Article ID: MTJPAM-D-20-00045

Title: On Function Spaces with Fractional Wavelet Transform


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00045; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 122-134

Document Type: Research Paper

Author(s): Muhammed Duman a , Öznur Kulak b

aGiresun University, Faculty of Sciences and Letters, Department of Mathematics, Giresun, Turkey

bAmasya University, Faculty of Sciences and Letters, Department of Mathematics, Amasya, Turkey

Received: 5 December 2020, Accepted: 1 January 2021, Published: 25 April 2021.

Corresponding Author: Öznur Kulak (Email address: oznur.kulak@amasya.edu.tr)

Full Text: PDF

Abstract

Let  \omega _{1}  and  \omega _{2}  be weight functions on  \mathbb{R}. In this paper, we define  \left( {FW}_{\omega _{1},\omega _{2}}^{\theta,p,q}\right) _{a}\left(\mathbb{R}\right)  to be the vector space of  f \in L_{\omega_{1}}^{p}\left(\mathbb{R}\right)  such that the fractional wavelet transform  W_{\psi }^{\theta }f  belongs to L_{\omega _{2}}^{q}\left(\mathbb{R}\right) for 1 ≤ p, q < ∞. We endow this space with a sum norm and show that  \left({FW}_{\omega _{1},\omega _{2}}^{\theta ,p,q}\right) _{a}\left( \mathbb{R}\right)  becomes a Banach space. Also we prove that  \left( {FW}_{\omega_{1},\omega _{2}}^{\theta ,p,q}\right) _{a}\left( \mathbb{R}\right)  is an essential Banach Module over  L_{\omega _{1}}^{1}\left( \mathbb{R}\right)  under some conditions. We obtain its approximate identities, dual space and multipliers space. At the end of this paper we discuss the inclusion properties, compact embeddings of these spaces.

Keywords: Fractional wavelet transform, Essential Banach module, Approximate identity, Compact embedding, Multipliers space

References:
  1. L. B. Almeida, Product and convolution theorems for the fractional Fourier transform, IEEE Signal Processing Letters 4 (1), 15-17, 1997.
  2. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF, SIAM, Philadelphia, 1992.
  3. R. S. Doran and J. Wichmann, Approximate Identity and Approximate Identity and Factorization in Banach Modules, Lecture Notes in Math., Heidelberg, New York, 1979.
  4. C. Duyar and A. T. Gürkanli, Multipliers and relative completion in weighted Lorentz space, Acta Math. Sci. 23B (4), 467-476, 2003.
  5. H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, Journal of Functional Analysis 86 (2), 307-340, 1989.
  6. H. G. Feichtinger and A. T. Gürkanli, On a family of weighted convolution algebras, Int. J. Math. Math. Sci. 13 (3), 517-525, 1990.
  7. R. H. Fischer, A. T. Gürkanli and T. S. Liu, On a family of weighted spaces, Math. Slovaca. 46 (1), 71-82, 1996.
  8. G. I. Gaudry, Multipliers of weighted Lebesgue and measure spaces, Proc. Lon. Math. Soc. 19 (3), 327-340, 1969.
  9. Ö. Kulak, On function spaces with wavelet transform in  L_{\omega }^{p}\left(\mathbb{R}^{d}\times\mathbb{R}_{+}\right), Hacet. J. Math. Stat. 40 (2), 163-177, 2011.
  10. T. S. Liu and A. Van Rooij, Sums and intersections of normed linear spaces, Math. Nachr. 42, 29-42, 1969.
  11. A. Prasad, S. Manna, A. Mahato and V. K. Singh, The generalized continuous wavelet transform associated with the fractional Fourier transform, J. Comput. Appl. Math. 259, 660-671, 2014.
  12. A. Prasad and P. Kumar, Fractional wavelet transform in terms of fractional convolution, Progr. Fract. Differ. Appl. 3, 201-210, 2015.
  13. H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Group, Clarendon Press, Oxford, 2000.
  14. M. A. Rieffel, Induced Banach representation of Banach algebras and locally compact groups, J. Funct. Anal. 1, 443-491, 1967.
  15. E. Toksoy and A. Sandıkçı, On function spaces with fractional Fourier transform in weighted Lebesgue spaces, J. Inequal. Appl. 2015 (87), 1-10, 2015.
  16. H. C. Wang, Homogeneous Banach Algebras, Marcel Dekker INC., New York, 1977.