Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator
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Advances in Applied Clifford Algebras
Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator Jinxia Li and Jianxun He∗ Communicated by Helmuth Robert Malonek Abstract. In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space L2 (R2 , H), such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations in the quaternionic settings. Finally, we prove Walnut’s and Janssen’s representation theorems and other boundedness results of the QGFO. Mathematics Subject Classification. Primary 42C15; Secondary 44A15, 16H05, 30G35. Keywords. Quaternion valued function, Quaternionic Fourier transform, Two-sided QGFT, QGFO, Walnut’s representation.
1. Introduction In 1843, Irish mathematician Hamilton firstly discovered quaternions when attempting to extend the complex numbers to 3-dimension. After his remarkable invention, quaternions have been studied extensively in both theory and applications. Quaternions are very efficient for analyzing situations, such as rotations in R3 . It is known that quaternions have important applications in image processing, computer graphics, robotic mechanics of satellites and crystallographic texture analysis etc. (see [8,11,12,23–25]). This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671414 and 12071229). ∗ Corresponding
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J. Li, J. He
Adv. Appl. Clifford Algebras
In harmonic analysis and applied mathematics, the Fourier transform is an essential tool so that the extension of the Fourier transform to the quaternion valued functions has become an interesting problem. Ernst et al. [13] and Delsuc [10] first applied quaternions to the Fourier transform, later many authors studied the quaternionic Fourier transform (QFT) (see [4–6,15,19–21]). Because of the non-commutative property of the quaternion multiplication, the QFT on R2 can be defined in at least 3 different ways, that is, the left-sided, right-sided and two-sided (windowed) Fourier transforms, which are different transforms and whose corresponding properties are obtained by many scholars. The two-sided QGFT is one of the most interesting cases of a quaternionic windowed Fourier transform because it is neither left- nor right-linear with respect to quaternionic constants (see [2,3,6,21]). It is obvious that the nonlinear property makes a challenge for the study of the two-sided QGFT and quaternionic Gabor frame operator (QGFO). In 2012, Fu et al. [16] obtained reconstruction formula, uncertainty principle and other results for the windowed quaternionic Fourier transform. In 2016, Akila et al. [2] extended the Gabor transform to the quaternion valued function on Rd (d ∈ N) in two different ways, one of which is the two-sided way. In 2018, Cerejeiras et al. [7] studied the qu
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