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Motivated by problems in robust control of power signal set H2 (e.g,, H2 is not a vector space, Fourier analysis cannot be carried out on H2 in general), we study those functions in H2 which we call strong limit power. We show that the set of all such functions is a sufficiently large C*-algebra. Fourier Analysis is carried out on the functions. In particular, the uniqueness of the Fourier expansion of a strong limit power function is established. Finally we point out how to analyze and reconstruct such functions.