MHB Can You Solve the Harmonic Function Challenge?

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    2017
Euge
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Here is this week's POTW:

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Let $\Bbb D$ be the open unit disc in the complex plane, and let $f$ be a continuous complex function on $\partial\Bbb D$. Consider the function

$$F(re^{i\phi}) \,\dot{=}\, \frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})\frac{1-r^2}{1-2r\cos(\theta-\phi) + r^2}\, d\theta\quad (re^{i\phi}\in \Bbb D)$$

Prove $F$ is harmonic on $\Bbb D$, and that for all $z_0\in \partial \Bbb D$, $\lim\limits_{z\to z_0} F(z) = f(z_0)$.
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I'll give one more week for users to attempt a solution. Keep in mind, the definition of $F$ involves the Poisson kernel. It's key to use the fact that the Poisson kernel is an approximate identity.
 
No one answered this week's problem. You can read my solution below.
Note that $F(z)$ is the real part of the holomorphic function $$\frac{1}{2\pi}\int_{0}^{2\pi} f(e^{i\theta})\frac{e^{i\theta}-z}{e^{i\theta} + z}\, d\theta$$ whence $F$ is harmonic. Now fix $z_0 = e^{i\phi_0}\in \partial \Bbb D$. By continuity of $u$, given $\epsilon > 0$ there corresponds an $\eta > 0$ such that for all $\theta$, $\lvert \theta - \phi_0\rvert < \eta$ implies $\lvert f(e^{i\theta}) - f(e^{i\phi_0})\rvert < 0.5\epsilon$. Fix $\phi$ such that $\lvert \phi - \phi_0\rvert < 0.25\eta$, and note

$$F(re^{i\phi}) - f(re^{i\phi_0}) = \frac{1}{2\pi} \int_{-\pi}^\pi [f(re^{i\theta}) - f(re^{i\phi_0})]P_r(\phi-\theta)\, d\theta$$ where $P_r$ is the Poisson kernel. Break up the latter integral as $J_1 + J_2$, where $J_1, J_2$ are integrals over regions $\lvert \theta - \phi_0\rvert < \eta$ and $\lvert \theta - \phi_0\rvert \ge \eta$, respectively. Then $\lvert J_1 \rvert< 0.5\epsilon$ and $\lvert J_2 \rvert < 2\max_{\lvert \phi - \theta\rvert \ge 0.75\eta} P_r(\phi - \theta) \cdot \max_{-\pi \le \theta \le \pi} \lvert u(re^{i\theta})\rvert$. Since, $\lim_{r\to 1} \max_{\lvert \phi - \theta\rvert \ge 0.75\eta} P_r(\phi - \theta) = 0$, then $|J_2| < 0.5 \epsilon$ for all $r$ sufficiently close to $1$. Hence, $\lvert F(re^{i\phi}) - F(re^{i\phi_0})\rvert < 0.5 \epsilon + 0.5 \epsilon = \epsilon$ whenever $\lvert \phi - \phi_0\rvert < 0.25\eta$ and $r$ is sufficiently close to $1$. Consequently, $\lim_{z \to z_0} F(z) = f(z_0)$.
 
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