All Questions
Tagged with fourier-transform cv.complex-variables
26 questions
1
vote
1
answer
190
views
Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
- 1,352
0
votes
0
answers
46
views
Function samplable from the past and Hardy spaces
What I am ultimately looking for is a $L^2$ function $f$ on the real line that can be sampled from the past, i.e. for each $x<0$ there are $L^2$ coefficients $c_n(x)$, $n\in \mathbb{N}$ such that, ...
- 1,352
2
votes
0
answers
116
views
Construction of an analytic function whose Fourier transformation has compact support [closed]
Is there a non-constant real analytic function $f$ on $\mathbb{R^2}$ satisfying the following properties?
$f$ vanishes on $x$-axis and $y$-axis;
the Fourier transformation $\hat{f}$ of $f$ has a ...
- 67
4
votes
2
answers
193
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Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
Consider the function $$\phi_{a,b}(t)=\frac{|t|}{e^{a|t|}-e^{-b|t|}}, \ \ t\in\mathbb{R},$$ where $0<a<b$. Can $\phi_{a,b}$ be the Fourier transform of a positive function for some $a<b$?
- 407
1
vote
1
answer
116
views
Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
- 424
1
vote
0
answers
43
views
Looking at a frequency reassignment rule as a Möbius transform
Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed.
I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
- 111
0
votes
0
answers
113
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Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$
$$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$
where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
- 151
2
votes
1
answer
203
views
Existence of the special entire Hardy space function with infinitely many zeros in the strip
Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...
- 577
0
votes
0
answers
62
views
To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
6
votes
2
answers
336
views
On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
- 4,143
0
votes
0
answers
102
views
How to construct non-abelian functions?
I have found some functions $t_g, g \in G$ for cyclic groups $G=C_n$ which seem to satisfy the following convolution identity:
$$t_g(x+y) = \sum_{h \in G} t_{gh^{-1}}(x) t_h(y)$$
Example of such ...
- 3,064
8
votes
1
answer
640
views
Rate of decrease of the Fourier transform of standard mollifiers
What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$,
$$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$
and
$$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
- 128k
1
vote
0
answers
38
views
Solving an equation containing Laplace transform
Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...
- 47
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$
where $y\in(0,\infty)$, $u\in(0,1)$, and
$$f(t):=t+\pi (1-t) t \cot (\pi t).$$
Here are the graphs of $f$ (black), ...
- 128k
1
vote
2
answers
2k
views
Fourier transform of a holomorphic function
Question: Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?
In order for my question to be well-...
- 547
0
votes
1
answer
344
views
Variance of spectral density is related to the gradient of signal?
Define the frequency variance as:
$$ \sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$
Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore,
$...
- 433
1
vote
0
answers
140
views
Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
- 435
2
votes
0
answers
74
views
Particular Ehrenpreis factorization for covariance function
Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite).
Is there a compactly supported function $g:\mathbb{R}^d\...
- 321
1
vote
0
answers
107
views
Comparison of two Fourier transforms
I am looking for $\delta>0$, such that
$$
\delta \int_{-\infty}^{\infty} \exp(its)
{ \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\
\int_{-\infty}^{\infty} \exp(its)
{ \Gamma (it+1)\over \...
- 93
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 ...
- 335
5
votes
4
answers
952
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
1
vote
0
answers
148
views
Fourier inversion formula for compactly supported distributions
I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies
$$
|\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1}
$$
...
- 424
3
votes
1
answer
244
views
How to compute $\int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega)$
I would like compute the following
$$I_{t,x,y} = \int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega); $$
where $\mathbb S^2$ is the two-...
- 650
6
votes
1
answer
752
views
Roots of characteristic function of "reciprocal gamma measure"
Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
- 133
2
votes
1
answer
850
views
Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)
Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right|...
0
votes
1
answer
410
views
About vertex algebra, mode expansion
A vertex operator is a linear map associating every state to a operator-valued distributions (quantum field) on a algebra curve, which is also called operator-state correspondence.
Chose a local ...
- 787