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 $f$?
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$\begingroup$ Refined and posted another question here, which adds computability condition, after reading the answer by Yuval. mathoverflow.net/q/362189/14414 $\endgroup$– Rajesh DCommented Jun 4, 2020 at 13:22
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1 Answer
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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.
answered Jun 4, 2020 at 12:54
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2$\begingroup$ That would be a different question. The paper you cite involves computability from finite samples. Rather than editing your question when it is answered, it is better to post a new question. $\endgroup$ Commented Jun 4, 2020 at 13:05