Abstract
In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley–Wiener type, namely the Paley–Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availibility Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Békollé, D., Bonami, A.: Hausdorff–Young inequalities for functions in Bergman spaces on tube domains. Proc. Edinb. Math. Soc. 41, 553–566 (1998)
Békollé, D., Bonami, A., Garrigós, G., Nana, C., Peloso, M., Ricci, F.: Lecture notes on Bergman projectors in tube domains over cones: an analytic geometric viewpoint. In: IMHOTEP: African Journal of Pure and Applied Mathematics 5, Exposé I (2012)
Bernstein, S.: A Paley–Wiener theorem and Wiener–Hopf integral equations in Clifford analysis. Adv. Appl. Cliff. Algebras 8, 31–46 (1998)
Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis, Research Notes in Mathematics, vol. 75. Pitman Advanced Publishing Company, Boston, London, Melbourne (1982)
Dang, P., Mai, W.X., Qian, T.: Fourier spectrum of Clifford \({H}^p\) spaces on \({\mathbf{R} }^{n+1}_+\) for \(1\le p\le \infty \). J. Math. Anal. Appl. 483(1), 123598 (2020)
Duren, P., Gallardo-Gutiérrez, E.A., Montes-Rodríguez, A.: A Paley–Wiener theorem for Bergman spaces with application to invariant subspaces. Bull. Lond. Math. Soc. 39, 459–466 (2007)
Garrigós, G.: Generalized Hardy spaces on tube domains over cones. Colloq. Math. 90, 213–251 (2001)
Genchev, .T.G.: Paley–Wiener type theorems for functions in Bergman spaces over tube domains. J. Math. Anal. Appl. 118, 496–501 (1986)
Gilbert, J., Murray, M.: Clifford algebras and Dirac operators in harmonic analysis
Hőrmander, L.: The analysis of linear partial differential operator, I. distribution theory and Fourier analysis, 2 edn. Springer, Berlin (2003)
Kou, K.I., Qian, T.: Shannon sampling in the Clifford analysis setting. J. Anal. Appl. 24, 853–870 (2005)
Kou, K.I., Qian, T.: The Paley–Wiener theorem in \({\mathbf{R} }^n\) with the Clifford analysis setting. J. Funct. Anal. 189, 227–241 (2002)
Li, C., McIntosh, A., Qian, T.: Clifford algebras, Fourier transform and singular convolution operators on Lipschitz surfaces. Revista Mathematica Iberoamericana 10, 665–721 (1994)
Li, H.C., Deng, G.T., Qian, T.: Fourier spectrum characterizations of Hp spaces on tubes over cones for \(1\le p\le \infty \). Complex Anal. Oper. Theory 12, 1193–1218 (2018)
Li, S., Leng, J., Fei, M.: Paley–Wiener type theorems for the Clifford Fourier transform. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.5707
Li, S., Leng, J., Fei, M.: On the supports of functions associated to the radially deformed Fourier transform. Adv. Appl. Clifford Algebras 30 (2020). https://doi.org/10.1007/s00006-020-01067-7
Mitrea, M.: Clifford Wavelets, Singular Integrals and Hardy Spaces. Lecture Note in Mathematics, Springer, Berlin Heidelberg (1994)
Paley, R.C., Wiener, N.: Fourier transforms in the complex plane, Colloquium Publications, vol. 19. American Mathematical Society (1934)
Peña Peña, D.: Cauchy–Kowalevski extensions, Fueter’s theorems and boundary values of special systems in Clifford analysis. Ph.D. thesis, Ghent University (2008)
Qian, T.: Characterization of boundary values of functions in Hardy spaces with applications in signal analysis. J. Integ. Equ. Appl. 17, 159–198 (2005)
Qian, T., Xu, Y.S., Yan, D.Y., Yan, L., Yu, B.: Fourier spectrum characterization of Hardy spaces and applications. Proc. Am. Math. Soc. 137, 971–980 (2009)
Schwartz, L.: Transformation de Laplace des distributions. Comm.Sém. Math. Univ. Lund pp. 196–206 (1952)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, New Jersey (1970)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, New Jersey (1971)
Acknowledgements
This work was supported in part by the Science and Technology Development Fund, Macau SAR: 154/2017/A3; NSFC Grant Nos. 11701597; NSFC Grant Nos. 11901594; The Science and Technology Development Fund, Macau SAR: 079/2016/A2, 0123/2018/A3.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
No potential conflict of interest was reported by the authors.
Ethical Statement
This paper complies to the Ethical rule applicable for the journal “Advances in Applied Clifford Algebras”.
Additional information
Communicated by Uwe Kaehler.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dang, P., Mai, W. & Qian, T. On Monogenic Reproducing Kernel Hilbert Spaces of the Paley–Wiener Type. Adv. Appl. Clifford Algebras 32, 50 (2022). https://doi.org/10.1007/s00006-022-01240-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-022-01240-0