Questions tagged [fourier-transform]

Question 1

I came across this problem when studying spectral multipliers concerning the Laplace operator on Euclidean spaces. Denote $L = - \Delta$ as the Laplace operator on $\mathbb{R}^n$. Given $F \in L^2_s(\...

Question 2

One proof of the statement that the image of Fourier transform $F: L^1(\mathbb R)\to L^{\infty} (\mathbb R)$ is contained in $C_0(\mathbb R)$ is as follows: First we use straight-forward computation ...

Question 3

Is there a way to define the Fourier transform of the hyperbolic cosine? $$ \cosh (x/2) $$ My only idea is to expand the exponential functions into taylor series and use the dirac delta derivatives

Question 4

In a review process, I was trying to work through the derivation of the inverse cosine transform from the forward cosine transform. However, I am shocked by how poorly my previous "understanding&...

Question 5

I'm trying to prove that $\mathcal{\underline{F}}(\tau_p \underline{f}) = \omega^{-p} \mathcal{\underline{F}}(\underline{f})$ version for the discrete Fourier Transform, but I'm getting stuck at the ...

Question 6

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 ...

Question 7

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}}...

Question 8

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 $C[A]$. ...

Question 9

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, $...

Question 10

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}...

Question 11

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\...

Question 12

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 . $...

Question 13

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 ...

Question 14

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 ...

Question 15

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}^{\...

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Questions tagged [fourier-transform]

L1 boundedness of some spectral multiplier

One doubt on the proof that the Fourier transform of a function is in $C_0(\mathbb R)$

Fourier transform of the hyperbolic cosine? [closed]

About the Derivation of Inverse Cosine Transform using Orthogonality of Cosine Functions on $[0,∞)$?

Discrete Fourier Transform Shift Theorem Help

Spectral measure associated with "translation by $z$" operator acting on $L^2(\mathbb{R})$

Inverse Fourier transform for the square root of the Ohmic bath spectral function

Fourier transform and the acoustic tensor

Computing Fourier transform for a real odd signal

Known properties of these generalized Cauchy distributions

Decomposition of a Schwartz function by Lacey and Thiele

How to show $\int_{\mathbb R^n} |Du|^2 dx = \int_{\mathbb R^n} |y^2| |\hat u|^2 dy$?

Is this “Elliptical Fourier Transform (R-Transform)” a real generalization or just nonsense?

Maximum domain of the Fourier Transform [closed]

Confused about distribution of ceiling function

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