Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the trigonometric polynomial) along any direction, being non decreasing, in such a way that the sequence converges pointwise to the function $f$?



Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus?
The multidimensional Fejer series, i.e the Cesaro averages of the Fourier series of f, will converge uniformly to f. See <https://arxiv.org/pdf/1206.1789.pdf> or <https://www.sciencedirect.com/science/article/pii/S0022247X12000546> for a lot more detailed information. 


