Monthly Archives: October 2018

Lemma 1. Suppose $f$ is continuous and of moderate decrease, i.e., $|f(x)| <= {A \over (1 + x^2)}$. Then its periodization $g(x) = \sum_{k\in\Bbb Z} f(x + k)$ is continuous and periodic, and its partial sums converge uniformly to $g$. … Continue reading →

Theorem (Minkowski’s Inequality). If $f, g \in L^p$ for $1 \leq p < \infty$, $\|f + g\|_p \leq \|f\|_p + \|g\|_p$. Proof. Start by noticing that $|f + g|^p \leq 2^{p-1}(|f|^p + |g|^q)$ by Jensen’s inequality, so that $\|f + … Continue reading →

Theorem (Argument Principle). Suppose $f$ is meromorphic on an open region $R$ enclosed by a contour $C$. Then $\displaystyle \frac{1}{2\pi i} \int_C \frac{f'(z)}{f(z)} dz = N – M$, where $N$ is the number of zeroes (with multiplicity) enclosed by $C$ and … Continue reading →