International Scholarly Research Network ISRN Mathematical Physics Volume 2012, Article ID 530473, 12 pages doi:10.5402/2012/530473 Research Article Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States Zouha¨ ır Mouayn Department of Mathematics, Faculty of Sciences and Technics (M’Ghila), Sultan Moulay Slimane University, BP 523, Beni Mellal, Morocco Correspondence should be addressed to Zouha¨ ır Mouayn, mouayn@fstbm.ac.ma Received 11 April 2012; Accepted 31 May 2012 Academic Editors: V. Moretti and W.-H. Steeb Copyright q 2012 Zouha¨ ır Mouayn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis. 1. Introduction The study of Hardy spaces, which originated during the 1910s in the setting of Fourier series and complex analysis in one variable, has over time transformed into a rich and multifaceted theory, providing basic insights into such topics as maximal functions, Hankel operators, Hilbert transforms, and wavelets analysis 1. In physics, Hardy spaces are central in the rigged Hilbert space or Gel’fand triplet theory and play a crucial role in time-asymmetric quantum mechanics 2. These spaces are usually involved in causality problems. Indeed, a Hardy function is important in signal processing because it may be used as signal filter 3. In this paper, our aim is to construct an integral transform that connects the classical Hardy space H 2 R of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semi-axis, with a one-parameter family of weighted Bergman spaces F σ C consisting of analytic functions on the complex plane, which are square integrable with respect to the measure z z σ-1/2 K 1/2-σ 2 √ z zdμz where 2σ 1, 2, 3,..., K ν · denotes the MacDonald function and dμ is the Lebesgue measure on C. These spaces have been considered by Barut and Girardello while introducing a class of coherent states associated with noncompact groups 4. The constructed integral transform