Questions tagged [fourier-transform]
The fourier-transform tag has no usage guidance.
516 questions
4
votes
0
answers
226
views
Any references on infinite-dimensional Fourier-Plancherel theory?
Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
- 4,010
4
votes
0
answers
349
views
Is the Fourier transform of $\frac{1}{\mu+|\xi|^{2\alpha}}$($\mu>0$) a bounded function?
Consider $m(\xi)=\frac{1}{\mu+|\xi|^{2\alpha}}$, where $\xi\in\mathbb{R}^n$, $\mu, \alpha>0$, I want to know that if $m(\xi)$ is a multiplier of $\mathcal{M_{1}^{\infty}}$,i.e., whether the ...
- 879
4
votes
0
answers
980
views
Inverse Fourier Transform involving a Bessel Function, Exponential, and Power
I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...
- 41
4
votes
3
answers
433
views
Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$
I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
- 81
3
votes
2
answers
590
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 4,058
3
votes
2
answers
1k
views
Behavior of the Fourier transform (FT) of a function and FT of its absolute function
Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}.$ We define the Fourier transform of $f$ as follows:
$$...
- 1,051
3
votes
2
answers
487
views
Where to find a table of fair Fourier transforms? [closed]
I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.
For ...
- 10.1k
3
votes
2
answers
869
views
How do functions operate in a Sobolev space $H^{s}$?
Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ ...
- 1,051
3
votes
1
answer
1k
views
Decay of the Fourier transform of a function with a discontinuty at zero
Let $f \colon \mathbb R^2 \to \mathbb R^2$ be a function from the Schwartz class and $f(0) \neq 0$. Define it's projection $g(x) = \langle f(x), \frac{x}{|x|} \rangle \frac{x}{|x|}$, where $\langle a, ...
- 1,329
3
votes
1
answer
244
views
How to compute $\int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega)$
I would like compute the following
$$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); $$
where $\mathbb S^2$ is the two-...
- 650
3
votes
2
answers
1k
views
Are the zeroes of the Fourier Transform of compactly supported functions isolated?
I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...
- 2,071
3
votes
1
answer
210
views
Minimal number of operations of a discrete Fourier transform
Is there somewhere a table of the smallest known number $F(n)$ of (infinite precision) operations on complex numbers needed to compute a discrete Fourier transform on vectors of a given length $n$, ...
- 3,873
3
votes
1
answer
328
views
Large Fourier submatrices with small operator norm
Consider a finite abelian group $G$ (I'm mostly interested in $\mathbb{Z}_2^n$).
For two subsets $A$ and $B$ of $G$, one can form a submatrix of the Fourier transform matrix on $G$ by keeping only ...
- 2,772
3
votes
2
answers
1k
views
Composition of Riesz potentials
For $0<\alpha<n$ and $n\geq 2$ we define the Riesz potential by
$$
(I_\alpha f)(x) = \frac{1}{\gamma(\alpha)}
\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, ,
\quad
\text{where}
\quad
\...
- 28k
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 ...
- 1,199
3
votes
1
answer
535
views
Fourier transform of the unit ball in L1 metric
The Fourier transform of the (indicator function of the) unit ball is well known to be given by the Bessel functions (see Fourier transform of the unit sphere).
What can be said about the $\ell_1$ ...
- 1,324
3
votes
1
answer
304
views
Existence of probability measure on the circle with given Fourier coefficients
We say that a Hermitian symmetric (i.e., $f_{-n} = f_n^*$ for any $n \in \mathbb{Z})$ sequence $(f_n)_{n\in \mathbb{Z}}$ is positive-definite if, for any $N \geq 0$ and any $z_0 , \ldots, z_N \in \...
- 2,306
3
votes
1
answer
763
views
2D Fourier transform of log function
I am studying the paper found here. Halfway in the paper (Equation 6), the inverse 2D Fourier transform of $1/(k_x^2+k_y^2)$ needs to be determined. Is is stated that this is straightforward, and that ...
- 230
3
votes
1
answer
319
views
Origin of the theorem related to the integral transform pair
The development of Fast Fourier transform is attributed to Cooley & Tukey, both have written a lot about it is historical development. Both Cooley and Tukey call it a re-discovery rather. However,...
- 879
3
votes
1
answer
404
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
- 178
3
votes
2
answers
196
views
Inverse Fourier of $\omega^{-1+{\rm i}\alpha} u(\omega-1)$
Let $\alpha$ be an arbitrary real number and define
\begin{align}
\widehat{f}(\omega)=\left\{\begin{array}{ll}
\omega^{-1+{\rm i}\alpha}, & \omega>1,\\
0, & \textrm{otherwise}.
\end{array}
\...
- 31
3
votes
1
answer
181
views
The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$
Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
- 4,058
3
votes
1
answer
187
views
General solution to a n-dimensional partial differential equation
$$
\begin{split}
\frac{\partial}{\partial t}P(x, t)& =\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,\bigg(c_i\frac{\partial P}{\partial x_i} - c_j\frac{\partial P}{\partial x_j}\...
- 131
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$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).$$
Here are the graphs of $f$ (black), ...
- 128k
3
votes
1
answer
305
views
What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?
I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
...
- 41.9k
3
votes
1
answer
2k
views
Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
- 879
3
votes
1
answer
473
views
Expected value of the maximum of the periodogram
Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...
- 423
3
votes
1
answer
2k
views
About Fourier transforms of piecewise linear functions. [closed]
Consider a function $f$ which is $0$ for $x< 1$ and is say $x-1$ for $x >1$.
Consider a function $g$ which is $0$ for $x <2$ and is say $x -2$ for $x>2$.
Now using some kind of ...
- 2,246
3
votes
1
answer
518
views
Connection between the Fourier transform of f and |f|
If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
\|\widehat{f}\|_{L^{p'}}\...
- 425
3
votes
3
answers
171
views
tranforms that lowers the number of variables of a function
Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...
- 183
3
votes
1
answer
943
views
Solving Stokes Equations using 3D Fourier transforms
How do you calculate the inverse Fourier transform of $\frac{k_ik_j}{k^4}$. I know it has to be a matrix of the form $=δ_{ij}A(r)+r_ir_jB(r)$, but how do you calculate the functions A(r) and B(r)?
I ...
- 31
3
votes
2
answers
413
views
A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
- 351
3
votes
1
answer
423
views
Is there (fast) fourier transform for vector convolution?
Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined
$$U*V(t)={\sum_{i}} u_iv_{t-i}.$$
Given a list of vectors $u_1,\dots,u_m\in\...
- 13.9k
3
votes
1
answer
243
views
Image of Fourier transform for finite non-abelian groups
I am working on the Fourier transform over finite non-abelian groups, specifically following Diaconis. He defines it as follows (p.7):
Let $P$ be a probability on a finite group $G$. The Fourier ...
- 33
3
votes
1
answer
164
views
Recovering information for $\sum_{n \leq x}a(n)$ from $\sum_{n \geq 1}a(n)e^{-nx}$
I am wondering if I could deduce the bound for the partial sums
\[
\sum_{n \leq x}a(n) \ll x^{A}, \quad x \to \infty
\]
from the relation
\[
\sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}.
\]...
3
votes
1
answer
162
views
Generalized Radon transform (Relaxed sufficient condition for invertibility)
The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace.
To be more ...
user45183
3
votes
2
answers
354
views
Bandwidth approximation for a nonlinear problem
Can anyone please help me with this problem.
I must let you know from the beginning that it's not an easy one.
"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...
3
votes
1
answer
158
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 807
3
votes
1
answer
262
views
Low/high-frequency estimates in $\mathrm{L}^\infty$ for Lipschitz nonlinearities
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a Lipschitz nonlinearity with $f(0) = 0$ and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. ...
- 63
3
votes
1
answer
348
views
Fourier transform of subgroups of $(\mathbf{Z}/n\mathbf{Z})^*$
Let $\zeta$ be a primitive $n$-th root of unity, and for each function $f : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ define its Fourier transform $\widehat{f} : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ by
...
- 51
3
votes
0
answers
136
views
Possible gaps for a function and its Fourier transform
This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.
A nice remark by Christian ...
- 1,352
3
votes
0
answers
144
views
Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
- 20.2k
3
votes
0
answers
75
views
Non-vanishing of a "push-forward" Fourier–Harish-Chandra transform on a compact set
Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
- 81
3
votes
0
answers
103
views
How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
- 1,955
3
votes
0
answers
93
views
Efficient multiplication of Cayley-Dickson numbers
The question was already asked here, but doesn't have any meaningful answer, hence I'd like to re-post it.
Assuming that we have an algebra with conjugation, we can use Cayley-Dickson construction to ...
- 1,171
3
votes
0
answers
86
views
Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$
Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial ,
so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that
$g(|x|^\gamma)$ is positive ...
- 31
3
votes
0
answers
308
views
Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
- 870
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
3
votes
0
answers
106
views
A new arranging of discrete sine transform
Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...
- 4,058
3
votes
0
answers
347
views
Is this FFT algorithm known?
Recently I've been thinking about alternatives to the usual Cooley-Tukey FFT for multiplying polynomials. I think I've come up with a pretty nifty algorithm for multiplying polynomials. So my question ...
- 31