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Tagged with fourier-transform schwartz-distributions
11 questions with no upvoted or accepted answers
6
votes
0
answers
159
views
Fourier transformation of a distribution
We have no idea how to tackle the following Fourier transformation of a distribution:
$$
\lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
- 373
6
votes
0
answers
203
views
Uniform estimates of Fourier transform of tempered functions with parameters
Consider the following function in $\mathbb{R}^3$:
$$
f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2},
$$
where $\frac{1}{2}<\alpha<1$, and ...
- 879
3
votes
0
answers
261
views
Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions
Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set
\begin{equation*}
{\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1)
\end{...
- 31
3
votes
0
answers
211
views
A question about Fourier transform of function of the type $Q(x)(1+P(x))^{z}$
For simplicity, consider in $\mathbb{R}^3$, and the Fourier transform of the following function
$$f=(x_1+x_2+x_3)(1+|x|^2+x_1^2(x_2^2+x_3^2)+x_2^2x_3^2)^{-t+is},~~ \frac12<t<1,~~s\in \mathbb{R}.$...
- 879
2
votes
0
answers
191
views
Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$
Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$.
For a fixed Littlewood-Paley decomposition $\chi \in \...
- 233
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
214
views
Bounds on functions pullbacked via exponential map
Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* ...
- 9,853
1
vote
0
answers
127
views
Does convergence of tempered distributions implies convergence in $\mathcal{S}(\mathbb{R}^4,\mathbb{R})/\mathcal{S}_{0}$?
We can define the following symmetric semi-definite positive bi-linear form on
$\mathcal{S}(\mathbb{R}^{4},\mathbb{R})$ with values in $\mathbb{C}$,
\begin{equation}\label{prodintespaciales}
(h_{...
- 424
1
vote
0
answers
148
views
Fourier inversion formula for compactly supported distributions
I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies
$$
|\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1}
$$
...
- 424
1
vote
0
answers
91
views
Support of functions on Minkowski space and their Fourier transform
Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x^2<0$ and its Fourier transform $\hat f(p)$ vanishes for $p^2<0$.
Are then $f$ ...
- 3,873
0
votes
0
answers
23
views
Is there a classification of 2D projective convolution kernels?
Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...
