= Solution
For fixed $r<1$, $\sum r^n\cos(n\theta)$ converges absolutely and uniformly. Thus for integrable $f$, one may integrate the kernel <series> term by term. The <cosine addition formula> gives
$$
\cos[n(\theta-t)]=\cos(n\theta)\cos(nt)+\sin(n\theta)\sin(nt).
$$
Substitute this into the defining <integral>. The constant kernel term gives $\tfrac1{2\pi}\int f=a_0/2$; the remaining terms give
$$
\boxed{y_f(r,\theta)=\frac{a_0}{2}+\sum_{n=1}^\infty r^n[a_n\cos(n\theta)+b_n\sin(n\theta)],}
$$
where
$$
\boxed{a_n=\frac1\pi\int_{-\pi}^{\pi}f(t)\cos(nt)\,dt,\qquad
b_n=\frac1\pi\int_{-\pi}^{\pi}f(t)\sin(nt)\,dt.}
$$
These are the same <Fourier coefficients> as in part (a). The kernel $2K$ is the <Poisson kernel for the unit disk>; the factors $r^n$ suppress the higher angular frequencies inside the <disk>.
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