By Giuliano Sorani

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Qian, The Paley–Wiener theorem in the non-commutative and non-associative octonions, Sci. China Ser. A 52 (2009), 129–141. [18] C. Nolder, Conjugate harmonic functions and Cliﬀord algebras, J. Math. , 302 (2005), 137–142. K. Pichorides, On the best values of constants in the theorems of M. , 44 (1972), 165–179. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. [21] P. Stein, On a theorem of M. Riesz, J. London Math. , 8 (1933), 242–247. edu Temple University 1801 N.

In fact, in the recent PhD-thesis by L. Shuang (cf. [31]) a comparison between the AFD (Adaptive Fourier Decomposition)-method and Basis Pursuit where made showing that there are indeed situations where a Basis Pursuit has an advantage. The mathematical justiﬁcation for the applicability of Basis Pursuit was given only by an asymptotic analysis and, therefore, is only valid for large scale matrices. Here, we will use a compressed sensing approach to the reconstruction of a given signal in terms of Takenaka–Malmquist systems.

The principles outlined in the last two sections are valid and the results available in a number of contexts for several real and complex variables, and with functions valued in vectors and matrices. We give a brief introduction to this circle of ideas in the following sections. 28 L. -X. Mai and T. Qian 4. Quaternionic and Cliﬀord contexts for functions of several real variables Denote by Q the quaternion algebra, and by Am the Cliﬀord algebra of linear dimension 2m . The analogues of AFD have been formulated for Hardy spaces in the setting of Q and of Am in [36] [42], respectively.