Criteria for absolute convergence of Fourier series

Abstract:

Let $f \in {\mathbf {L}^1}({\mathbf {T}})$. Define ${f_\alpha }$ by ${f_\alpha }(x) = f(x + \alpha )$. Then the Fourier series of $f$ is absolutely convergent if and only if there exists a Lebesgue point $\alpha$ for $f$ such that both sequences ${\langle {({{\mathbf {R}}_{\text {e}}}{\hat f_\alpha }(n))^ - }\rangle _{n \in {\mathbf {Z}}}}{\langle {({\mathcal {I}_{\text {m}}}{\hat f_\alpha }(n))^ - }\rangle _{n \in {\mathbf {Z}}}}$ belong to ${l^1}$. The theorem remains true if the sentence “there exists a Lebesgue point $\alpha$ for $f$” is replaced by “there is $\alpha \in {\mathbf {R}}$ such that $f$ is essentially bounded in some neighborhood of $\alpha$".