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This research is motivated by the program of Reverse Mathematics.
We investigate some theorems for the convergence of Fourier series within some
weak subsystems of second order arithmetic, in order to determine which set
existence axioms are needed to prove these theorems.
We show that uniformly convergence of Fourier series for C^1-functions and
L^2-convergence of Fourier series for continuous functions are equivalent to
WKL_0 over RCAo_0. We also show that L^2-convergence of Fourier series for
bounded continuous functions is equivalent to WWKL_0 over RCA_0.