Category Archives: Uncategorized

Theorem. For $p\in [1, \infty)$, $L^p$ is complete. Proof. Pick a Cauchy sequence $f_n$. We can pick a subsequence $f_{n_k}$ such that $\|f_{n_{k+1}}- f_{n_k}\| \leq 2^{-k}$. Set $S_K(f) = f_{n_1} + \sum_{i=1}^{K} (f_{n_{i+1}} – f_{n_i})$ and also $S_K(g) = |f_{n_1}| … Continue reading →

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 →

Statement. $F[e^{-\pi x^2}] = e^{-\pi x^2}$ Proof. $$ \begin{eqnarray} F[e^{-\pi x^2}](s) &=& \int_{\Bbb R} e^{-\pi x^2} e^{-2\pi i sx} dx \\ &=& \int_{\Bbb R} e^{-\pi x^2 -2\pi i sx} dx \\ &=& e^{\pi (is)^2} \int_{\Bbb R} e^{-\pi x^2 -2\pi i sx – … Continue reading →

Given a periodic $f$, we know we can approximate it in $L^2$ with trigonometric polynomials. In particular, $S_n$ converges to $f$ in $L^2$. Now, $\langle \cdot, c \rangle$ is a linear operator, and by the Cauchy-Schwarz inequality, it’s bounded, hence … Continue reading →

A sequence of kernels $K_n(x)$ on $[-\pi, \pi]$ is called *good* if $\frac{1}{2\pi} \int_{-\pi}^\pi K_n(x) \, dx = 1$, $\frac{1}{2\pi} \int_{-\pi}^\pi |K_n(x)| \, dx$ is bounded, and $\frac{1}{2\pi} \int_{|x| \geq c} |K_n(x)| \, dx \to 0$ as $n \to \infty$ … Continue reading →

Theorem. Suppose $\lim_{|z|\to\infty} |f(z)| = 0$ and $a > 0$. Then $\lim_{R\to \infty} \int_{C_R} e^{iaz} f(z) \, dz = 0$ where $C_R$ is the semicircle in the upper half-plane with radius $R$ centered on $0$. Proof. Our assumption implies that … Continue reading →

Let $\hat f(n) = \frac{1}{2\pi} \int_{-\pi}^\pi f(t)e^{-int} \, \text{d}t$ be the Fourier coefficients of $f$. Theorem. $\hat f(n) = 0$ implies $f(t) = 0$ when $f$ is continuous at $t$. Proof. By linearity, this is equivalent to showing that $\hat … Continue reading →