Analytic signal in many dimensions

In this paper we extend analytic signal method to the functions in many dimensions. First it is shown how to obtain separate phase-shifted components and how combine them to obtain signal's envelope, instantaneous frequencies and phases in many dimensions. Second, we show that phase-shifted components may be obtained by positive frequency restriction of the Fourier transform defined in the algebra of commutative elliptic hypercomplex numbers. Finally we prove that for $d>2$ there is no corresponding Clifford-Fourier transform that allows to recover phase-shifted components correctly.