# Uniform convergence of a trigonometric series

Let $L>0$ be a constant. With what coefficients $\alpha_k$ and $\beta_k$ does the trigonometric series $$\alpha_0 +\sum_{k=1}^{\infty} \left[\alpha_k \cos\left( \frac{k\pi x}{L} \right) +\beta_k \sin\left(\frac{k\pi x}{L}\right)\right]$$ converge uniformly?

Using the triangle inequality we have

$$\left|\alpha_k\cos\left(\frac{k\pi x}{L}\right) + \beta_k\sin\left(\frac{k\pi x}{L}\right)\right| \le \left|\alpha_k\cos\left(\frac{k\pi x}{L}\right)\right| + \left|\beta_k\sin\left(\frac{k\pi x}{L}\right)\right| \le |\alpha_k| + |\beta_k|.$$

Now by the Weierstrass criterion the trigonometric series converges uniformly if the series $$\sum_{k=1}^{\infty}\left(|\alpha_k|+|\beta_k|\right)$$ converges.

Is there anything we can say about the coefficients $\alpha_k$ and $\beta_k$ other than the fact that $|\alpha_k|+|\beta_k|$ must approach zero as $k$ approaches infinity?