A Note on Function Spaces with Fractional Fourier Transforms in Wiener-type Spaces
Abstract
The purpose of this paper is to introduce and study a function space A_(α,w)^(B,Y) (R^d ) to be a linear space of functions h∈L_w^1 (R^d ) whose fractional Fourier transforms F_α h belong to the Wiener-type space W(B,Y)(R^d ), where w is a Beurling weight function on R^d. We show that this space becomes a Banach algebra with the sum norm 〖‖h‖〗_(1,w)+〖‖F_α h‖〗_(W(B,Y)) and Θ convolution operation under some conditions. We find an approximate identity in this space and show that this space is an abstract Segal algebra with respect to L_w^1 (R^d ) under some conditions.
Keywords
Fractional Fourier transform, convolution, Wiener-type spaces
References
- W. Rudin, Real and complex analysis. New York: MacGraw-Hill, 1966.
- H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups. Clarendon Press, Oxford, 2000.
- H. G. Feichtinger, “Banach convolution algebras of Wiener type”, Functions, Series, Operators, Proc. Conf. Budapest, 1980, vol. 38, pp. 509-524.
- H. G. Feichtinger, “On a class of convolution algebras of functions”, Annales de l’institut Fourier, 1977, vol. 27, no. 3, pp. 135-162.
- W. Rudin, Functional analysis. McGraw-Hill, New York, 1973.
- H. Feichtinger, C. Graham and E. Lakien, “Nonfactorization in commutative, weakly selfadjoint Banach algebras”, Pac. J. Math., vol. 80, no. 1, pp. 117-125, 1979.
- J. T. Burnham, “Closed ideals in subalgebras of Banach algebras. I”, Proc. Am. Math. Soc., vol. 32, no. 2, pp. 551-555, 1972.
- H. G. Feichtinger and K. H. Gröchenig, “Banach spaces related to integrable group representations and their atomic decompositions, I”, J. Funct. Anal., vol. 86, no. 2, pp. 307-340, 1989.
- H. Wang, Homogeneous Banach algebras. Marcel Dekker Inc., New York and Basel, 1977.
- L. B. Almeida, “The fractional Fourier transform and time-frequency representations”, IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3084-3091, 1994.