Using the following theorem: The trigonometric polynomials ($\mathbb{C} \to \mathbb{C}$) are uniformly dense in $C(\mathbb{T})$ (functions $\mathbb{C} \to \mathbb{C}$ that are continuous and periodic with a period of $2\pi$.

Prove: For every continuous function $f:[-\pi,\pi]\to\mathbb{R}$ there exists a series of polynomials $\{P_n: \mathbb{R} \to \mathbb{R}\}$, such that: $P_n \to f$ uniformly.

Any help would be appreciated.

  • 1
    $\begingroup$ It suffices you show the Taylor series of the sine and cosine converge uniformly on compact intervals. $\endgroup$ – Pedro Tamaroff Jun 2 '18 at 20:47
  • $\begingroup$ @PedroTamaroff Why is that true? $\endgroup$ – Um Shmum Jun 2 '18 at 20:49
  • $\begingroup$ you will have shown that there exists a series of polynomial that converge to a trigonometric polynomial, and since they're dense in periodic functions of period $2\pi$ and $f$ is in the interval $[-\pi,\pi]$ you can conclude $\endgroup$ – user438666 Jun 2 '18 at 20:52
  • $\begingroup$ The Cesaro sums of the sequence of partial sums of the Fourier series are just averages of trigonometric polynomials, and therefore themselves trigonometric polynomials. Fejer tells you that they converge uniformly to the function. $\endgroup$ – user565560 Jun 2 '18 at 20:53
  • $\begingroup$ @UmShmum Check the formula for the remainder. $\endgroup$ – Pedro Tamaroff Jun 2 '18 at 22:19

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