Quaternionic Fractional Fourier Transform for Boehmians
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QUATERNIONIC FRACTIONAL FOURIER TRANSFORM FOR BOEHMIANS R. Roopkumar
UDC 517.9
We construct a Boehmian space of quaternion-valued functions by using the quaternionic fractional convolution. By applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the case of Boehmians and its properties are established.
1. Introduction It is well known that the classical Fourier transform on square integrable functions is of order four. In order to generalize Fourier transforms to the case of fractional orders, Namias [26] introduced fractional Fourier transforms, and numerous works on fractional Fourier transforms were performed with different objectives in pure and applied mathematics. In particular, the properties, applications, and generalizations of the fractional Fourier transforms were discussed from the viewpoint of classical analysis (see [7, 17, 20–22, 25, 26, 29, 30, 35, 39, 40]) following the introduction of Fourier transforms for quaternion-valued functions. The fractional Fourier transforms for quaternion-valued functions on R2 and their properties, including inversion formula and Parseval identity, were discussed and derived in [13, 38]. Recently, the fractional Fourier transform for quaternion-valued functions on R has been introduced in [32] and all its properties, including the inversion formula, the Parseval identity, and the convolution and product theorems, have been proved. Note that the convolution theorem for quaternionic fractional Fourier transform in [32] is quite similar to the convolution theorem for the fractional Fourier transform of complex-valued functions from [39]. Hence, in the present paper, following the techniques employed in [40], we extend the quaternionic fractional Fourier transform to a suitable Boehmian space of quaternion-valued functions. It is clear that the fractional Fourier transform on Boehmians of quaternion-valued functions generalizes both the theory of fractional Fourier transform on quaternion-valued L2 -functions [32] and the theory of fractional Fourier transform on Boehmians of complex-valued functions [40]. For the sake of convenience of the reader, we recall the definitions of the division algebra of quaternions and Lp -spaces of quaternion-valued functions, as well as the theory of fractional Fourier transform in Section 2. The general theory of Boehmians and the construction of two suitable Boehmian spaces are discussed in Section 3. The last section is devoted to the definition and properties of the extended quaternionic fractional Fourier transform. 2. Preliminaries As usual, by R and C we denote the sets of all real and complex numbers, respectively. The set of all quaternions is defined as � H = q1 + jq2 : q1 , q2 2 C ,
where j is an imaginary number other than the imaginary complex number i such that j 2 = −1 and jz = zj Central University of Tamil Nadu, Neelakudi, Thiruvarur, India; e-mail: [email protected].
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 812–821, June, 2020. Ukrainian DOI
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