The question is the following:
Let $f$ be a piecewise $\mathcal{C}^1(\mathbb{R})$, $2\pi$-periodic function, and let $a_{n}[f],b_{n}[f]$ be its real Fourier coefficients. Show that the series \begin{align*} a_{0}[f]+ \sum_{n=1}^{\infty} \rho(x, y)^n(a_{n}[f]\cos n\theta(x, y)+b_n[f]\sin n\theta(x, y)), \end{align*} converges uniformly in any $\Omega_{\rho_0}=\{(x,y)\in\mathbb{R}^2:x^2+y^2<\rho_0^2\},\,\forall \rho_0\in(0,1)$ to a function $u\in\mathcal{C}^{\infty}(\Omega_1)$, solution to \begin{align*} &u_{xx}+u_{yy}=0,\quad (x, y)\in\Omega_1 \end{align*} which satisfies the boundary condition \begin{align*} &\lim_{\rho\to 1-}u(\rho\cos\theta, \rho\sin\theta)=\frac{f(\theta-)+f(\theta+)}{2},\,\forall \theta\in\mathbb{R}. \end{align*}
I have indeed shown that the series converges uniformly in each $\Omega_{\rho_0}$ to a solution of Laplace's equation $u\in\mathcal{C}^{\infty}(\Omega_1)$. However, I'm struggling to show that the boundary condition is satisfied. I have also shown that \begin{align*} \lim_{m\to\infty}&\sum_{n=0}^{m}u_n(\cos\theta, \sin\theta)=\lim_{m\to\infty}\sum_{n=0}^{m}a_{n}[f]\cos(n\theta)+b_n[f]\sin(n\theta)=\\ &=\lim_{m\to\infty} T_m[f](\theta)=\frac{f(\theta-)+f(\theta+)}{2},\;\forall \theta\in\mathbb{R}, \end{align*} but as the convergence is not uniform in $\Omega_1$, the infinite sumation and the limit cannot be exchanged. I have also tried to fix $\theta^*$ and show that $$v(\rho)=u(\rho\cos\theta^*,\rho\sin\theta^*)$$ is continuous at $\rho=1$, but I cannot make it work. Do you have any ideas on how should I focus the proof or any references I could follow? I have seen some things based on the Poisson kernel, but I am not supposed to use complex numbers.
It seems like Abel's theorem for power series applies due to the pointwise convergence of $\sum_{n = 0}^{\infty} a_n[f]\cos(n\theta) + b_n[f]\sin(n\theta)$, meaning that you do actually have continuity of your function $v$ at $\rho = 1$.
And even if you were not supposed to know about that theorem, maybe you could adapt its proof to your case?