Home Nonlinear operations and factorizations on a class of affine modulation spaces
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Nonlinear operations and factorizations on a class of affine modulation spaces

  • Md Hasan Ali Biswas EMAIL logo and Ramesh Manna
Published/Copyright: November 30, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 5

Abstract

In the first part of the paper, we obtain a necessary and sufficient condition on a complex-valued function 𝐹 on R 2 such that F ⁢ ( f ) ∈ M θ r , s ⁢ ( R n ) for all f ∈ M θ r , 1 ⁢ ( R n ) , 1 ≤ r < ∞ , 1 ≤ s < 2 , where M θ r , s ⁢ ( R n ) is the affine modulation space associated with the fractional Fourier transform. In this context, we obtain that if 𝐹 acts on M θ r , 1 ⁢ ( R n ) by composition, then 𝐹 is necessarily real analytic and F ⁢ ( 0 ) = 0 . As M θ r , 1 ⁢ ( R n ) is neither a Banach algebra nor closed under complex conjugation, we prove sufficiency of the condition for a subclass of M θ r , 1 ⁢ ( R n ) under some additional hypothesis on 𝐹. We provide an example to show that we cannot drop the additional hypothesis. Thus we characterize the functions that operate in the affine modulation space setting. In the second part, we study the factorization problem in the context of M θ r , s ⁢ ( R n ) . We obtain factorization of M θ r , s ⁢ ( R 2 ⁢ n ) , viewing it as an L 1 ⁢ ( R 2 ⁢ n ) module with respect to convolution and 𝜆-twisted convolution associated with the fractional Fourier transform. The problems mentioned above originated in the work of Helson, Kahane, Katznelson, Rudin, Cohen, and Hewitt respectively.

Keywords: Nonlinear operations; factorizations; affine modulation spaces; special affine Fourier transform
MSC 2020: 42B35; 46H25; 42B10; 42A85

Funding source: Department of Science and Technology, Ministry of Science and Technology, India

Award Identifier / Grant number: DST/INSPIRE/04/2019/001914

Funding statement: Ramesh Manna is thankful for the research grants (DST/INSPIRE/04/2019/001914).

A Appendix

In Example 3.4, we outlined the key steps. Although the verification is elementary, here we provide a detailed justification for the convenience of the reader.

Example A.1

Let 1 < r < 2 and define

F ⁢ ( x , ω ) = 1 | x | 1 / r ⁢ ( 1 + ( ln ⁡ | x | ) 2 ) ⋅ 1 1 + ω 2 .

Then F ∈ L r , 1 ⁢ ( R 2 ) . Let g ⁢ ( x ) = e − π ⁢ x 2 and h = V g ∗ ⁢ F . Then, using Proposition 2.13, we get that h ∈ M r , 1 ⁢ ( R ) . Set f = C − θ ⁢ h . Then f ∈ M θ r , 1 ⁢ ( R ) , appealing to Proposition 2.20. Further, using (2.4) and (2.6), we obtain

V C − θ ⁢ g ⁢ f ⁢ ( u , η ) = V C − θ ⁢ g ⁢ C − θ ⁢ h ⁢ ( u , η ) = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ V g ⁢ h ⁢ ( u , η + u ⁢ cot ⁡ θ ) = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ ∫ R 2 F ⁢ ( x , ω ) ⁢ V g ⁢ g ⁢ ( u − x , η + u ⁢ cot ⁡ θ − ω ) ⁢ e − 2 ⁢ π ⁢ i ⁢ x ⁢ ( η + u ⁢ cot ⁡ θ − ω ) ⁢ d x ⁢ d ω = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ ∫ R 2 F ⁢ ( u − x , η + u ⁢ cot ⁡ θ − ω ) ⁢ V g ⁢ g ⁢ ( x , ω ) ⁢ e − 2 ⁢ π ⁢ i ⁢ ( u − x ) ⁢ ω ⁢ d x ⁢ d ω = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ ∫ R 2 F ⁢ ( u + x , η + u ⁢ cot ⁡ θ − ω ) ⁢ V g ⁢ g ⁢ ( ω , x ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ω ⁢ d x ⁢ d ω ,

applying a change of variables. Similarly,

V g ⁢ C θ ⁢ f ⁢ ( u , η ) = V g ⁢ h ⁢ ( u , η ) = ∫ R 2 F ⁢ ( u + x , η − ω ) ⁢ V g ⁢ g ⁢ ( ω , x ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ω ⁢ d x ⁢ d ω .

Thus, using Remark 2.8 (i), we obtain

V C − θ ⁢ g 2 ⁢ C θ ⁢ f 2 ⁢ ( u , η ) = ( C θ ⁢ f 2 ⁢ T u ⁢ ( C − θ ⁢ g 2 ̄ ) ) ⁢ ̂ ⁢ ( η ) = ( C θ ⁢ f ⁢ T u ⁢ ( g ̄ ) ⁢ f ⁢ T u ⁢ ( C − θ ⁢ g ̄ ) ) ⁢ ̂ ⁢ ( η ) = ( ( C θ ⁢ f ⁢ T u ⁢ ( g ̄ ) ) ⁢ ̂ ⋆ ( f ⁢ T u ⁢ ( C − θ ⁢ g ̄ ) ) ⁢ ̂ ) ⁢ ( η ) = ∫ R ( C θ ⁢ f ⁢ T u ⁢ ( g ̄ ) ) ⁢ ̂ ⁢ ( y ) ⁢ ( f ⁢ T u ⁢ ( C − θ ⁢ g ̄ ) ) ⁢ ̂ ⁢ ( η − y ) ⁢ d y = ∫ R V g ⁢ C θ ⁢ f ⁢ ( u , y ) ⁢ V C − θ ⁢ g ⁢ f ⁢ ( u , η − y ) ⁢ d y .

Now, taking

F 1 ⁢ ( x ) = 1 | x | 1 / r ⁢ ( 1 + ( ln ⁡ | x | ) 2 ) and F 2 ⁢ ( ω ) = 1 1 + ω 2 ,

we obtain

V C − θ ⁢ g ⁢ C θ ⁢ f 2 ⁢ ( u , η ) = ∫ R 2 ∫ R 2 V g ⁢ g ⁢ ( ω 1 , x 1 ) ⁢ V g ⁢ g ⁢ ( ω 2 , x 2 ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ( ω 1 + ω 2 ) ⁢ F 1 ⁢ ( u + x 1 ) ⁢ F 1 ⁢ ( u + x 2 ) × ∫ R F 2 ( η − y + u cot θ − ω 1 ) F 2 ( y − ω 2 ) d y d x 1 d ω 1 d x 2 d ω 2 = ∫ R 2 ∫ R 2 V g ⁢ g ⁢ ( ω 1 , x 1 ) ⁢ V g ⁢ g ⁢ ( ω 2 , x 2 ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ( ω 1 + ω 2 ) ⁢ F 1 ⁢ ( u + x 1 ) ⁢ F 1 ⁢ ( u + x 2 ) × F 2 ⋆ F 2 ⁢ ( η + u ⁢ cot ⁡ θ − ω 1 − ω 2 ) ⁢ d ⁢ x 1 ⁢ d ⁢ ω 1 ⁢ d ⁢ x 2 ⁢ d ⁢ ω 2 .

Hence, for s > r , appealing to Minkowski’s integral inequality, we have

I = ∫ R ( ∫ R | V C − θ ⁢ g ⁢ C θ ⁢ f 2 ⁢ ( u , η ) | s ⁢ d u ) 1 / s ⁢ d η = ∫ R ( ∫ R | ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ u ⁢ ( ω 1 + ω 2 ) F 1 ( u + x 1 ) F 1 ( u + x 2 ) × F 2 ⋆ F 2 ( η + u cot θ − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 | s d u ) 1 / s d η = ∫ R ( ∫ R | ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 | s d u ) 1 / s d η ≥ ( ∫ R ( ∫ R | ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 | d η ) s d u ) 1 / s ≥ ( ∫ R | ∫ R ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η | s d u ) 1 / s ≥ ( ∫ R ( ∫ R ∫ R 2 ∫ R 2 Re ( V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η ) s d u ) 1 / s = 1 2 ( ∫ R ( ∫ R ∫ R 2 ∫ R 2 e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) Re ( e 2 ⁢ π ⁢ i ⁢ ( ω 1 ⁢ x 1 + ω 2 ⁢ x 2 − ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) ) × F 1 ( ( u − η ) tan θ + x 1 ) F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η ) s d u ) 1 / s = 1 2 ( ∫ R ( ∫ R ∫ R 2 ∫ R 2 e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − ( u − η ) ( ω 1 + ω 2 ) tan θ ) ) × F 1 ( ( u − η ) tan θ + x 1 ) F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η ) s d u ) 1 / s ∼ 1 2 ( ∫ R ( ∫ R 2 ∫ R 2 ∫ R e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − η ( ω 1 + ω 2 ) ) ) F 1 ( η + x 1 ) × F 1 ( η + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s = 1 2 ( ∫ R ( ∫ R 2 ∫ R 2 ∫ R e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − ( η − x 2 ) ( ω 1 + ω 2 ) ) ) × F 1 ( η + x 1 − x 2 ) F 1 ( η ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s ,

using | x + i ⁢ y | ≥ x and applying a change of variables. Our aim is to show that the integral 𝐼 is not finite.

In order to prove the claim, let

X = { ( x 1 , ω 1 , x 2 , ω 2 ) ∈ R 4 : 0 ≤ ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 < 1 3 , ω 1 + ω 2 > 0 ,  0 ≤ x 2 ≤ x 1 ≤ 1 } .

and for fixed N ∈ N , ( x 1 , ω 1 , x 2 , ω 2 ) ∈ X , set

Y N = { η ∈ R : | η ⁢ ( ω 1 + ω 2 ) − ω 1 ⁢ ( x 1 + x 2 ) − 2 ⁢ ω 2 ⁢ x 2 | < 1 3 , x 1 − x 2 < 1 N } .

Then, for any ( x 1 , ω 1 , x 2 , ω 2 ) ∈ X and η ∈ Y N , we have

1 ω 1 + ω 2 ⁢ ( − 1 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 ) < η < 1 ω 1 + ω 2 ⁢ ( 1 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 )

and

1 ω 1 + ω 2 ⁢ ( − 1 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 ) < 0 .

Taking η 0 = 1 ω 1 + ω 2 ⁢ ( 1 / 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 ) , we obtain

∫ Y N F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ cos ⁡ ( 2 ⁢ π ⁢ ( ω 1 ⁢ x 1 + ω 2 ⁢ x 2 − ( η − x 2 ) ⁢ ( ω 1 + ω 2 ) ) ) ⁢ d η ≥ 1 2 ⁢ ∫ Y N F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ≥ 1 2 ⁢ ∫ 0 η 0 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = 1 2 ⁢ ∫ 0 η 0 1 ( η + x 1 − x 2 ) 1 / r ⁢ ( 1 + ( ln ⁡ ( η + x 1 − x 2 ) ) 2 ) ⋅ 1 η 1 / r ⁢ ( 1 + ( ln ⁡ η ) 2 ) ⁢ d η ≥ 1 2 ⁢ ∫ 0 η 0 1 ( η + x 1 − x 2 ) 2 / r ⁢ ( 1 + ( ln ⁡ ( η + x 1 − x 2 ) ) 2 ) 2 ⁢ d η = 1 2 ⁢ ∫ ln ⁡ ( x 1 − x 2 ) ln ⁡ ( η 0 + x 1 − x 2 ) e ( 1 − 2 / r ) ⁢ y ( 1 + y 2 ) 2 ⁢ d y ,

applying a change of variables. Since ( 1 − 2 / r ) < 0 , the integral

∫ Y N F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ cos ⁡ ( 2 ⁢ π ⁢ ( ω 1 ⁢ x 1 + ω 2 ⁢ x 2 − ( η − x 2 ) ⁢ ( ω 1 + ω 2 ) ) ) ⁢ d η

diverges to infinity as N → ∞ .

Now, for any δ > 0 , we compute

I δ = ( ∫ R ( ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − ( η − x 2 ) ( ω 1 + ω 2 ) ) ) × F 1 ( η + x 1 − x 2 ) F 1 ( η ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s ≤ ( ∫ R ( ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) F 1 ( η + x 1 − x 2 ) F 1 ( η ) × F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s ≤ ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) ⁢ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) × ( ∫ R ( F 2 ⋆ F 2 ⁢ ( u − ω 1 − ω 2 ) ) s ⁢ d u ) 1 / s ⁢ d ⁢ η ⁢ d ⁢ x 1 ⁢ d ⁢ ω 1 ⁢ d ⁢ x 2 ⁢ d ⁢ ω 2 ≲ ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) ⁢ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ⁢ d x 1 ⁢ d ω 1 ⁢ d x 2 ⁢ d ω 2 ,

using Minkowski’s integral inequality. For x 1 , x 2 ∈ R , we get

∫ | δ | > 0 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = ∫ − ∞ − δ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η + ∫ δ ∞ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = ∫ − ∞ x 2 − x 1 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η + ∫ x 2 − x 1 − δ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η + ∫ δ ∞ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η .

We note that the second integral in the above line is zero if δ > x 1 − x 2 . Now, we estimate each of the three integrals separately. Consider

I δ 1 = ∫ − ∞ x 2 − x 1 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = ∫ − ∞ x 2 − x 1 1 ( x 2 − x 1 − η ) 1 / r ⁢ ( 1 + ( ln ⁡ ( x 2 − x 1 − η ) ) 2 ) ⋅ 1 ( − η ) 1 / r ⁢ ( 1 + ( ln ⁡ ( − η ) ) 2 ) ⁢ d η
≲ ∫ x 1 − x 2 ∞ 1 ( x 2 − x 1 + η ) 1 / r ⁢ η 1 / r ⁢ d η = ∫ x 1 − x 2 x 1 − x 2 + 1 1 ( x 2 − x 1 + η ) 1 / r ⁢ η 1 / r ⁢ d η + ∫ x 1 − x 2 + 1 ∞ 1 ( x 2 − x 1 + η ) 1 / r ⁢ η 1 / r ⁢ d η
≤ 1 ( x 1 − x 2 ) 1 / r ⁢ ∫ x 1 − x 2 x 1 − x 2 + 1 1 ( x 2 − x 1 + η ) 1 / r ⁢ d η + ∫ x 1 − x 2 + 1 ∞ 1 ( x 2 − x 1 + η ) 2 / r ⁢ d η
≤ 1 ( x 1 − x 2 ) 1 / r ⁢ ∫ 0 1 1 η 1 / r ⁢ d η + ∫ 1 ∞ 1 η 2 / r ⁢ d η ≲ 1 ( x 1 − x 2 ) 1 / r .
Similarly,
I δ 2 = ∫ x 2 − x 1 − δ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ≤ ∫ x 2 − x 1 − δ 1 ( η + x 1 − x 2 ) 1 / r ⁢ ( − η ) 1 / r ⁢ d η ≤ 1 δ 1 / r ⁢ ∫ x 2 − x 1 − δ 1 ( η + x 1 − x 2 ) 1 / r ⁢ d η = 1 δ 1 / r ⁢ ∫ 0 − δ + x 1 − x 2 1 η 1 / r ⁢ d η ≲ 1 δ 1 / r ⁢ ( δ + x 1 − x 2 ) 1 − 1 / r ,
I δ 3 = ∫ δ ∞ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ≤ ∫ δ ∞ 1 η 2 / r ⁢ d η ≲ δ 1 − 2 / r .
Since e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) ∈ S ⁢ ( R 4 ) , the integral I δ < ∞ . This proves that the integral 𝐼 is not finite. In other words, C θ ⁢ f 2 ∉ M s , 1 , which is the same as f 2 ∉ M θ s , 1 . Since s > r , M θ r , 1 ⊂ M θ s , 1 . Thus, in particular, f 2 ∉ M θ r , 1 .

Acknowledgements

The authors thank the National Institute of Science Education and Research Bhubaneswar for providing an excellent research facility. We thank the referee for meticulously reading our manuscript and giving us several valuable suggestions.

  1. Communicated by: Anke Pohl

References

[1] S. Abe and J. T. Sheridan, Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation, Optics Lett. 19 (1994), no. 22, 1801–1803. 10.1364/OL.19.001801Search in Google Scholar

[2] W. Baoxiang, Z. Lifeng and G. Boling, Isometric decomposition operators, function spaces E p , q λ and applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006), no. 1, 1–39. 10.1016/j.jfa.2005.06.018Search in Google Scholar

[3] A. Bényi and T. Oh, Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math. 228 (2011), no. 5, 2943–2981. 10.1016/j.aim.2011.07.023Search in Google Scholar

[4] A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc. 41 (2009), no. 3, 549–558. 10.1112/blms/bdp027Search in Google Scholar

[5] A. Bhandari and A. I. Zayed, Shift-invariant and sampling spaces associated with the special affine Fourier transform, Appl. Comput. Harmon. Anal. 47 (2019), no. 1, 30–52. 10.1016/j.acha.2017.07.002Search in Google Scholar

[6] D. G. Bhimani, Composition operators on Wiener amalgam spaces, Nagoya Math. J. 240 (2020), 257–274. 10.1017/nmj.2019.4Search in Google Scholar

[7] D. G. Bhimani and P. K. Ratnakumar, Functions operating on modulation spaces and nonlinear dispersive equations, J. Funct. Anal. 270 (2016), no. 2, 621–648. 10.1016/j.jfa.2015.10.017Search in Google Scholar

[8] D. Bhimani and J. Toft, Factorizations in quasi-Banach modules and applications, preprint (2023), https://arxiv.org/abs/2307.01590. Search in Google Scholar

[9] M. H. A. Biswas, H. G. Feichtinger and R. Ramakrishnan, Modulation spaces, multipliers associated with the special affine Fourier transform, Complex Anal. Oper. Theory 16 (2022), no. 6, Paper No. 86. 10.1007/s11785-022-01264-1Search in Google Scholar

[10] M. H. A. Biswas, F. Filbir and R. Radha, The system of translates and the special affine Fourier transform, preprint (2024), https://arxiv.org/abs/2212.05678. Search in Google Scholar

[11] W. Chen, Z. Fu, L. Grafakos and Y. Wu, Fractional Fourier transforms on L p and applications, Appl. Comput. Harmon. Anal. 55 (2021), 71–96. 10.1016/j.acha.2021.04.004Search in Google Scholar

[12] P. J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199–205. 10.1215/S0012-7094-59-02620-1Search in Google Scholar

[13] E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl. 353 (2009), no. 2, 583–591. 10.1016/j.jmaa.2008.12.027Search in Google Scholar

[14] E. Cordero and L. Rodino, Time-Frequency Analysis of Operators, De Gruyter Stud. Math. 75, De Gruyter, Berlin, 2020. 10.1515/9783110532456Search in Google Scholar

[15] C. F. Dunkl, Functions that operate in the Fourier algebra of a compact group, Proc. Amer. Math. Soc. 21 (1969), 540–544. 10.2307/2036416Search in Google Scholar

[16] H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983. Search in Google Scholar

[17] H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Wavelets and Their Applications, Allied Publishers, New Delhi (2003), 99–140. Search in Google Scholar

[18] Z. Fu, L. Grafakos, Y. Lin, Y. Wu and S. Yang, Riesz transform associated with the fractional Fourier transform and applications in image edge detection, Appl. Comput. Harmon. Anal. 66 (2023), 211–235. 10.1016/j.acha.2023.05.003Search in Google Scholar

[19] K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmo. Anal., Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0003-1Search in Google Scholar

[20] K. Gröchenig and G. Zimmermann, Spaces of test functions via the STFT, J. Funct. Spaces Appl. 2 (2004), no. 1, 25–53. 10.1155/2004/498627Search in Google Scholar

[21] J. J. Healy, M. A. Kutay, H. M. Ozaktas and J. T. Sheridan, Linear Canonical Transforms, Springer Ser. Optical Sci. 198, Springer, New York, 2015. 10.1007/978-1-4939-3028-9Search in Google Scholar

[22] H. Helson, J.-P. Kahane, Y. Katznelson and W. Rudin, The functions which operate on Fourier transforms, Acta Math. 102 (1959), 135–157. 10.1007/BF02559571Search in Google Scholar

[23] E. Hewitt, The ranges of certain convolution operators, Math. Scand. 15 (1964), 147–155. 10.7146/math.scand.a-10738Search in Google Scholar

[24] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups, Grundlehren Math. Wiss. 152, Springer, New York, 1970. 10.1007/978-3-662-26755-4_3Search in Google Scholar

[25] B. E. Johnson, Continuity of centralisers on Banach algebras, J. Lond. Math. Soc. 41 (1966), 639–640. 10.1112/jlms/s1-41.1.639Search in Google Scholar

[26] J.-P. Kahane, Séries de Fourier absolument convergentes, Ergeb. Math. Grenzgeb. (3) 50, Springer, Berlin, 1970. 10.1007/978-3-662-59158-1_2Search in Google Scholar

[27] T. Kato, M. Sugimoto and N. Tomita, Nonlinear operations on a class of modulation spaces, J. Funct. Anal. 278 (2020), no. 9, Article ID 108447. 10.1016/j.jfa.2019.108447Search in Google Scholar

[28] M. Kobayashi and E. Sato, Operating functions on modulation and Wiener amalgam spaces, Nagoya Math. J. 230 (2018), 72–82. 10.1017/nmj.2017.3Search in Google Scholar

[29] M. Kobayashi and E. Sato, A note on operating functions of modulation spaces, J. Pseudo-Differ. Oper. Appl. 13 (2022), no. 4, Paper No. 61. 10.1007/s11868-022-00494-3Search in Google Scholar

[30] M. Kobayashi and E. Sato, Operating functions on A s q ⁢ ( T ) , J. Fourier Anal. Appl. 28 (2022), no. 3, Paper No. 42. 10.1007/s00041-022-09925-7Search in Google Scholar

[31] P. Lévy, Sur la convergence absolue des séries de Fourier, Compos. Math. 1 (1935), 1–14. Search in Google Scholar

[32] D. Rider, Functions which operate in the Fourier algebra of a compact group, Proc. Amer. Math. Soc. 28 (1971), 525–530. 10.1090/S0002-9939-1971-0276792-3Search in Google Scholar

[33] M. A. Rieffel, On the continuity of certain intertwining operators, centralizers, and positive linear functionals, Proc. Amer. Math. Soc. 20 (1969), 455–457. 10.2307/2035676Search in Google Scholar

[34] W. Rudin, Factorization in the group algebra of the real line, Proc. Natl. Acad. Sci. USA 43 (1957), 339–340. 10.1073/pnas.43.4.339Search in Google Scholar PubMed PubMed Central

[35] W. Rudin, Representation of functions by convolutions, J. Math. Mech. 7 (1958), 103–115. 10.1512/iumj.1958.7.57009Search in Google Scholar

[36] W. Rudin, Fourier Analysis on Groups, Wiley Class. Libr., John Wiley & Sons, New York, 1990. 10.1002/9781118165621Search in Google Scholar

[37] W. Rudin, Functional Analysis, 2nd ed., Int. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991. Search in Google Scholar

[38] M. Ruzhansky, M. Sugimoto, J. Toft and N. Tomita, Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr. 284 (2011), no. 16, 2078–2092. 10.1002/mana.200910199Search in Google Scholar

[39] R. Salem, Sur les transformations des séries de Fourier, Fund. Math. 33 (1939), 108–114. 10.4064/fm-33-1-108-114Search in Google Scholar

[40] M. Sugimoto, N. Tomita and B. Wang, Remarks on nonlinear operations on modulation spaces, Integral Transforms Spec. Funct. 22 (2011), no. 4–5, 351–358. 10.1080/10652469.2010.541054Search in Google Scholar

[41] H. Triebel, Modulation spaces on the Euclidean 𝑛-space, Z. Anal. Anwend. 2 (1983), no. 5, 443–457. 10.4171/zaa/79Search in Google Scholar

[42] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific, Hackensack, 2011. 10.1142/9789814360746Search in Google Scholar

[43] N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. 10.2307/1968102Search in Google Scholar
Received: 2024-04-25
Revised: 2024-10-15
Published Online: 2024-11-30
Published in Print: 2025-06-01

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https://doi.org/10.1515/forum-2024-0191
Keywords for this article
Nonlinear operations; factorizations; affine modulation spaces; special affine Fourier transform

Articles in the same Issue

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(𝔽𝑞) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
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