On certain distributions met in signal theory and other domains of systems science

• Autorzy: Vallee R.
• Języki publikacji: EN

Abstrakty

EN

The introduction in mathematics of concepts obtained as limits of already accepted ones has been a factor of progress. It has been the case with delta-pseudo-function (or delta-distribution). Here, we present what we call "epsilon-distribution", its theory and some applications. In a way epsilon-distribution is the anti-thesis of delta-distribution (or Dirac's delta). Heuristically it is a real pseudo-function epsilon (t) defined on R, "equal to zero everywhere but having an integral over R equal to one". It can be considered to have the limit of certain generative functions. It can also be presented, as usual in distribution theory, as a linear operator e acting on functions u in such a way that it gives < epsilon,u> = m(u), the mean value of u on R. Applications are given in the fields of probability distributions, Schrodinger equation, heat equation, white noise, generalized harmonic analysis and pseudo-random functions.

Słowa kluczowe

EN

signal theory; signal processing; harmonic analysis; probability distribution; Schrödinger equation; heat equation; pseudorandom function

• Wydawca: Oficyna Wydawnicza Politechniki Wrocławskiej
• Czasopismo: Systems Science
• Rocznik: 2004
• Tom: Vol. 30, no 2
• Strony: 5--10
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Vallee R.

• Universite Paris-Nord and World Organisation of Systems and Cybernetics, 2 rue de Vouille, 75015 Paris, France,

Bibliografia

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• [2] Bass J., Stationary functions and their applications to the theory of turbulence, Journal of Mathematical Analysis, Vol. 47, No. 2-3, 1974.
• [3] Bertrandias J.-P., Espaces de fonctions continues et bornées en moyenne asymptotique d'ordre p. Mémorial de la Société Mathématique de France, No. 5, 1986.
• [4] Vallee R., Expression asymptotique, pour les grandes valeurs du temps, de l'information associée à la fonction d'onde dans le cas d'un corpuscule libre. Comptes Rendus de l'Académie des Sciences, Tome. 267, s. B. (1968), pp. 1968-seq.
• [5] Vallee R., Information entropy and systems. Proceedings of the 8th International Congress of Cybernetics and Systems, Vol. 1, The New Jersey Institute of Technology Press, New Jersey 1990.
• [6] Vallee R., The "epsilon-distribution" or the antithesis of Dirac's delta, [in:] Cybernetics and Systems Research'92, Vol. 1, World Scientific, Singapore, 1992, 97-102.
• [7] Vallee R., About Wiener's generalized harmonic analysis, Kybemetes, Vol. 23, No. 6-7, 1994, 65-70.
• [8] Wiener N., Generalized harmonic analysis, Acta Mathematica, Vol. 55, 1930, 117-258.
• [9] Wiener N., Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley and Sons, New York, 1949.