Articles of analysis

Uniform continuity and translation invariance

Consider the function $a(s)=\dfrac{1}{1+s^2}$ and the space $X=\{f:\mathbb{R}\to \mathbb{R}$ such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous$ \}$. I want to show that $X$ is translation invariant, i.e. if $f\in X$ then $f_t\in X$ for all $t\in \mathbb{R}$, where $f_t$ is the $t$-translation of $f$ defined by $f_t(s)=f(t+s)$. I showed that if the function $s\mapsto […]

Prob. 4, Chap. 4 in Baby Rudin: A continuous image of a dense subset is dense in the range.

Here is Prob. 4, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$. If $g(p) = f(p)$ […]

Fourier series simplification

I want to show that $$\frac{1}{\pi} \int_{-\pi}^{\pi} f(x)g(x)dx = \frac{a_0\alpha_0}{2} + \sum_{n=1}^{\infty} (a_n\alpha_n + b_n\beta_n)$$ where $f,g: [-\pi,\pi] \to \mathbb{R}$ are integral functions and that the Fourier series of $f$ and $g$ are uniformly convergent to $f,g$ respectively. $a_0,a_n,b_n,\alpha_0,\alpha_n,\beta_n$ are the fourier coefficients of $f,g$ respectively. I thought we could simply expand the LHS, simplify, […]

Compute the limit of $\int_\mathbb{R} f(x)\sin (nx)$ when $n\to\infty$, for $f \in L^1$

Let $f \in L^1(\mathbb{R})$. Find $$ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty f(x)\sin(nx) dx \,. $$ LDCT is a no go, as well as MCT and FL, which are really the only integration techniques we’ve developed thus far in the course. Integration by parts isn’t applicable either since we just have $f \in L^1$. I’ve tried splitting […]

Uniqueness of Bounded Solutions to Dirichlet's Problem in the Half-Space

Title basically says everything. Prove that if $u\in C^{2}(\mathbb{R}^{n}_{+})\cap C(\bar{\mathbb{R}^{n}_{+}})$ is a bounded solution of the BVP $$\left\{\begin{array} -\Delta u=0&\text{in}\;\mathbb{R}^{n}_{+}\\ u=g&\;\text{on}\;\partial\mathbb{R}^{n}_{+}, \end{array}\right.$$ then it is unique. Various tools I have in mind are maximum principle, mean value formulas, Liouville’s theorem, “energy” functionals, and Harnack’s inequality, uniqueness of Green’s function, Hopf’s lemma, etc….but in all my […]

An inequality of integrals

Let $f\in C^1([a,b])$ with $f(a)=0$. How can I show that there exists a positive constant $M$ independent of $f$ such that $\int^b_a|f(x)|^2dx\leq M\int^b_a|f^\prime(x)|^2dx$?

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive harmonic function $u(z)$ in $\Omega$ satisfies $u(z_2) \leq M u(z_1)$ for any two points $z_1, z_2 \in E$. This […]

Is this map uniformly continuous? continuous?

Let $s$ denote the metric space of all sequences of complex numbers with the metric $$d(x,y) \colon= \sum_{j=1}^\infty \frac{1}{2^j} \frac{\vert \xi_j(x) – \xi_j(y)\vert}{1+\vert \xi_j(x) – \xi_j(y) \vert} $$ for all $x, y \in s$. Here $\xi_j(x)$ denotes the $j$-th term of $x$. Let $k$ be a fixed natural number. Then is the map $T_k \colon […]

Weierstrass product form

How to show the Weierstrass product form of the entire function $f(z)= \sinh z$ This question seem so interesting. I would like to write my some ideas, but I dont want to direct incorrectly. Please help me to learn correctly and explicitly. I asked to question in fact in order to learn the topic precisely.

Determine best possible Lipschitz constant

I’m slightly confused by a homework problem here…I’ve been given the function: $ f(u) = log(u) $ With the bounds: $ 2 \leq u \lt \infty $ Now I thought I understood what the Lipschitz Condition required, but that was a function of two variables, but here I only have $u$. Believe it or not, […]