Abstract
Pseudodifferential equations of the formv(Dx)y=f (here,v is a function holomorphic at zero andD x is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to the construction of a generalized translation operatorT χy =χ(<y, Dχ>) on the spaces indicated and to the investigation of its properties. In particular, we prove the associativity, commutativity, and other properties ofT χy , which are analogs of the classical properties of a generalized translation operator.
Similar content being viewed by others
References
Yu. L. Daletskii, “A biorthogonal analog of Hermite polynomials and inversion of Fourier transformations, with respect to non-Gaussian measure,”Funkts. Anal. Prilozh.,25, No. 2, 68–70 (1991).
Yu. M. Berezanskii and Yu. G. Kondrat’ev, “Non-Gaussian analysis and hypergroups,”Funkts. Anal. Prilozh.,29, No. 3, 51–55 (1995).
G. F. Us, “Dual Appell systems in Poissonian analysis,”Meth. Funct. Anal. Topol.,1, No. 1, 93–108 (1995).
Yu. G. Kondrat’ev, L. Streit, W. Westerkamp, and J. Yan,Generalized Functions in Infinite-Dimensional Analysis, IIAS Preprint No. 1995-002, Kyoto (1995).
S. Albeverio, Yu. L. Daletskii, Yu. G. Kondrat’ev, and L. Streit, “Non-Gaussian infinite-dimensional analysis,”J. Funct. Anal.,138, No. 2, 311–350 (1996).
Yu. M. Berezanskii and Yu. G. Kondrat’ev, ”Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis,”Meth. Funct. Anal. Topol.,2, No. 2, 1–50 (1996).
Yu. M. Berezanskii, “Infinite-dimensional non-Gaussian analysis and generalized translation operators,”Funkts. Anal. Prilozh.,30, No. 4, 61–65 (1996).
N. A. Kachanovskii, “Biorthogonal Appell-like systems in a Hilbert space,”Meth. Funct. Anal. Topol.,2, No. 3,4, 36–52 (1996).
N. A. Kachanovskii, “Dual Appell system and Kondrat’ev spaces in analysis on Schwartz spaces,”Ukr. Mat. Zh., No. 4, 527–534 (1997).
Yu. M. Berezanskii, “Infinite-dimensional analysis related to generalized translation operators,”Ukr. Mat. Zh.,49, No. 3, 364–409 (1997).
N. A. Kachanovskii and G. F. Us, “Biorthogonal Appell systems in analysis on dually nuclear spaces,”Funkts. Anal. Prilozh.,32, No. 1, 69–72 (1998).
Yu. M. Berezanskii, “Construction of generalized translation operators from the system of Appell characters,”Trans. Amer. Math. Soc.,184, No. 2, 7–21 (1998).
R. P. Boas and R. C. Buck,Polynomial Expansions of Analytic Functions, Springer, Berlin (1964).
Additional information
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1334–1341, October, 1999.
Rights and permissions
About this article
Cite this article
Kachanovskii, N.A. Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis. Ukr Math J 51, 1503–1511 (1999). https://doi.org/10.1007/BF02981683
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02981683