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

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 squar...

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Bibliographic Details
Published inBoletín de la Sociedad Matemática Mexicana Vol. 26; no. 2; pp. 599 - 616
Main Authors Achak, A., Bouhlal, A., Daher, R., Safouane, N.
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.07.2020
Springer Nature B.V
Subjects
Estimates
Fourier transforms
Integrals
Mathematics
Mathematics and Statistics
Operators (mathematics)
Original Article
Quaternions
Theorems
Dini–Lipschitz class
Lipschitz class
42B10
43A62
42B37
Titchmarsh theorem
Quaternion Fourier transform
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Summary: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 ( R 2 , H ) , where H a quaternion algebra which will be specified in due course.
ISSN:1405-213X
2296-4495
DOI:10.1007/s40590-019-00274-y