Abstract
The aim of this work is to introduce quasi-monogenic functions, to prove their properties and to discuss some examples. Quasi-monogenic functions are null solutions of a differential operator with Fourier symbol \(|\underline{\xi }| m(\underline{\xi }),\) where \(m(\underline{\xi })\) is an \(L^p\)- multiplier. Furthermore, \(m(\underline{\xi })\) is the Fourier symbol of a generalized Riesz–Hilbert transform. As examples we investigate the Riesz–Hilbert transform, higher Riesz–Hilbert transforms, and the linearized Riesz transforms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Lecture Notes in Mathematics, vol. 2044. Springer, Berlin (2012)
Bernstein, S.: Fractional Riesz–Hilbert-type transforms and associated monogenic signals. Complex Anal. Oper. Theory 11, 995–1015 (2017). https://doi.org/10.1007/s11785-017-0667-3
Bernstein, S., Bouchot, J.-L., Reinhardt, M., Heise, B.: Generalized analytic signals in image processing: comparison, theory and applications. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics, pp. 221–246. Birkhäuser, Basel (2013)
Bernstein, S., Heise, B., Reinhardt, M., Häuser, S., Schausberger, S., Stifter, D.: Fourier plane filtering revisited—analogies in optics and mathematics. Sampl. Theory Signal Image Process. 13(3), 231–248 (2014)
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics, vol. 76. Pitman, London (1982)
Delanghe, R.: Clifford analysis: history and perspective. Comput. Methods Funct. Theory 1(1), 107–153 (2001)
Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions, A Function Theory for the Dirac Operator, Mathematics and Its Applications, vol. 53. Springer, Amsterdam (1992)
Duoandikoetxea, J.: Fourier Analysis, Graduate studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)
Felsberg, M., Sommer, G.: The monogenic signal. IEEE Trans. Signal Proc. 49(12), 3136–3144 (2001)
Gabor, D.: Theory of communication. J. Inst. Electr. Eng. Part III Radio Commun. Eng. 93(26), 429–457 (1946)
Grafakos, L.: Classical Fourier Analysis, 1st edn. Springer, New York (2009)
Gürlebeck, K., Sprößig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel (1990)
Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser, Basel (2008)
Hahn, S.L.: Multidimensional complex signals with single-orthant spectra. Proc. IEEE 80(8), 1287–1300 (1992)
Häuser, S., Heise, B., Steidl, G.: Linearized Riesz transform and quasi-monogenic shearlets. Int. J. Wavelets Multiresolut. Inf. Process. 12, 1450027 (2014). https://doi.org/10.1142/S0219691314500271
Held, S.: Monogenic Wavelet Frames of Image Analysis, Ph.D. thesis. TU München, Fakultät für Mathematik, Munich (2012)
Larkin, K.G., Bone, D.J., Oldfield, M.A.: Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform. J. Opt. Soc. Am. A 18(8), 1862–1870 (2001)
McIntosh, A.: Fourier theory, singular integrals and harmonic functions on Lipschitz domains. In: Ryan, J. (ed.) Clifford Algebras in Analysis and Related Topics, pp. 33–88. CRC Press, Boca Raton (1996)
Mikhlin, S.G.: Multidimensional Singular Integrals and Integral Equations, International Series of Monographs in pure and applied Mathematics, vol. 83. Pergamon Press, Oxford (1965)
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, PMS 32. Princeton University Press, Princeton (1971)
Unser, M., Sage, D., Van De Ville, D.: Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform. IEEE Trans. Image Process. 18(11), 2402–2418 (2009). https://doi.org/10.1109/TIP.2009.2027628
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.
Rights and permissions
About this article
Cite this article
Bernstein, S. Quasi-monogenic Functions. Adv. Appl. Clifford Algebras 28, 91 (2018). https://doi.org/10.1007/s00006-018-0908-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-018-0908-1