Operator Topologies and Strong Convergence

Find the radius of convergence of the Taylor series \[ \text {for the function } f(x)=\frac {1}{\cos ^2 x+4} \quad \text {around the origin } x_0=0. \] {You do not need to find the Taylor series itself.}

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be a domain. Let \(\Omega _1\) be a bounded subdomain with \(\overline {\Omega }_1\subset \Omega \).

1. State the Harnack Inequality for harmonic functions.
2. Let \(\{u_n\}\) be a monotone increasing sequence of nonnegative harmonic functions on \(\Omega \). Suppose that for some point \(\xi \in \Omega _1\), the sequence of real numbers \(\{u_n(\xi )\}\) is bounded. Prove that \(\{u_n\}\) converges uniformly on \(\Omega _1\) to a harmonic function. You may use standard theorems about convergence of sequences of harmonic functions without proof.

Let \(\Omega \subset \mathbb {R}\) be open, and let \(C_0^\infty (\Omega )\) be the space of \(C^\infty \) functions with support in \(\Omega \). Define convergence of a sequence \(\varphi _n\) as follows: \(\varphi _n\to 0\) iff \(\exists \) a compact set \(K\subset \Omega \) such that

1. Support \(\varphi _n\subset K\) for all \(n\)
2. \(\displaystyle \lim _{n\to \infty }\sup _{x\in K}\lvert \varphi _n^{(p)}(x)\rvert =0\) for \(p\in Z^+\).

1. Define what is meant by a distribution on \(C_0^\infty (\Omega )\).
2. Define \[ \mu _f(\varphi )=\int _{-\infty }^{\infty } f(x)\bar {\varphi }(x)\,dx \] where \(f\in L^1_{\mathrm {loc}}(\Omega )\) and \(\varphi \in C_0^\infty (\Omega )\). Does this define a distribution on \(C_0^\infty (\Omega )\)? Give reasons.
3. Give a concrete example of a regular distribution and verify that is is a distribution.
4. Show how to define the derivatives of a regular distribution.

State and prove the Classical Fourier pointwise convergence theorem for the representation of a function by its Fourier series.