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We characterize in terms of Fourier spectrum the boundary values of functions in the complex Hardy Space Hp(C+) with 0 < p < 1.First,we prove that the function f in the Lp(R) can be decomposed into a sum g + h,where g and h are the non-tangential boundary limits of a function in Hp(C+) and Hp(C_??) in the sense of Lp(R),where Hp(C+) and Hp(C_??) are the Hardy spaces in the upper and lower complex plane C+ and C_??,respectively.Secondly,as an application,we extend the Paley-Wiener Theorem,originally for square-integrable functions,to the Hp(C+) cases with 0 < p < 1.Finally,we give a Fourier spectrum characterization for Lp(R) functions.