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23 votes
6 answers
4k views

Anti-delta function?

Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property: its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
Anixx's user avatar
  • 10.1k
18 votes
3 answers
7k views

Eigenvectors of the Fourier transformation

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$ by $ \hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx. $ It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the ...
Bazin's user avatar
  • 16.2k
11 votes
2 answers
8k views

About the Fourier transform of the logarithm function

I want to calculate / simplify: $$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$ where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
Bertrand's user avatar
  • 1,199
10 votes
1 answer
658 views

Are functions of moderate growth a bornological space?

I was thinking a bit about distribution theory the last weeks and stumbled across the following question: There are two natural locally convex topologies on the space of smooth functions of moderate ...
Johannes Hahn's user avatar
9 votes
1 answer
4k views

Fourier transform of a bounded function

This should really be well-known, but I was not able to find a definite answer to this question: Is the Fourier transform of a bounded function always a borel measure (i.e. an order 0 distribution)? ...
Matthias Ludewig's user avatar
7 votes
2 answers
469 views

Eigenstates of Fourier transformation

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by $$ (\mathcal F u)(\xi)=\int e^{-2iπ x\...
Bazin's user avatar
  • 16.2k
6 votes
2 answers
458 views

Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?

Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
J. Swail's user avatar
  • 437
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^{-\...
Y.Okuyama's user avatar
  • 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 ...
Tomas's user avatar
  • 879
5 votes
3 answers
2k views

Fourier transform of periodic distributions

Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
spaceman's user avatar
  • 595
4 votes
1 answer
466 views

Fourier transform of tempered distribution

I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t_+^{1/2}$? (Here $t_+=\max (t,0)$.)
flavio's user avatar
  • 450
4 votes
2 answers
405 views

Fourier transform of a Lorentz invariant generalized function

Consider on $\mathbb{R}^{n+1}$ the indefinite quadratic form defining the Minkowski metric $$B(p)=(p^0)^2-(p^1)^2-\dots-(p^n)^2.$$ Let $\mu$ be a generalized function on $\mathbb{R}^{n+1}$ which is ...
asv's user avatar
  • 21.8k
4 votes
1 answer
255 views

Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?

Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as $$ H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
Adrien Wohrer's user avatar
4 votes
1 answer
199 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
Dima's user avatar
  • 335
3 votes
2 answers
1k views

Fourier transform inversion theorem for a function not in L1 or L2

For $\frac{1}{4}<a<1$ consider the following function: $$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$ If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be ...
Bertrand's user avatar
  • 1,199
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{...
Alex Pereira's user avatar
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}.$...
Tomas's user avatar
  • 879
2 votes
1 answer
190 views

Half Poisson summation

Suppose $f$ is a Schwartz function on $\mathbb{R}$. Is there a closed formula for $$\sum_0^\infty \hat{f}(n)$$ where $\hat{f}$ is the $n$-th Fourier coefficient of $f$?
Qijun Tan's user avatar
  • 587
2 votes
2 answers
539 views

Fourier transform of $e^{it|\xi|^{\alpha}}$

Consider the fourier transform of $e^{it|\xi|^{2\alpha}}$ ($\alpha>0$)in $\mathbb{R}^n$,let $K_{\alpha}=\mathcal{F}(e^{it|\xi|^{\alpha}})$,so $K$ is a tempered distribution.Now I want to know if ...
user23078's user avatar
  • 1,644
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 \...
Desura's user avatar
  • 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 ...
Math604's user avatar
  • 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^* ...
Matthias Ludewig's user avatar
1 vote
1 answer
1k views

Fourier transform of delta function restricted to sphere [duplicate]

I want to compute $\mathcal{F}^{-1}\{\delta(|\cdot|-1)\}(x)$, which exactly means the following computation: $$f(x) = (2\pi)^{-n/2} \int_{|\xi|=1}e^{ix\cdot\xi}\mathrm{d}\xi, \mbox{ where }~ \xi \in \...
henrysupercool's user avatar
1 vote
1 answer
385 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
Hans's user avatar
  • 2,239
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_{...
Gabriel Palau's user avatar
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 $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...
Gabriel Palau's user avatar
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 $f$ ...
Arnold Neumaier's user avatar
0 votes
2 answers
227 views

Does this distribution exist?

Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...
Nicolas Medina Sanchez's user avatar
0 votes
1 answer
127 views

Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?

Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere. Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$. $\hat{f}$ is the Fourier transform fora function f.
Edward's user avatar
  • 9
0 votes
1 answer
334 views

Fourier transform of a Radon measure [closed]

Let $\mu$ be a Radon measure on $\mathbb R^d$ with finite total mass: I guess that it is a tempered distribution on $\mathbb R^d$ and thus one may consider its Fourier transform. Now I guess that its ...
Bazin's user avatar
  • 16.2k
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\...
Nicolas Medina Sanchez's user avatar
-4 votes
1 answer
370 views

Is delta function symmetric against real axis? [closed]

Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$? I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis. We can write Delta function as $$\delta(z) = \...
Anixx's user avatar
  • 10.1k