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Titchmarsh’s theorem and some remarks concerning the right-sided quaternion Fourier transform

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Abstract

This paper is based mainly on Titchmarsh’s theorem (Introduction to the theory of Fourier integrals. Clarendon Press, Oxford, 1937, Theorem 84) in the one-dimensional case. Abilov et al. (Comput Math Math Phys 48:2146, 2008) proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilovs for only two variables, using a translation operator. The purpose of this paper is to study these estimates for Quaternion Fourier transforms, also the functions satisfy Lipschitz conditions of certain orders. Thus we study the Quaternion Fourier transforms of Lipschitz function in the functions space \(L^r({\mathbb {R}}^{2},{\mathcal {H}})\), where \({\mathcal {H}}\) a quaternion algebra which will be specified in due course.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Aïn Chock University of Hassan II, 20100, Casablanca, Morocco

    A. Achak, R. Daher & N. Safouane

  2. Labo Math Appli, Faculty of Sciences, B.P. 20, El Jadida, Morocco

    A. Bouhlal

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  1. A. Achak
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  2. A. Bouhlal
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  3. R. Daher
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  4. N. Safouane
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Correspondence to A. Achak.

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Achak, A., Bouhlal, A., Daher, R. et al. Titchmarsh’s theorem and some remarks concerning the right-sided quaternion Fourier transform. Bol. Soc. Mat. Mex. 26, 599–616 (2020). https://doi.org/10.1007/s40590-019-00274-y

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