# 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

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