Abstract
This paper establishes a real Paley-Wiener theorem to characterize the quaternion-valued functions whose quaternion Fourier transform has compact support by the partial derivative and also a Boas theorem to describe the quaternion Fourier transform of these functions that vanish on a neighborhood of the origin by an integral operator.
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Pei S.C., Ding J.J., Chang J.H.: Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Transactions on Signal Processing 49(11), 2783–2797 (2001)
T. Bülow, M. Felsberg, G. Sommer, Non-commutative hypercomplex Fourier transforms of multidimensional signals. G. Sommer (Ed.), Geometric Computing with Clifford Algebras, Springer, Heidelberg, 2001, pp. 187–207.
Hitzer E.: Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras 17(3), 497–517 (2007)
Hitzer E.: Directional uncertainty principle for quaternion Fourier transform. Advances in Applied Clifford Algebras 20, 271–284 (2010)
Sangwine S.J., Ell T.A.: Hypercomplex Fourier transforms of color images. IEEE Transactions on Image Processing 16(1), 22–35 (2007)
Bayro-Corrochano E., Trujillo N., Naranjo M.: Quaternion Fourier descriptors for preprocessing and recognition of spoken words using images of spatiotemporal representations. Journal of Mathematical Imaging and Vision 28(2), 179–190 (2007)
Bahri M., Hitzer E., Hayashi A., Ashino R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56(9), 2398–2410 (2008)
Bang H.H.: A property of infinitely differentiable functions. Proc. Amer. Math. Soc. 108, 73–76 (1990)
Tuan V.K.: Spectrum of signals. J. Foureir Anal. Appl. 7(3), 319–323 (2001)
Tuan V.K.: Paley-Wiener-Type theorems. Frac. Cal. Appl. Anal. 2, 135–143 (1999)
Tuan V.K., Zayed A.I.: Paley-Wiener-Type theorems for a Class of Integral Transforms. J. Math. Anal. Appl. 266, 200–226 (2002)
Andersen N.B., De Jeu M.: Real Paley-Wiener theorems and local spectral radius formulas. Trans. Amer. Math. Soc. 362, 3613–3640 (2010)
Andersen N.B.: Real Paley-Wiener theorems for the Hankel transform. J. Fourier Anal. Appl. 12, 17–25 (2006)
Q. H. Chen, L. Q. Li, G. B. Ren, Generalized Paley-Wiener theorems. Int. J. Wavelets, Multiresolut. Inf. Process. 10(2) (2012), 1250020 (7 pages).
Chettaoui C., Othmani Y., Triméche K.: On the range of the Dunkl transform on \({\mathbb{R}}\). Anal. Appl. (Singap.) 2(3), 177–192 (2004)
Mejjaoli H., Triméche K.: Spectrum of functions for the Dunkl transform on \({{\mathbb{R}}^d}\). Fract. Calc. Appl. Anal. 10(1), 19–38 (2007)
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The research was partially supported by NSFC under grant 11071058.
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Fu, Y., Li, L. Paley-Wiener and Boas Theorems for the Quaternion Fourier Transform. Adv. Appl. Clifford Algebras 23, 837–848 (2013). https://doi.org/10.1007/s00006-013-0412-6
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DOI: https://doi.org/10.1007/s00006-013-0412-6