Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the trigonometric polynomial) along any direction, being non decreasing, and additionally each element of the sequence being computable from a finite set of samples of $f$, in such a way that the sequence converges pointwise to the function $f$? (clarity: The finite set of samples of $f$ for computability of each element of the sequence could be different).

Without this additional computability from finite samples condition, an example given by Yuval Peres here, the multi dimensional Fejer series would be an example. But the computability from finite samples condition would exclude this, due to this non-computability theorem.

This question was refined from here, after the answer from Yuval there.

  • $\begingroup$ Is it correct that for approximation of a continuous function on $[0, 1]$ by polynomials Bernstein polynomials do this work? $\endgroup$ – Fedor Petrov Jun 6 '20 at 19:38
  • $\begingroup$ @FedorPetrov : Although, they do approximate, Bernstein polynomials aren't band limited, I looked up this page en.wikipedia.org/wiki/…. $\endgroup$ – Rajesh D Jun 7 '20 at 0:49
  • $\begingroup$ Can you approximate your continuous function by a piecewise linear function (this requires a finite number of samples), then approximate that with a sufficiently far out term of the Fejer series? $\endgroup$ – Sam Zbarsky Jun 12 '20 at 6:06
  • $\begingroup$ Its been done here with trigonometric polynomials like Bernstein polynomials : hindawi.com/journals/tswj/2014/174716 $\endgroup$ – Rajesh D Jun 23 '20 at 16:51

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