# Continuity of Fourier series with decay condition on coefficients

Let $$\theta \in \ell^2(\mathbb{N})$$ satisfy $$\sum_{j=1}^n \theta_j^2 a_j^2 \leq Q$$ for some $$Q > 0$$ where $$a_j = \begin{cases} j^\beta & \text{j even}\\ (j-1)^\beta & \text{j odd} \end{cases}$$ for some $$\beta > 1/2$$. Let $$\{\varphi_j\}_{j=1}^\infty$$ denote the trigonometric basis of $$[0,1]$$, i.e. $$\varphi_1(x) = 1$$, $$\varphi_{2k}(x) = \sqrt{2} \cos(2\pi k x)$$ and $$\varphi_{2k+1}(x) = \sqrt{2} \sin(2\pi k x)$$ for $$k \in \mathbb{N}$$. Show that $$f = \sum_{j=1}^\infty \theta_j \varphi_j$$ is continuous.

I know that $$\varphi_j$$ is Lipschitz with constant $$\pi\lfloor j+1 \rfloor$$ but I don't think that helps very much. Any ideas?

Note first that the hypothesis is just a silly way of writing $$\sum_{j=1}^\infty\theta_j^2j^{2\beta}<\infty.$$

## Proof by Looking it Up

The hypothesis says precisely that $$f$$ is in the Sobolev space often denoted $$H^\beta$$; since $$\beta>1/2$$ the Sobolev Embedding Theorem shows that $$f$$ is continuous.

## Actual Proof

$$\sum|\theta_j|=\sum|\theta_j|j^\beta j^{-\beta}\le\left(\sum\theta_j^2j^{2\beta}\right)^{1/2}\left(\sum j^{-2\beta}\right)^{1/2}<\infty,$$since $$2\beta>1$$. So the series converges uniformly, hence $$f$$ is continuous.

## Proof of a Stronger Result

Say $$x and $$y-x$$ is small. Choose an integer $$N$$ with $$1/2\le N(y-x)\le 2$$. Now $$f(y)-f(x)=\left(\sum_{j=1}^N+\sum_{j=N+1}^\infty\right)\theta_j(\phi_j(y)-\phi_j(x)):=I+II.$$Now the Lipschitz constant for $$\phi_j$$ shows that $$I\le c(y-x)\sum_{j=1}^N\theta_jj^\beta j^{1-\beta}\le c(y-x)\left(\sum_{j=1}^N\theta_j^2j^{2\beta}\right)^{1/2}\left(\sum_{j=1}^Nj^{2-2\beta}\right)^{1/2}\le c(y-x)N^{\frac32-\beta}\le c(y-x)^{\beta-\frac12}.$$To estimate $$II$$ you just use the fact that $$|\phi_j|\le1$$: $$II\le 2\sum_{j=N+1}^\infty\theta_j j^\beta j^{-\beta}\le c\left(\sum_{j=N+1}^\infty j^{-2\beta}\right)^{1/2}\le cN^{\frac12-\beta}\le c(y-x)^{\beta-\frac12}.$$Since $$\beta-\frac12>0$$ this shows that $$f\in Lip_{\beta-\frac12}$$.

## Note

It's worthwhile spending some time understanding this argument, because the outline "use a trivial uniform estimate for the high-order terms and the Lipschitz condition on the low-order terms" comes up a lot in this sort of thing.