Jorge L. deLyra1
Department of Mathematical Physics,
Physics Institute,
University of São Paulo
April 8, 2018
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 behavior of the corresponding integrals
constitutes such a necessary and sufficient condition.
