Abstract
Let m be a v-moderate function defined on R d and let g ∈ L 2(R d). In this work, we defineΩ p m (R d) to be the vector space of f ∈ L 2m (R d) such that the Gabor transform V f belongs to L p(R 2d), where 1 ≤ p < ∞. We equip it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of Ω p m (R d). We also discuss inclusion properties and obtain the dual space of Ω p m (R d). At the end of this work, we study multipliers from L 1w (R d) into Ω p w (R d) and from Ω p w (R d) into L ∞ w−1 (R d), where w is the Beurling weight function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston (2001).
H. Rieter, Classical Harmonic Analysis and Local Compact Groups, Clarendon Press, Oxford (1968).
G. N. K. Murthy and K. R. Unni, “Multipliers on weighted spaces, functional analysis and its applications,” Int. Conf. Modras. Lect. Notes Math., 399 (1973).
M. A. Rieffel, “Induced Banach representation of Banach algebras and locally compact groups,” J. Funct. Anal., 443–491 (1967).
E. H. Lieb, “Integral bounds for radar ambiguity functions and Wigner distributions,” J. Math. Phys., 31, No. 3, 594–599 (1990).
R. H. Fischer, A. T. Gürkanli, T. S. Liu, “On family of weighted spaces,” Math. Slovaca, 46, No. 1, 71–82 (1996).
H. Feichtinger, “Generalized amalgams, with applications to Fourier transform,” Can. J. Math., 42, No. 3, 395–409 (1990).
G. I. Gaudry, “Multipliers of weighted Lebesgue and measure spaces,” Proc. London Math. Soc., 3, No. 19 (1969).
R. S. Doran and J. Wichman, “Approximate identities and factorization in Banach modules,” Lect. Notes Math., 768 (1979).
T. Liu and A. van Rooij, “Sums and intersections of normed linear spaces,” Math. Nachr., 42, 9–42 (1969).
C. Duyar and A. T. Gurkanli, “Multipliers and relative completion in weighted Lorentz spaces,” Acta Math. Sci. Ser. B, 23, No. 4, 467–476 (2003).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 139–145, January, 2006.
Rights and permissions
About this article
Cite this article
Sandikçi, A., Gürkanli, A.T. The space Ω p m (R d) and some properties. Ukr Math J 58, 155–162 (2006). https://doi.org/10.1007/s11253-006-0058-9
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0058-9