0
$\begingroup$

This is what I know so far:

  1. The given series uniformly converges by the M-test and that I can swap the integration and the sum when calculating the coefficients.
  2. Apparently I am supposed to use the trigonometric polynomial form of $f(x)$ to find the coefficients, but I am unsure as to how to go about proving that there is some Trigonometric polynomial that converges (uniformly?) to $f(x)$ (rather than just and approximation of it).

That said, please consider when you give an answer that this question was taken from an exam for first year math B.Sc. , and that I would love an answer that gives me pointers as to how to approach these kinds of questions.

  • 5
    Hint: uniqueness...you can read off the Fourier coefficients directly.2012-08-08
  • 0
    Thomas please elaborate - my main problem isn't finding the coefficients after i show that: $f(x) = A_0+\sum_{n=1}^{\infty} A_ncos(nx)+\sum_{n=1}^{\infty} B_nsin(nx)$, but rather what arguments to make so i could say there is a real equality there and it is not just the function's approximation.2012-08-08
  • 2
    Just compare your representation of $f$ in your comment with the definition of $f$ in your question. You are already given a Fourier series representation. Just make sure you take into account the normalizing coefficients of $1/\pi$ or $1/(2\pi)$ or $1/\sqrt(\pi)$ -- depending on the conventions in your class -- to write down the Fourier coefficients.2012-08-08

0 Answers 0