Articles of analysis

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. and a discrete Fourier transformation with discrete $n$ and discrete $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta_{x_1-x_2,0} $$ with Kronecker delta function $\delta$. Question: what is the discrete Fourier transformation with discrete $n$ and discrete $x_1,x_2$ below: […]

Prove partial derivatives exist, but not all directional derivatives exists.

During my analysis course my teacher explained the difference between partial derivatives and directional derivatives using the notion that a partial derivatives looks at the function as approaching a point along the axes (in case of of the plane), and a directional derivative as approaching a point from any direction in the plane. He also […]

Theorem 3.55 Rudin (rearrangement and convergence)

If $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement of $\sum a_n$ converges, and they all converge to the same sum. Proof: Let $\sum a_n’$ be a rearrangement , with partial sums $s_n’$. Given $\epsilon > 0$ there exist an integer $N$ such that $m \geq m \geq N$ […]

Derivative of $f(x) \cdot g(x)^{(n)} = \sum_{k = 0}^{n}(-1)^k \cdot {n \choose k}\cdot (f(x)^{(k)}\cdot g(x))^{(n-k)}$

can someone please help me with this rule: $$ f(x) \cdot g(x)^{(n)} = \sum_{k=0}^{n}(-1)^k \cdot {n \choose k}\cdot (f(x)^{(k)}\cdot g(x))^{(n-k)} $$ or you can write: $$ f(x) \cdot \frac{d^n g(x)}{dx^n} = \sum_{k=0}^{n}(-1)^k \cdot {n \choose k}\frac{d^{n-k}}{dx^{n-k}}(f(x)^{(k)}\cdot g(x)) $$ Where the powers with $()$ are derivations of the function. Where does it come from and how […]

How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then $\exists~\delta>0$ such that $|u(x,y)-l|<\epsilon~\forall~(x,y)\in (B(z_0,\delta)-\{(x_0,y_0\})\cap D$ ($D$ being the domain of $u$). Of course the domain of $u(x,y_0)$ is $D_1=\{x:(x,y_0)\in D\}$ Thus for all $x\in((x_0-\delta,x_0+\delta)-\{x_0\})\cap D_1, |u(x,y_0)-l|<\epsilon.$ This far is […]

A sequentially compact subset of $\Bbb R^n$ is closed and bounded

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this claim about your sequence. Let $U$ be a subset of $\mathbb{R}^n$ such that $U$ is not closed. Construct a […]

Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?

I’m trying to show that $f(x)$ is of bounded variation where $f(x)=x^{3/2}\sin(\frac{1}{x})$ on $[0,1]$. I think that it is but I can’t show it explicitly. Any help will be appreciated.

Possible matrix-determinant identity

Is $$\det(I+ ABB^*A^*C^{-1})=\det(I+ B^*A^*ABC^{-1})$$ where $I$ is identity matrix, $A,B,C$ are complex valued matrices. And $C$ is $(I+X)$ where $X$ is PSD. I know that this makes $ABB^*A^*$ and $B^*A^*AB$ positive semi/definite. And I know for square matrices $X,Y,Z$ we have $\det(XYZ)=\det(YXZ)$ so $\det (ABB^*A^*C^{-1})=\det(B^*A^*ABC^{-1})$. And also $\det (I + ABB^*A^*C^{-1})=\det(I + B^*A^*ABC^{-1})$. Please comment […]

Find the limit of a recursive sequence

Let $(u_n)_n$ be a real sequence such that $$ u_{n+2}=\sqrt{u_{n+1}}+\sqrt{u_{n}},\,u_0>0,\,u_1>0. $$ Fisrt, it is easy to check that $(u_n)_n$ is well defined and $u_n>0$ for all $n\in\mathbb{N}$. The question now is show that $$ \exists p\in \mathbb{N}\,;\,\forall n\in\mathbb{N},\,n\geq p\implies u_n>1. $$ From this, we can deduce the limit of the sequence $(u_n)_n$.

Proving the last part of Nested interval property implying Axiom of completeness

I took a non-empty set A that is bounded above. And I went on with the regular algorithm, which either gave us a LUB or gave us an infinite chain of nested intervals $I_1$ $\supseteq$ $I_2$ $\supseteq$ $I_3$ $\dots$ Now I used the nested interval property to say that there is at least a member […]