Fejér kernels for Fourier series on \(\mathbb{T}^ n\) and on compact Lie groups (Q1340926)

We discuss two different generalizations of the one-dimensional Fejér kernel to the polyhedral summability of multiple Fourier series and we obtain fairly explicit extensions of the well-known identity \[ \sum^ N_{k=-N}\left(1- {| k|\over N+1}\right) e^{ikt}= {1\over N+1} \left({\sin ((N+ 1)t/2) \over \sin (t/2)}\right)^ 2. \] We then apply these results to the study of Fourier series on a compact simple simply connected Lie group \(G\). For each definition of Fejér kernel on \(G\) we compute the Lebesgue constants (i.e., the \(L^ 1\)-norms), which in one case turn out to be unbounded even if we substitute the Fejér kernel with a Cesàro kernel of arbitrary order.