Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T19:58:21.322Z Has data issue: false hasContentIssue false
  • English
  • Français

One-parameter Groups of Operators and Discrete Hilbert Transforms

Published online by Cambridge University Press:  20 November 2018

Laura De Carli  and
Gohin Shaikh Samad
Laura De Carli
Affiliation:
Department of Mathematics, Florida International University, Miami, FL 33199, USA e-mail: decarlil@fiu.edu
Gohin Shaikh Samad
Affiliation:
Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA e-mail: shaikhgohin-samad@uiowa.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the discrete Hilbert transform and the discrete Kak–Hilbert transform are infinitesimal generators of one-parameter groups of operators in ${{\ell }^{2}}$.

Keywords

42A4542A5041A44discrete Hilbert transformgroups of operatorsisometries
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 497 - 507
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Aigner, M. and Ziegler, G. M., Proofs from The Book. Fourth éd.,Springer-Verlag, Berlin, 2010. http://dx.doi.org/10.1007/978-3-642-00856-6 Google Scholar
[2] Belov, Y., Mengestie, T., and Seip, K., Unitary discrete Hilbert transforms. J. Anal. Math. 112(2010), 383–393. http://dx.doi.org/10.1007/s11854-010-0035-y Google Scholar
[3] Grafakos, L., An elementary proof of the square summability of the discrete Hilbert transform. Amer. Math. Monthly 101(1994), no. 5, 456–458. http://dx.doi.org/10.2307/2974910 Google Scholar
[4] Grafakos, L., Best bounds for the Hilbert transform on L^R1). Math. Res. Lett. 4(1997), no. 4, 469–471. http://dx.doi.org/10.4310/MRL.1997.v4.n4.a3 Google Scholar
[5] Grafakos, L., Classical and modern Fourier analysis. Pearson Education, Inc., Upper Saddle River, NJ, 2004.Google Scholar
[6] Hille, E. and Phillips, R., Functional analysis and semigroups. American Mathematical Society Colloquium Publications, 31, American Mathematical Society, Providence, RI, 1957.Google Scholar
[7] Kak, S., The discrete Hilbert transform. Proc. IEEE 58(1970), 585–586.Google Scholar
[8] Laeng, E., Remarks on the Hilbert transform and some families of multiplier operators related to it. Collect. Math. 58(2007), no. 1, 25–44.Google Scholar
[9] Pichorides, S. K., On the best values of the constants in the Theorems of M. Riesz, Zygmund and Kolmogorov.Studia Math. 46(1972), 165–179.Google Scholar
[10] Schreier, P. and Scharf, L., Statistical signal processing of complex-valued data: the theory of improper and noncircular signals. Cambridge University Press, Cambridge, 2010. http://dx.doi.org/10.1017/CBO9780511815911 Google Scholar
[11] Schur, J., Bemerkungen zur Theorie der beschrànkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math. 140(1911), 1–28. http://dx.doi.org/10.1515/crll.1911.140.1 Google Scholar
[12] Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971.Google Scholar
[13] Weyl, H., RegulàreIntegralgleichungenmit besondererBerùcksichtigung des Fourierschen Integraltheorems. Doctoral Dissertation, University of Gottingen, 1908.Google Scholar
[14] Yosida, K., Functional analysis. Springer-Verlag, 1968.Google Scholar