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We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms of Dirichlet integrals. The expression for the remainder is written in terms of a similar type of integral. Since the asymptotic limit of the remainder being zero is a necessary and sufficient condition for the convergence of the series, this same condition on the asymptotic behaviour of the corresponding integrals constitutes such a necessary and sufficient condition. 

analytic functions integrable real functions, Fourier coefficients, Taylor series, Fourier series, convergence.