0
$\begingroup$

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 $f$?

$\endgroup$
1
  • $\begingroup$ Refined and posted another question here, which adds computability condition, after reading the answer by Yuval. mathoverflow.net/q/362189/14414 $\endgroup$
    – Rajesh D
    Jun 4, 2020 at 13:22

1 Answer 1

5
$\begingroup$

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.

$\endgroup$
1
  • 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. $\endgroup$ Jun 4, 2020 at 13:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.