Articles of analysis

Approximation by smooth function while preserving the zero set

Let $\mathbb{T}$ denote the unit circle. Given $f \in \mathcal{C}(\mathbb{T})$, can we approximate $f$ by smooth functions having the same zero set ? i.e. for $\varepsilon >0$, can we find $g \in \mathcal{C}^\infty(\mathbb{T})$ such that f(x) = 0 if and only if $g(x) = 0$; $\|f-g\|_\infty < \varepsilon$. Both tasks can easily be performed separately. […]

Solving a challenging differential equation

How would one go about finding a closed form analytic solution to the following differential equation? $$\frac{d^2y}{dx^2} +(x^4 +x^2+x+c)y(x) =0 $$ where $c\in\mathbb{R}$

A limit about $a_1=1,a_{n+1}=a_n+$

Let the sequence $\{a_n\}$ satisfy $$a_1=1,a_{n+1}=a_n+[\sqrt{a_n}]\quad(n\geq1),$$ where $[x]$ is the integer part of $x$. Find the limit $$\lim\limits_{n\to\infty}\frac{a_n}{n^2}$$. Add: By the Stolz formula, we have \begin{align*} &\mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{n^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{a_{n + 1}} – {a_n}}}{{2n + 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{\left[ […]

Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$

Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the sets $S_1$ and $S_2$ given that $S_1 \subset S_2$? Intuitively I can see that the the fact that $S_1$ and $S_2$ lie in $[-M,M]^n$ […]

Prob. 2, Chap. 6, in Baby Rudin: If $f\geq 0$ and continuous on $$ with $\int_a^bf(x)\ \mathrm{d}x=0$, then $f=0$

Here is Prob. 2, Chap. 6, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f \geq 0$, $f$ is continuous on $[a, b]$, and $\int_a^b f(x) \ \mathrm{d} x = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$. My Attempt: Let $c$ and $d$ be any […]

Do such sequences exist?

I wish to know if there are real sequences $(a_k)$, $(b_k)$ (and if there are, how to construct such sequences) such that: $b_k<0$ for each $k \in \mathbb{N}$ with $\lim\limits_{k \rightarrow \infty} b_k=-\infty,$ $$\sum_{k=1}^\infty |a_k| |b_k|^n< \infty, \space\forall n\in\mathbb{N}\cup\{0\}$$ $$\sum_{k=1}^\infty a_k b_k^n=1, \space \forall n\in\mathbb{N}\cup\{0\}$$

Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz’ lemma and set $\|\pi\| = 1$, that’s as far as I got. I could also use the fact […]

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard analysis result, but cannot remember how to prove it. Thanks!

How to solve this derivative of f proof?

A function $f$ satisfies: $$f”(x) + f'(x)g(x) – f(x) = 0$$ for some function $g$. Prove that if $f$ is $0$ at two points, then $f$ is $0$ on the interval between them. Can someone verify my proof? Scratchwork: So let $I = [a, b]$ and $f(a) = f(b) = 0$. $g(x)$ is some function, […]

Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f$ is closed.

If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, then $f[A]$ is also closed. I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I’m not sure that the converse is true, I can’t find a counterexample either.