Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników

Czasopismo

1998 | 128 | 1 | 55-69

Tytuł artykułu

Tauberian theorems for vector-valued Fourier and Laplace transforms

Autorzy

Treść / Zawartość

Języki publikacji

EN

Abstrakty

EN
Let X be a Banach space and $f ∈ L^1_loc(ℝ;X)$ be absolutely regular (i.e. integrable when divided by some polynomial). If the distributional Fourier transform of f is locally integrable then f converges to 0 at infinity in some sense to be made precise. From this result we deduce some Tauberian theorems for Fourier and Laplace transforms, which can be improved if the underlying Banach space has the analytic Radon-Nikodym property.

Czasopismo

Rocznik

Tom

128

Numer

1

Strony

55-69

Daty

wydano
1998
otrzymano
1997-03-13
poprawiono
1997-08-21

Twórcy

autor
  • Abteilung Mathematik V, Universität Ulm, 89069 Ulm, Germany

Bibliografia

  • [1] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852.
  • [2] W. Arendt and C. J. K. Batty, Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half line, J. London Math. Soc., to appear.
  • [3] W. Arendt and C. J. K. Batty, Almost periodic solutions of first and second order Cauchy problems, J. Differential Equations 137 (1997), 363-383.
  • [4] W. Arendt and J. Prüss, Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. Math. Anal. 23 (1992), 412-448.
  • [5] C. J. K. Batty, J. M. A. M. van Neerven and F. Räbiger, Local spectra and individual stability of uniformly bounded $C_0$-semigroups, Trans. Amer. Math. Soc., to appear.
  • [6] C. J. K. Batty, J. M. A. M. van Neerven and F. Räbiger, Tauberian theorems and stability of solutions of the Cauchy problem, ibid., to appear.
  • [7] E. J. Beltrami and M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions, Academic Press, New York, 1966.
  • [8] A. V. Bukhvalov, Hardy spaces of vector-valued functions, J. Soviet Math. 16 (1981), 1051-1059 (English transl.).
  • [9] A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in Banach space, Math. Notes 31 (1982), 104-110 (English transl.).
  • [10] R. Chill, Taubersche Sätze und Asymptotik des abstrakten Cauchy-Problems, Diplomarbeit, Universität Tübingen, 1995.
  • [11] R. Chill, Stability results for individual solutions of the abstract Cauchy problem via Tauberian theorems, Ulmer Sem. Funktionalanal. Differentialgleichungen 1 (1996), 122-133.
  • [12] P. L. Duren, Theory of $H^p$-Spaces, Academic Press, New York, 1970.
  • [13] J. Esterle, E. Strouse et F. Zouakia, Stabilité asymptotique de certains semi-groupes d'opérateurs et idéaux primaires de $L^1 (ℝ_+)$, J. Operator Theory 28 (1992), 203-227.
  • [14] E. Hille and R. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, 1957.
  • [15] S. Huang and J. M. A. M. van Neerven, B-convexity, the analytic Radon-Nikodym property and individual stability of $C_0$-semigroups, J. Math. Anal. Appl., to appear.
  • [16] A. E. Ingham, On Wiener's method in Tauberian theorems, Proc. London Math. Soc. 38 (1933), 458-480.
  • [17] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.
  • [18] J. Korevaar, On Newman's quick way to the prime number theorem, Math. Intelligencer 4 (1982), 108-115.
  • [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, 1982.
  • [20] Y. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42.
  • [21] J. M. A. M. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Oper. Theory Adv. Appl. 88, Birkhäuser, 1996.
  • [22] G. Pisier, Une propriété de stabilité de la classe des espaces ne contenant pas $l^1$, C. R. Acad. Sci. Paris Sér. A 286 (1978), 747-749.
  • [23] C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. 299, Springer, Berlin, 1992.
  • [24] R. Ryan, The F. and M. Riesz theorem for vector measures, Indag. Math. 25 (1962), 558-562.
  • [25] Z. Szmydt, Characterization of regular tempered distributions, Ann. Polon. Math. 41 (1983), 255-258.

Identyfikator YADDA

bwmeta1.element.bwnjournal-article-smv128i1p55bwm