My advisor asked me to describe the spectral properties of the operator $T_z: L^2(\mathbb{R})\longrightarrow L^2(\mathbb{R})$ that is defined by $T_z(f)(x) = f(x+z)$.It was fairly direct to verify ...

I think this is a bit hopeless but let me ask just in case. Consider the real and positive function:$$\hat{f}(\omega) = \sqrt{\frac{\omega}{1-e^{-\frac{\omega}{T}}}} e^{- \frac{\omega^2}{4\Lambda^2}}...

Let $C$ the fourth-order tensor of elastic constants which can be seen as as a linear transformation from Sym into Sym (matrices). We denote the action of $C$ on a symmetric matrix $A$ as $C[A]$. ...

Given a real and odd signal $x(t)$, such that $\vert X(\omega)\vert = e^{-\vert \omega\vert}$ is the magnitude of Fourier transform.Question: Find the Fourier Transform $X(w)$.My attempt:We know, $...

Consider the following family of normalized probability densities parametrized by the strictly positive integer $k$:$$\begin{align}\begin{aligned}&f_k(x) = \frac{k}{\pi}\sin\left(\frac{\pi}{2k}...

Suppose we have a Schwartz function $\varphi\colon \mathbb{R}\to \mathbb{R}$ supported in (0,1) such that $||\varphi||_{L^2}\leq 1$ satisfying that for all $\xi\in \mathbb{R}$,\begin{align}\sum_{l\...

Pictures below is from Evans' PDE, I want to calculate the red line. $\hat u$ is the Fourier transform of $u$, namely$$\hat u(y) = \frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n} e^{-i x\cdot y} u(x) dx .$...

I’m trying to sanity-check an idea I came up with about 10 years ago during my engineering undergrad.Back then, I noticed the standard Fourier kernel $\boldsymbol{e^{i\omega t}}$ traces a circle in ...

There are many functions that can be transformed by the Fourier Transform, as the Schwartz functions, the $L^2$ functions, and also less behaved [generalized] functions as the Dirac's delta function ...

Let $x \in \mathbb{R} \setminus \mathbb{Z}$. Then $x \in (k, k+1)$ for exactly one integer $k$, and$\lfloor x \rfloor = k, \ \lceil x \rceil = k + 1.$$$\mathrm{III}_p(x) := \sum_{k=-\infty}^{\...

I am trying to find a solution to the following integral:$$I = \int^{r_0}_{-r_0} \bigg [ \int^{r_0}_x \frac{dm(r(\tau))}{dr} \frac{1}{r(\tau)} d\tau \bigg ] m(r(x)) dx$$where:$$ m(r) = B [1-(r/r_0)^...

Im a furloughed federal employee just passing time. Below I describe how to replicate the findings for Invariant Patterns in the Geometric Analysis of Primes. I have code and a longer write up, but ...

Question: Is there a (simpler) closed form of $\color{blue}{C(t) = \operatorname{Ei}(-t) \theta(t) \star \operatorname{Ei}(t) \theta(-t)}$?Definitions:Exponential Integral:$$\operatorname{Ei}(x) = \...

A well known positive definite function is $f:x \mapsto e^{-a\sin(x)^2}$, with $a>0$.My question is, what does the inverse Fourier transform of this function look like? According to Bochner's ...

I came across the following integral $$I=\int_0^\infty e^{-a^2x-\frac{b^2}{x}}x^{-\frac{1}{2}}dx$$ for real parameters $a>0$ and $b\geq0$. My notes say that the solution is $$I=\frac{\sqrt{\pi}}{a}...