized Functions, Pasadena, 1959.
[Er4] A. Erdélyi, General asymptotic expansions of Laplace integrals, Arch. Rational Mech. Anal. 7 (1961), 1-20.
[Er5] A. Erdélyi, Operational Calculus and Generalized Functions, Holt, Rinehart and Winston, New York, 1962.
[Fen] S. Fenyö, Über den Zusammenhang zwischen den Mikusińskischen Operatoren und den Distributionen, Math. Nachr. 19 (1958), 161-164.
[Fis] B. Fisher, The product of distributions, Quart. J. Math. Oxford (2) 22 (1971), 291-298.
[Fl] H. G. Flegg, A survey of the development of operational calculus, Internat. J. Math. Ed. Sci. Tech. 2 (1971), 329-335.
[Fo] C. Foiaş, Approximation des opérateurs de J. Mikusiński par des fonctions continues, Studia Math. 21 (1961), 73-74.
[Fr] H. Freudenthal, Operatoren van Heaviside tot Mikusiński, Simon Stevin 33 (1959), 3-19.
[Ge1] E. Gesztelyi, Generalized functions defined by fields which are isomorphic to the field of Mikusiński operators, Publ. Math. Debrecen 16 (1969), 265-296.
[Ge2] E. Gesztelyi and Á. Száz, On generalized convolution quotients, ibid. 16 (1969), 297-305.
[Gi1] J. Gilly, Comparaison entre la théorie de la composition et la transformation de Laplace-Carson, C. R. Acad. Sci. Paris 218 (1944), 382-384.
[Gi2] J. Gilly, Sur une extension de la théorie de la composition de première espèce, C. R. Acad. Sci. Paris 218 (1944), 100-102.
[Gi3] J. Gilly, Les parties finies d'intégrales et la transformation de Laplace-Carson, Revue Scientifique 83 (1945), 259-270.
[Gl] H.-J. Glaeske, Über die Konstruktion von Faltungen für Integraltransformationen mit orthogonalen Polynomen als Kern, Wiss. Z. Friedrich-Schiller-Univ. Jena Math.-Naturwiss. R. 29 (1980), 213-220.
[Had] J. Hadamard, Recherches sur les solutions fondamentales et l'intégration des équations linéaires aux dérivées partielles, Ann. Ecole Norm. Sup. (3) 22 (1905), 101-141.
[Hau] F. Hausdorff, Momentprobleme für ein endliches Interval, Math. Z. 16 (1923), 220-248.
[He1] O. Heaviside, Operators in mathematical physics, Proc. Roy. Soc. A I, 52 (1893), 504-529 and II, 54 (1894), 105-143.
[He2] O. Heaviside, Electromagnetic Theory, London, 1899.
[Ich] S. Ichikawa and K. Akira, Matrix operational calculus and its applications, Mem. Eng. Kyoto Univ. 32 (1970), 210-222; 33 (1971), 80-93.
[Ka] G. K. Kalisch, A functional analysis proof of Titchmarsh's theorem on convolution, J. Math. Anal. Appl. 5 (1962), 176-183.
[Ka1] A. Kamiński, Remarks on delta- and unit-sequences, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1978), 25-30.
[Ka2] A. Kamiński, On characterization of topological convergence, in: Proc. Conf. on Convergence (Szczyrk 1979), Polska Akademia Nauk, Katowice, 1980, 50-70.
[Ka3] A. Kamiński, Convolution, product and Fourier transform of distributions, Studia Math. 74 (1982), 83-96.
[Kan] S. Kantorovitz, On operational calculus and spectral operators, University of Minnesota, 1962.
[Ki1] W. Kierat, A remark on convergent power series of the differential operator in the field of Mikusiński operators, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 843-845.
[Ki2] W. Kierat, Applications of quadratic forms generated by Sturm-Lioville operators in the theory of generalized functions, Prace Nauk. Uniw. Śląskiego 388 (1980).
[Ki3] W. Kierat, A sequential characterizations of distributions, in: Proc. Second Conference on Convergence, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1981, 32-38.
[Ki4] W. Kierat, Some Fourier transform inversion theorem, Studia Math. 77 (1984), 485-488.
[Ki5] W. Kierat and J. Mikusiński, Sur un théorème de calcul opérationnel d'intervalle fini, Studia Math. 25 (1964), 93-96.
[Ki6] W. Kierat and K. Skórnik, A remark on the Foiaş theorem on convolution, Bull. Polish Acad. Sci. Math. 34 (1986), 15-17.
[Ki7] W. Kierat and K. Skórnik, A remark on the operational exponential functions, ibid. 35 (1987), 383-386.
[Ki8] W. Kierat and K. Skórnik, On continuity of perfect operators in the space $C₀^{(∞)}$, in: Complex Analysis and Applications (Sofia 87), 1989, 267-271.
[Kit] T. Kitagama, The composition of permutable functions and the operational calculus, Proc. Imp. Acad. Japan 13 (1937), 143-151 and 176-179.
[Koi] S. Koizumi, On Heaviside's operational solution of a Volterra integral equation when its nucleus is a function of (x-ξ), Philos. Mag. 11 (1931), 432-441.
[Ko1] H. Komatsu, An introduction to the theory of hyperfunctions, in: Hyperfunctions and Pseudo-Differential Equations, Lecture Notes in Math. 287, Springer, Berlin, 1973, 3-40.
[Ko2] H. Komatsu, Laplace transforms of hyperfunctions-A new foundation of the Heaviside calculus, J. Fac. Sci. Univ. Tokyo I A 34 (1987), 805-820.
[Ko3] H. Komatsu, Operational calculus in Japan, in: Generalized Functions and Convergence, Memorial Volume for Professor Jan Mikusiński, World Scientific, Singapore, 1990, 59-70.
[Kön] H. König, Neue Begründung der Theorie der Distributionen von L. Schwartz, Math. Nachr. 9 (1953), 129-148.
[KJ1] J. Korevaar, Distributions defined from the point of view of applied mathematics. I. Fundamental sequences, Proc. Koninkl. Nederl. Akad. Wetensch. A 58 (1955), 368-378.
[KJ2] J. Korevaar, Distributions defined by fundamental sequences. III. Convergence. Convolution. Definite integral. Inverse Laplace transformation, ibid., 483-493.
[KJ3] J. Korevaar, Distributions defined by fundamental sequences. IV. Multiplication and division. Substitution, ibid., 494-503.
[KJ4] J. Korevaar, Distributions defined by fundamental sequences. ibid., 379-389, 663-674.
[Kr] G. L. Krabbe, Operational Calculus, Springer, Berlin, 1970.
[Krä] E. Krätzel, Bemerkungen zur Meijer-Transformation und Anwendungen, Math. Nachr. 30 (1965), 327-334.
[La] J. Lavoine, Calcul symbolique. Distributions et pseudo-fonctions avec une table de nouvelles transformées de Laplace, Centre Nat. Rech. Sci., Paris, 1959.
[Let] A. V. Letnikov, The Theory of Differentiation with an Arbitrary Index, Moscow, 1868 (in Russian).
[Lév] P. Lévy, La calcul symbolique d'Heaviside, Bull. Sci. Math. 50 (1926), 174-192.
[Li] T. L. Lions, Supports de produit de composition, C. R. Acad. Sci. Paris 232 (1951), 1622-1624.
[Má] L. Máté, Multiplier operators and quotient algebra, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 523-526.
[Mc1] N. W. McLachlan, Historical note on Heaviside's operational method, Math. Gaz. 22 (1938), 255-260.
[Mc2] N. W. McLachlan, Modern Operational Calculus, with Applications in Technical Mathematics, revised ed., Dover, New York, 1962.
[Me] N. A. Meller, Operational calculus for the operator $B_α = t^{-α} d/dt t^{α+1}d/dt$, Numerical Mathematics, Collection 6 (1960), 161-168 (in Russian).
[MJ1] J. G. M. (J. Mikusiński), Hiperliczby, Cracow, 1944 (Edition of 7 copies); English transl.: Hiperliczby-Hypernumbers, University of California, Santa Barbara, 1983 and Studia Math. 77 (1983), 3-16.
[MJ2] J. Mikusiński, L'anneau algébrique et ses applications dans l'analyse fonctionnelle (Première partie), Ann. Univ. Mariae Curie-Skłodowska Sect. A 2 (1947), 1-48.
[MJ3] J. Mikusiński, Sur la méthode de généralisation de Laurent Schwartz et sur la convergence faible, Fund. Math. 36 (1948), 235-239.
[MJ4] J. Mikusiński, L'anneau algébrique et ses applications dans l'analyse fonctionnelle (Deuxième partie), Ann. Univ. Mariae Curie-Skłodowska Sect. A 3 (1949), 1-82.
[MJ5] J. Mikusiński, Sur les fondaments du calcul opératoire, Studia Math. 11 (1950), 41-70.
[MJ6] J. Mikusiński, Remarks on the moment problem and a theorem of Picone, Colloq. Math. 2 (1951), 138-141.
[MJ7] J. Mikusiński, Sur les équations différentielles du calcul opératoire et leurs applications aux équations classiques aux dérivées partielles, Studia Math. 12 (1951), 227-270.
[MJ8] J. Mikusiński, Sur les fonctions exponentielles du calcul opératoire, ibid. 12 (1951), 208-224.
[MJ9] J. Mikusiński, A new proof of Titchmarsh's theorem on convolution, ibid. 13 1953), 56-58.
[MJ10] J. Mikusiński, Operational Calculus, PWN, Warszawa, 1953 (in Polish, 1st ed.).
[MJ11] J. Mikusiński, O rachunku operatorów, Zastos. Mat. 1 (1953), 28-40.
[MJ12] J. Mikusiński, Sur un type de conditions mixtes pour les équations aux dérivées partielles, Studia Math. 13 (1953), 277-286.
[MJ13] J. Mikusiński, Rachunek operatorów na tle aktualnych kierunków matematyki, Prace Mat. 1 (1955), 344-370 (in Polish).
[MJ14] J. Mikusiński, Une définition de distribution, Bull. Acad. Polon. Sci. Cl. III 3 (1955), 589-591.
[MJ15] J. Mikusiński, Le calcul opérationnel d'intervalle fini, Studia Math. 15 (1956), 225-251.
[MJ16] J. Mikusiński, Operational Calculus, IL, Moscow, 1956 (Russian translation from Polish).
[MJ17] J. Mikusiński, Problèmes initiaux et mixtes pour les équations aux dérivées partielles considérées du point de vue du calcul opérationnel, Cheskoslov. Mat. Zh. 6 (1956), 311-317.
[MJ18] J. Mikusiński, Sur la croissance de la fonction opérationelle $exp(-s^{α}λ)$, Bull. Acad. Polon. Sci. Cl. III 4 (1956), 423-425.
[MJ19] J. Mikusiński, Sur la fonction opérationnelle $exp(-s^{α}λ)$, ibid., 415-417.
[MJ20] J. Mikusiński, Sur quelques équations intégro-différentielles, Studia Math. 15 (1956), 182-187.
[MJ21] J. Mikusiński, The work of Polish mathematicians on generalized functions and operational calculus, Uspekhi Mat. Nauk 11 (6) (1956), 169-172 (in Russian).
[MJ22] J. Mikusiński, Sur la fonction dont la transformée de Laplace est $e^{-s^{α}}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 6 (1958), 691-693.
[MJ23] J. Mikusiński, On the function whose Laplace transform is $e^{-s^{α}}$, Studia Math. 18 (1959), 191-198.
[MJ24] J. Mikusiński, Operational Calculus, Pergamon Press and PWN, London-Warszawa, 1959 (transl. from the second Polish ed.).
[MJ25] J. Mikusiński, Operatorenrechnung und ihre Beziehungen zur modernen Mathematik, Z. Angew. Math. Mech. 39 (1959), 381-384.
[MJ26] J. Mikusiński, Sur la convolution par exp(t²), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 7 (1959), 669-671.
[MJ27] J. Mikusiński, Une simple démonstration du théorème de Titchmarsh sur la convolution, ibid., 715-717.
[MJ28] J. Mikusiński, On the convolution of functions of class l₁, Bull. Inst. Politehn. Iasi 6 (3-4) (1960), 31-34.
[MJ29] J. Mikusiński, Remarks on the algebraic derivative in the operational calculus, Studia Math. 19 (1960), 187-192.
[MJ30] J. Mikusiński, Convolution of functions of several variables, ibid. 20 (1961), 301-312.
[MJ31] J. Mikusiński, Operations on distributions, in: Proc. Internat. Sympos. Linear Spaces, Jerusalem, 1960, Academic Press and Pergamon Press, Jerusalem-Oxford, 1961, 335-339.
[MJ32] J. Mikusiński, Sur la convolution, in: 2ème Congr. Math. Hongrois (Budapest, 1960), Budapest, 1961, 80-84.
[MJ33] J. Mikusiński, Operators and distributions, Studia Math. Ser. Spec. 1 (1963), 77-78.
[MJ34] J. Mikusiński, An approximation theorem and its applications in operational calculus, Studia Math. 27 (1966), 141-143.
[MJ35] J. Mikusiński, Convolution approximation and shift approximation, ibid. 28 (1966), 1-8.
[MJ36] J. Mikusiński, A constructive theory of tempered distributions, Bull. Acad. Pol. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 727-732.
[MJ37] J. Mikusiński, The Bochner Integral, Birkhäuser, 1978.
[MJ38] J. Mikusiński, On the Foiaş theorem on the convolution, Bull. Polish. Acad. Sci. Math. 33 (1985), 285-288.
[MJ39] J. Mikusiński and C. Ryll-Nardzewski, Sur le produit de composition, Studia Math. 12 (1951), 51-57.
[MJ40] J. Mikusiński and C. Ryll-Nardzewski, A theorem on bounded moments, ibid. 13 (1953), 51-55.
[MJ41] J. Mikusiński and C. Ryll-Nardzewski, Un théorème sur le produit de composition des fonctions de plusieurs variables, ibid., 62-68.
[MJ42] J. Mikusiński and S. Świerczkowski, Twierdzenie Titchmarsha o splocie i teoria Dufresnoy, Prace Mat. 4 (1960), 59-76 (in Polish).
[MP1] P. Mikusiński, Convergence of Boehmians, Japan J. Math. 9 (1983), 159-179.
[MP2] P. Mikusiński, Boehmians and generalized functions, Acta Math. Hungar. 51 (1988), 271-281.
[Mu] P. A. Murav'ev, The elements of α-operational calculus, An. Sţiint. Univ. "Al. I. Cuza" Iaşi 10 (1964), 287-302.
[Ne] P. A. Nekrasov, Generalized differentiation, Mat. Sb. 14 (1888), 45-168 (in Russian).
[Ni1] M. S. Nikol'skiĭ, On a method of escape, Dokl. Akad. Nauk SSSR 15 (1974), 134-137 (in Russian).
[Ni2] M. S. Nikol'skiĭ, On a linear escape problem, ibid., 1462-1466 (in Russian).
[Ni3] M. S. Nikol'skiĭ, On a linear problem of tracking motion, ibid. 45 (1992), 193-197 (in Russian).
[No1] D. O. Norris, A topology for Mikusiński operators, PhD Diss., Ohio State Univ., 1963.
[No2] D. O. Norris, A topology for Mikusiński operators, Studia Math. 24 (1964), 245-255.
[OK1] J. O'Keefe, Singularities of Hadamard's finite part of improper integrals in the distributions of Schwartz, Rend. Circ. Mat. Palermo 6 (1957), 65-82.
[OK2] J. O'Keefe, Distribution theory of the operational calculus, ibid., 157-170.
[Os1] S. Osipov, Concerning the demonstration of Mikusiński's theorem on convolution with exp(t²), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 209-210.
[Os2] S. Osipov, The expansion of polynomials in $B_α$ operators, Zh. Vychisl. Mat. i Mat. Fiz. 3 (1963), 191-194 (in Russian).
[Os3] S. Osipov, The expansion of polynomials in Bessel operators, ibid. 4 (1964), 149-155 (in Russian).
[Pe1] J. Pérès, Calcul symbolique d'Heaviside et calcul de composition de Vito Volterra, C. R. Acad. Sci. Paris 217 (1943), 517-520.
[Pe2] J. Pérès, Quelques applications du calcul de composition de Volterra, ibid., 585-588.
[Pi1] S. Pilipović, A characterization of the space D', Mat. Vesnik 6 (19) (34) (1982).
[Pi2] S. Pilipović, The a-tempered derivative and some spaces of exponential distributions, Publ. Inst. Math. (Beograd) 34 (48) (1983), 139-192.
[Pl] A. I. Plesner, The inclusion of the Heaviside operational method in the spectral theory of maximal operators, Dokl. Akad. Nauk SSSR 26 (1940), 10-12 (in Russian).
[Po1] L. S. Pontryagin, A linear differential evasion game, Dokl. Akad. Nauk SSSR 11 (1970), 359-361 (in Russian).
[Po2] L. S. Pontryagin and E. F. Mishchenko, The problem of evasion in linear differential games, Differentsial'nye Uravneniya 7 (1971), 335-342 (in Russian).
[Po3] L. S. Pontryagin and E. F. Mishchenko, A linear differential evasion game, Trudy Mat. Inst. Steklov. 112 (1973), 27-60 (in Russian).
[Pot] A. A. Potapenko, Some applications of J. Mikusiński's operational calculus in the automatic control of systems subjected to random disturbances, Vestnik Leningrad. Univ. 19 (1957), 129-134 (in Russian).
[Pr1] A. P. Prudnikov, A contribution to operational calculus theory, Dokl. Akad. Nauk SSSR 142 (1962), 794-797 (in Russian).
[Pr2] A. P. Prudnikov, The properties of functions satisfying the differential equation $x²y''' + 3xy'' + y' + x²y = 0$, Dokl. Akad. Nauk SSSR. 144 (1962), 56-57 (in Russian).
[Pr3] A. P. Prudnikov and I. I. Ryabtsev, Operational calculus of perfect operators, Computing Center of the USSR Academy of Sciences, Moscow, 1991 (in Russian).
[Pr4] A. P. Prudnikov and I. I. Ryabtsev, Operational calculus of perfect operators (special problems), Computing Center of the USSR Academy of Sciences, Moscow, 1992 (in Russian).
[PR1] D. Przeworska-Rolewicz, Equations with Transformed Argument. An Algebraic Approach, PWN-Polish Scientific Publishers and Elsevier, Warszawa-Amsterdam, 1973.
[PR2] D. Przeworska-Rolewicz, Algebraic theory of right invertible operators, Studia Math. 48 (1973), 129-144.
[PR3] D. Przeworska-Rolewicz, Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1988.
[PR4] D. Przeworska-Rolewicz, Prodigious mystery of genuine analysis, in: Generalized Functions and Convergence, Memorial Volume for Professor Jan Mikusiński, World Scientific, Singapore, 1990, 51-58.
[Ry1] I. I. Ryabtsev, On variable delay, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (8) (1959), 164-173 (in Russian).
[Ry2] I. I. Ryabtsev, Local properties of Mikusiński operators, ibid. 3 (1962), 143-150 (in Russian).
[RN1] C. Ryll-Nardzewski, Sur la convergence des séries d'opérateurs, Studia Math. 13 (1953), 37-40.
[RN2] C. Ryll-Nardzewski, Sur les séries de puissances de l'opérateur différentiel, ibid., 41-47.
[RN3] C. Ryll-Nardzewski, Sur le corps des opérateurs de Mikusiński, ibid. 14 (1954), 247-248.
[Sa] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Reading, Mass., 1993.
[Sch] M. Schreiber, Remark on a paper of G. K. Kalisch "A functional analysis proof of Titchmarsh's theorem on convolution", J. Math. Anal. Appl. 7 (1963), 62-63.
[Sc1] L. Schwartz, Généralisation de la notion de fonction, de dérivation, de transformation de Fourier, et applications mathématiques et physiques, Ann. Univ. Grenoble 21 (1945), 57-74.
[Sc2] L. Schwartz, Théorie des distributions I-II, Hermann, Paris, 1950-51.
[Sc3] L. Schwartz, Convolution. Les cours de Sorbonne, Méthodes Math. de la Physique III, Paris, 1955.
[Sc4] L. Schwartz, Transformation de Laplace, Sorbonne, Paris, 1955.
[Sc5] L. Schwartz, Méthodes mathématiques pour les sciences physiques, Hermann, Paris, 1961.
[Se1] J. Sebastião e Silva, Sui fondamenti delle teoria dei funzionali analitici, Portug. Math. 12 (1953), 1-47.
[Se2] J. Sebastião e Silva, Le calcul opérationnel au point de vue des distributions, ibid. 14 (3-4) (1955), 105-132.
[Se3] J. Sebastião e Silva, Sur une construction axiomatique de la théorie des distributions, Révista Fac. Ci. Lisboa 2 (1955), 79-186.
[Sh] H.S. Shultz, Operational calculus for functions of two variables, SIAM J. Math. Anal. 9 (1978), 660-666.
[Si2] M. D a Silveira, General operational calculus in N variables, Portug. Math. 15 (1956), 49-69.
[Si1] M. D a Silveira, Correction to the article "General operational calculus in N variables", ibid. 16 (1-2) (1957), 41-42.
[Sk1] K. Skórnik, The form of locally integrable function whose m-th local derivative vanishes almost everywhere, Zeszyty Naukowe WSP w Katowicach, Sekcja Mat., Zeszyt 5 (1966), 127-152 (in Polish).
[Sk2] K. Skórnik, An estimation of Fourier coefficients of periodic distributions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 581-585.
[Sk3] K. Skórnik, Hereditarily periodic distributions, Studia Math. 43 (1972), 245-272.
[Sk4] K. Skórnik, On fractional integrals and derivatives of a class of generalized functions, Dokl. Akad. Nauk SSSR 254 (1980), 541-543 (in Russian).
[Sk5] K. Skórnik, On tempered integrals and derivatives of nonnegative orders, Ann. Polon. Math. 40 (1981), 47-57.
[Sk6] K. Skórnik, On local derivatives, Dokl. Akad. Nauk SSSR 257 (2) (1981), 286-288 (in Russian).
[Sk7] K. Skórnik, On local derivatives, Ann. Polon. Math. 41 (1983), 283-292.
[Sk8] K. Skórnik, On full derivatives, ibid. 43 (1983), 323-325.
[Sk9] K. Skórnik, Sobolev's and local derivatives, Studia Math. 77 (1984), 509-517.
[Sk10] K. Skórnik, On functions with vanishing local derivative, Ann. Polon. Math. 47 (1986), 179-188.
[Sk11] K. Skórnik, On the Foiaş theorem on convolution of continuous functions, in: Complex Analysis and Applications (Sofia 85), 1986, 604-608.
[Sk12] K. Skórnik, On the Foiaş theorem on convolution, in: Complex Analysis and Applications (Sofia 87), 1989, 460-464.
[Sl1] W. Słowikowski, A generalization of the theory of distributions, Bull. Acad. Polon. Sci. Cl. III 1 (1955), 3-6.
[Sl2] W. Słowikowski, On the theory of operator systems, ibid. 3 (1955), 137-142.
[Sl3] W. Słowikowski, A generalization of Mikusiński's operational calculus, ibid. 4 (1956), 633-638, 643-647.
[Sm] I. I. Smith and P. L. Alger, A derivation of Heaviside's operational calculus based on the generalized functions of Schwartz, Trans. AIEE 68 (1949), 939-944.
[Sob] S. Soboleff, Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales, Mat. Sb. 1 (1936), 39-71.
[St1] B. Stanković, Abbildung gewisser Operationen durch die zweidimensionale Laplace Transformation, Publ. Inst. Math. Acad. Serbe Sci. 11 (1957), 1-8.
[St2] B. Stanković, Otvoreni problemi u teoriji operatora J. Mikusińskog, Mat. Bibl. 25 (1963), 73-82.
[St3] B. Stanković, On the solvability of algebraic equations in the field of Mikusiński's operators, Mat. Vesnik 8 (23) (1971), 391-394.
[St4] B. Stanković and M. Skendžić, Opérateurs parfaits, Publ. Inst. Math. (Beograd) 4 (1964), 57-62.
[Sto] F. Stopp, Zur funktionentheoretischen Auffassung der Mikusińskischen Operatorenrechnung, Math. Nachr. 32 (1966), 87-205.
[SR1] R. A. Struble, An algebraic view of distributions and operators, Studia Math. 37 (1971), 103-109.
[SR2] R. A. Struble, Analytical and algebraic aspects of the operational calculus, SIAM Rev. 19 (1977), 403-436.
[SR3] R. A. Struble, A two-sided operational calculus, Studia Math. 60 (1977), 239-254.
[Sz1] Á. Száz, Krabbe's generalized functions as convolution quotients, Publ. Math. 19 (1972), 287-290.
[Sz2] Á. Száz, The multiplier extensions of admissible vector modules and the Mikusiński-type convergence, Serdica 3 (1977), 82-87.
[Te] G. Temple, Theories and applications of generalized functions, J. London Math. Soc. 28 (1953), 134-148.
[Tic] A. Tikhonov, Theorem of the uniqueness of the solution of the heat conduction equation, Dokl. Akad. Nauk SSSR 5 (1935), 294-300 (in Russian).
[Tit] E. C. Titchmarsh, The zeros of certain integral functions, Proc. London Math. Soc. 25 (1926), 283-302.
[Ur] K. Urbanik, Sur la structure non-topologique du corps des opérateurs, Studia Math. 14 (1954), 243-246.
[VB] B. Van der Pol and H. Bremmer, Operational Calculus Based on the Two-sided Laplace Integral, University Press, Cambridge, 1950.
[VZ] M. Vashchenko-Zakharchenko, Symbolic Calculus and its Application to the Integration of Linear Differential Equations, Kiev, 1862 (in Russian).
[Vl] V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir, 1979.
[Vo1] V. Volterra et J. Pérès, Leçons sur la composition et les fonctions permutables, Paris, 1924.
[Vo2] V. Volterra et J. Pérès, Théorie générale des fonctionnelles, Paris, 1936.
[We1] J. D. Weston, An extension of the Laplace-transform calculus, Rend. Circ. Mat. Palermo 6 (1957), 325-333.
[We2] J. D. Weston, Operational calculus and generalized functions, Proc. Roy. Soc. A 250 (1263) (1959), 460-471.
[We3] J. D. Weston, Characterization of Laplace transforms and perfect operators, Arch. Rational. Mech. Anal. 3 (1959), 348-354.
[We4] J. D. Weston, On the representation of operators by convolution integrals, Pacific J. Math. 10 (1960), 1453-1468.
[We5] J. D. Weston, Time-invariant linear systems and a theorem of Titchmarsh, in: Topics in Analysis, Colloquium on Mathematical Analysis, Jyväskylä 1970, Springer, Berlin, 1974, 376-383.
[WP] N. Wiener and R. Paley, Fourier Transforms in the Complex Plane, Nauka, Moscow, 1964 (Russian translation).
[Wl1] L. Włodarski, Sur une formule de Efros, Studia Math. 13 (1953), 183-187.
[Wl2] L. Włodarski, Une remarque sur une classe de fonctions exponentielles du calcul opérationnel, ibid., 188-189.
[Wlo] J. Wloka, Distributionen und Operatoren, Math. Ann. 140 (1960), 227-244.
[Wo] W. Wolibner, Sur certains corollaires du théorème de Titchmarsh, Studia Math. 14 (1953), 107-110.
[Ya] M. Yamazaki, Operational calculus, Soryuiron Kenkyu 7 (1955), 5-17.
[Yo] K. Yosida, Operational Calculus-a Theory of Hyperfunctions, Springer, Berlin, 1984.
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