On function spaces with fractional Fourier transform in weighted Lebesgue spaces
- Erdem Toksoy1 and
- Ayşe Sandıkçı1Email author
https://doi.org/10.1186/s13660-015-0609-4
© Toksoy and Sandıkçı; licensee Springer. 2015
Received: 13 October 2014
Accepted: 24 February 2015
Published: 6 March 2015
Abstract
Let w and ω be weight functions on \(\mathbb{R} ^{d}\). In this work, we define \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \) to be the vector space of \(f\in L_{w}^{1} ( \mathbb{R}^{d} ) \) such that the fractional Fourier transform \({F_{\alpha}}f\) belongs to \(L_{\omega}^{p} ( \mathbb{R}^{d} ) \) for \(1\leq p<\infty\). We endow this space with the sum norm \({\Vert f\Vert _{A_{\alpha,p}^{w,\omega}}}={\Vert f\Vert _{1,w}}+ {\Vert {{F_{\alpha}}f} \Vert _{p,\omega}}\) and show that \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \) becomes a Banach space and invariant under time-frequency shifts. Further we show that the mapping \(y\rightarrow{T_{y}f}\) is continuous from \(\mathbb{R} ^{d}\) into \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \), the mapping \(z\rightarrow{M_{z}}f\) is continuous from \(\mathbb{R}^{d}\) into \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \) and \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \) is a Banach module over \(L_{w}^{1} ( \mathbb{R}^{d} ) \) with Θ convolution operation. At the end of this work, we discuss inclusion properties of these spaces.
Keywords
1 Introduction
Let \({w_{1}}\) and \({w_{2}}\) are two weight functions. We say that \({w_{1}}\prec{w_{2}}\) if there exists \(c>0\), such that \({w_{1}} ( x ) \leq c{w_{2}} ( x ) \) for all \(x\in \mathbb{R}^{d}\) [1, 2].
In this work we define the function spaces with fractional Fourier transform in weighted Lebesgue spaces and discuss some properties of these spaces.
2 On function spaces with fractional Fourier transform in weighted Lebesgue spaces
Definition 1
Theorem 2
\(( {A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) ,{{\Vert \cdot \Vert }_{A_{\alpha,p}^{w,\omega}}}} ) \) is a Banach space for \(1\leq p<\infty\).
Proof
The following proposition is generalization of the one-dimensional and two-dimensional versions. The proof of this proposition is very similar to the proofs of one-dimensional and two-dimensional versions in [3, 5, 10, 11], and we omit the details.
Proposition 3
Theorem 4
- (1)
Let \(1\leq p<\infty\), w and ω be weight functions on \(\mathbb{R} ^{d}\). Then the space \(A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) \) is translation invariant.
- (2)
Let ω be a bounded weight function on \(\mathbb{R} ^{d}\). Then the mapping \(y\rightarrow{T_{y}}f\) of \(\mathbb{R}^{d}\) into \(A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) \) is continuous.
Proof
Theorem 5
- (1)
Let \(1\leq p<\infty\), w and ω be weight functions on \(\mathbb{R}^{d}\). Then \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \) is invariant under modulations.
- (2)
Let ω be a bounded weight function on \(\mathbb{R}^{d}\). Then the mapping \(z\rightarrow{M_{z}}f\) is continuous from \(\mathbb{R}^{d}\) into \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \).
Proof
(2) The proof technique of this part is the same as that of Theorem 4(2). So, for the sake of brevity, we will not prove it. □
The following definition is an extension of the convolution in [13, 14] of two functions to n dimensions.
Definition 6
Theorem 7
Proof
Theorem 8
Let \(\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} ) \), where \({\alpha_{i}}\neq k\pi\) for each index i with \(1\leq i\leq d\) and \(k\in \mathbb{Z} \). \(L_{w}^{1} ( \mathbb{R} ^{d} ) \) is a Banach algebra under Θ convolution.
Proof
Theorem 9
Let \(\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} ) \), where \({\alpha_{i}}\neq k\pi\) for each index i with \(1\leq i\leq d\) and \(k\in \mathbb{Z}\). \(A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} ) \) is a Banach Θ-convolution module over \(L_{w}^{1} ( \mathbb{R} ^{d} ) \).
Proof
3 Inclusion properties of the space \(A_{\alpha ,p}^{w,\omega} (\mathbb{R}^{d} )\)
Proposition 10
Proof
Lemma 11
Let \({w_{1}}\), \({w_{2}}\), \({\omega_{1}}\) and \({\omega_{2}}\) be weight functions on \(\mathbb{R}^{d}\). If \(A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) \), then \(A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \) is a Banach space under the norm \(|\!|\!|f|\!|\!|={\Vert f\Vert _{A_{\alpha ,p}^{{w_{1}}, {\omega_{1}}}}}+{\Vert f\Vert _{A_{\alpha ,p}^{{w_{2}},{\omega _{2}}}}}\).
Proof
Let \(( f_{n} ) _{n\in \mathbb{N}}\) is a Cauchy sequence in \(( A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) ,|\!|\!|\cdot|\!|\!|) \). Then \(( f_{n} ) _{n\in \mathbb{N}}\) is a Cauchy sequence in \(( A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) ,{\Vert \cdot \Vert _{A_{\alpha,p}^{{w_{1}},{\omega _{1}}}}} ) \) and \(( A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) ,{\Vert \cdot \Vert _{A_{\alpha,p}^{{w_{2}},{\omega _{2}}}}} ) \). As these spaces are Banach spaces, there exist \(f\in A_{\alpha ,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \) and \(g\in A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) \) such that \({\Vert {{f_{n}}-f}\Vert _{A_{\alpha ,p}^{{w_{1}},{\omega_{1}}}}}\rightarrow0\), \({\Vert {{f_{n}}-g} \Vert _{A_{\alpha,p}^{{w_{2}},{\omega_{2}}}}}\rightarrow0\). Using the inequalities \({\Vert \cdot \Vert _{1}}\leq{ \Vert \cdot \Vert _{1,{w_{1}}}}\leq{ \Vert \cdot \Vert _{A_{\alpha ,p}^{{w_{1}},{\omega _{1}}}}}\) and \({\Vert \cdot \Vert _{1}}\leq{ \Vert \cdot \Vert _{1,{w_{2}}}}\leq{ \Vert \cdot \Vert _{A_{\alpha ,p}^{{w_{2}},{\omega _{2}}}}}\), we obtain \({\Vert {{f_{n}}-f}\Vert _{1}}\rightarrow0\) and \({\Vert {{f_{n}}-g}\Vert _{1}}\rightarrow0\). Also \({\Vert {f-g}\Vert _{1}}\leq{ \Vert {{f_{n}}-f} \Vert _{1}}+{\Vert {{f_{n}}-g}\Vert _{1}}\), we have \(f=g\). Hence \(|\!|\!|{{f_{n}}-f}|\!|\!|\rightarrow0\) and \(f\in A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \). That means \(( A_{\alpha,p}^{{w_{1}},{\omega_{1}}} (\mathbb{R}^{d} ) , |\!|\!|\cdot|\!|\!|) \) is a Banach space. □
Theorem 12
Let \({w_{1}}\) and \({w_{2}}\) be weight functions on \(\mathbb{R} ^{d}\). Then \(A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},1} ( \mathbb{R} ^{d} ) \) if and only if \({w_{2}}\prec{w_{1}}\).
Proof
Proposition 13
Let \({w_{1}}\), \({w_{2}}\), \({\omega_{1}}\) and \({\omega_{2}}\) be weight functions on \(\mathbb{R} ^{d}\). If \({w_{2}}\prec{w_{1}}\) and \({\omega_{2}}\prec{\omega_{1}}\), then \(A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) \).
Proof
Assume that \({w_{2}}\prec{w_{1 }}\) and \({\omega_{2}}\prec {\omega _{1}}\). Then there exist \({c_{1}},{c_{2}}>0\) such that \({w_{2}} ( x ) \leq{c_{1}}{w_{1}} ( x ) \) and \({\omega_{2}} ( x ) \leq{c_{2}}{\omega_{1}} ( x ) \) for all \(x\in \mathbb{R} ^{d}\). Let \(f\in A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \). As \(f\in L_{{w_{1}}}^{1} ( \mathbb{R} ^{d} ) \) and \(\mathcal{F}_{\alpha}f\in L_{{\omega _{1}}}^{p} ( \mathbb{R} ^{d} ) \), we have \({\Vert f\Vert _{1,{w_{2}}}}\leq {c_{1}}{\Vert f\Vert _{1,{w_{1}}}}<\infty\) and \({\Vert \mathcal {F}{{_{\alpha}}f} \Vert _{p,{\omega_{2}}}}\leq{c_{2}}{\Vert \mathcal{F}{_{\alpha}f} \Vert _{p,{\omega_{1}}}}<\infty\). Hence, we obtain \(f\in A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) \), and then \(A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) \). □
4 Duality
The following proposition is proved by the duality theorem, Theorem 1.7 in [15].
Proposition 14
Let \(1\leq p<\infty\), and w and ω be weight functions on \(\mathbb{R}^{d}\). The dual space of \(A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) \) is isomorphic to \({{L_{{w^{-1}}}^{\infty} ( \mathbb{R} ^{d} ) \times L_{{\omega^{-1}}}^{p^{\prime}} ( \mathbb{R} ^{d} ) }}/ {K} \) where \(\frac{1}{p}+\frac {1}{{p^{\prime}}} =1\).
Notes
Declarations
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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