I’m asking for clarification regarding the proof of a theorem characterizing the solutions to the steady state heat equation on the 2D unit disc. This proof appeared in Stein’s analysis textbook from Princeton’s notes.
In proof to theorem 5.7, chapter 2, the author says that for the function $u(r, \theta)$ represented by the Fourier series, which is just a convolution of $f$ (defined on the boundary of the disk) and the Poisson kernel, “fix $ \rho>0$, inside each disc of radius $r<\rho<1$ centered around the origin, the series for $u$ can be differentiated term by term and the differentiated series is uniformly and absolutely convergent.” I’m confused and wants to know what’s the reasoning behind this, because I haven’t yet seen theorems on differentiating Fourier series in this chapter. Any suggestion about the analysis result that the author implicitly used will be helpful. Thanks!