Abstract
There have been several attempts in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra. The first part of this chapter features certain properties of the asymptotic behaviour of the quaternion Fourier transform. In the second part we introduce the quaternion Fourier transform of a probability measure, and we establish some of its basic properties. In the final analysis, we introduce the notion of positive definite measure, and we set out to extend the classical Bochner–Minlos theorem to the framework of quaternion analysis.
Mathematics Subject Classification (2010). Primary 30G35; secondary 42A38; tertiary 42A82.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Bahri. Generalized Fourier transform in real clifford algebra cl(0, n). Far East Journal of Mathematical Sciences, 48(1):11–24, Jan. 2011.
P. Bas, N. Le Bihan, and J.M. Chassery. Color image watermarking using quaternion Fourier transform. In Proceedings of the IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP, volume 3, pages 521–524. Hong-Kong, 2003.
E. Bayro-Corrochano, N. Trujillo, and M. Naranjo. Quaternion Fourier descriptors for preprocessing and recognition of spoken words using images of spatiotemporal representations. Mathematical Imaging and Vision, 28(2):179–190, 2007.
F. Brackx, N. De Schepper, and F. Sommen. The Clifford–Fourier transform. Journal of Fourier Analysis and Applications, 11(6):669–681, 2005.
F. Brackx, N. De Schepper, and F. Sommen. The Fourier transform in Clifford analysis. Advances in Imaging and Electron Physics, 156:55–201, 2009.
F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman, Boston, 1982.
T. Bülow. Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD thesis, University of Kiel, Germany, Institut für Informatik und Praktische Mathematik, Aug. 1999.
T. Bülow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fourier transforms of multidimensional signals. In G. Sommer, editor, Geometric computing with Clifford Algebras: Theoretical Foundations and Applications in Computer Vision and Robotics, pages 187–207, Berlin, 2001. Springer.
J. Ebling and G. Scheuermann. Clifford Fourier transform on vector fields. IEEE Transactions on Visualization and Computer Graphics, 11(4):469–479, July/Aug. 2005.
T.A. Ell. Quaternion-Fourier transforms for analysis of 2-dimensional linear timeinvariant partial-differential systems. In Proceedings of the 32nd Conference on Decision and Control, pages 1830–1841, San Antonio, Texas, USA, 15–17 December 1993. IEEE Control Systems Society.
T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEE Transactions on Image Processing, 16(1):22–35, Jan. 2007.
S. Georgiev. Bochner–Minlos theorem and quaternion Fourier transform. In Gürlebeck [14].
S. Georgiev, J. Morais, and W. Sprößig. Trigonometric integrals in the framework of quaternionic analysis. In Gürlebeck [14].
K. Gürlebeck, editor. 9th International Conference on Clifford Algebras and their Applications, Weimar, Germany, 15–20 July 2011.
K. Gürlebeck and W. Sprößig. Quaternionic Analysis and Elliptic Boundary Value Problems. Berlin: Akademie-Verlag, Berlin, 1989.
K. Gürlebeck and W. Sprößig. Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Aug. 1997.
E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.
E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n = 2(mod 4) and n = 3(mod4). Advances in Applied Clifford Algebras, 18(3–4):715–736, 2008.
V.V. Kravchenko. Applied Quaternionic Analysis, volume 28 of Research and Exposition in Mathematics. Heldermann Verlag, Lemgo, Germany, 2003.
B. Mawardi. Generalized Fourier transform in Clifford algebra Cℓ 0,3. Far East Journal of Mathematical Sciences, 44(2):143–154, 2010.
B. Mawardi, E. Hitzer, A. Hayashi, and R. Ashino. An uncertainty principle for quaternion Fourier transform. Computers and Mathematics with Applications, 56(9):2411–2417, 2008.
B. Mawardi and E.M.S. Hitzer. Clifford Fourier transformation and uncertainty principle for the Clifford algebra Cℓ 3,0. Advances in Applied Clifford Algebras, 16(1):41– 61, 2006.
S.-C. Pei, J.-J. Ding, and J.-H. Chang. Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Transactions on Signal Processing, 49(11):2783–2797, Nov. 2001.
M. Shapiro and N.L. Vasilevski. Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems I. ψ-hyperholomorphic function theory. Complex Variables, Theory and Application, 27(1):17–46, 1995.
M. Shapiro and N.L. Vasilevski. Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems II. Algebras of singular integral operators and Riemann type boundary value problems. Complex Variables, Theory Appl., 27(1):67–96, 1995.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Georgiev, S., Morais, J., Kou, K.I., Sprößig, W. (2013). Bochner–Minlos Theorem and Quaternion Fourier Transform. In: Hitzer, E., Sangwine, S. (eds) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0603-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0603-9_6
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0602-2
Online ISBN: 978-3-0348-0603-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)