1

Given a sequence $ \{a_n\}_{n=1}^{\infty} $, such that the series $ \sum_{n=1}^{\infty} a_n $ converges absolutely, show that the series $$ \sum_{n=1}^{\infty} a_n \cos(nx) $$ converges for all $ x \in \mathbb{R} $. Additionally, if the sum is a function $ f(x) $, prove that $ f(x) $ is continuous, and that its Fourier series is $$ \sum_{n=1}^{\infty} a_n \cos(nx). $$

For the first part, I used the M-test, since $ |\cos(nx)| \leq 1 $ and the fact that the series $ \sum_{n=1}^{\infty} a_n $ is absolutely convergent. It follows that the series $ \sum_{n=1}^{\infty} a_n \cos(nx) $ is uniformly convergent to some function $ f(x) $. The function $ f(x) $ is continuous because of the uniform convergence of the series and the fact that $ a_n \cos(nx) $ is continuous for all $ n $.

I am not sure about the last part: how do I show that the Fourier series of $ f(x) $ is the series $ \sum_{n=1}^{\infty} a_n \cos(nx) $?

Edit: Thanks to Srini for the comment, I substituted f(x) with $$\sum_{n=1}^\infty a_n \cos(n x)$$, when calculating the Fourier coefficients and it worked out.

mpavlov23
  • 177
  • 1
    You can show that the coefficients $B_{n}$ for the sine become $0$ by showing that the integral used to compute $B_{n}$ vanishes due to orthogonality of cos and sin. – Srini Sep 12 '24 at 20:29
  • Thanks that worked, I substituted f(x) with the given series that converges uniformly to f and I worked out the coefficients an and bn for the Fourier representation. – mpavlov23 Sep 13 '24 at 13:20

1 Answers1

1

The Fourier coefficients are coefficients of a vector to an orthogonal basis. They are unique. As the function already has the form of a Fourier series, there is nothing to calculate. It is similar to ask if the Cartesian coordinates of the vector $(1,2,3)$ are $(1,2,3)$. Or to determine the Laurent series at $z_0=0$ of $f(z)=\frac1{z^2}$.

Lutz Lehmann
  • 131,652