All Questions
Tagged with fourier-transform ap.analysis-of-pdes
16 questions with no upvoted or accepted answers
5
votes
0
answers
304
views
Similarity in Navier-Stokes equation and convolution in finite abelian groups?
Let $G$ be a finite abelian group, $X = (x_g)_{g \in G}$ be a vector of variables.
Set for $g \in G$:
$$\tau_g(X) := \frac{1}{|G|} \sum_{\rho \text{ irred. }} \chi_{\rho}(-g) \exp(\sum_{s \in G} \chi_{...
- 3,064
4
votes
0
answers
146
views
Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
- 1,755
3
votes
0
answers
272
views
A generalization of Weierstrass transform
As stated in this article, the Weierstrass transform of $f(x)$ is defined as:
\begin{equation}
W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy
\end{equation}
which can be ...
- 350
2
votes
0
answers
144
views
Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...
- 1,101
2
votes
0
answers
149
views
An oscillatory integral
Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates?
\begin{...
- 351
2
votes
0
answers
95
views
Fourier Transform ; half space elliptic baby problem
I am attempting to look at some Liouville type theorems via a Fourier analysis approach and
after looking at a baby problem I seem to be very confused. I assume this doesn't count as a research ...
- 1,385
2
votes
0
answers
480
views
A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
- 356
1
vote
0
answers
55
views
Comparison principle for porous medium equation in Fourier variables
Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
- 1,101
1
vote
0
answers
108
views
Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
- 151
1
vote
0
answers
180
views
A potential wrong proof of a Lemma
Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
- 159
1
vote
0
answers
282
views
Fourier Transform; half space baby problem (new)
This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem.
Essentially I am asking the same question now but taking a lot more care.
So lets ...
- 1,385
1
vote
0
answers
74
views
Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 339
1
vote
0
answers
353
views
Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
vote
0
answers
440
views
A problem about Joint sine and cosine fourier transform
There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
- 11
0
votes
0
answers
75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
0
votes
0
answers
223
views
Solvable PDEs and their Green's functions
I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
- 1,687