The fourier-transform tag has no usage guidance.

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### Spectrum of a convolution operator in the unit disk

Let $T: L^{2}(D)\to L^{2}(D)$ be the convolution operator $T(u)(x)=(J*u)(x)=\int_{D} J(x-y)u(y)\,dy$ on $D=\{(x_1,x_2) \in R^2\ |\ x_1^2+x_2^2\le 1\}$.
How do you obtain eigenvalues and ...

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691 views

### Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...

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103 views

### Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...

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48 views

### Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of $\...

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41 views

### A question about Fourier transform of function of the type $(1+P(x))^{z}$

Let $$f= (1+P(x))^{z},$$
where $P(x)\ge 0$ is a real polynomial in $\mathbb{R}^n$ of degree $2m>0$, and $z=a+ib$ with $a<0$. I want to study the behavior near the origin of the Fourier ...

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74 views

### Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$

I am trying to find the inverse of the following kernel in 3 dimensions
$$
\nabla^2-x^2,
$$
where,
$$
x^2=\vec{x}.\vec{x}
$$
It seems quit simple and one would think there should already be solutions ...

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68 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...

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267 views

### How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& c_k\...

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211 views

### Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...

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1k 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 ...

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519 views

### Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...

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402 views

### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...

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66 views

### Restricted Isometry Property for Discrete Fourier Transform Matrix

I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\...

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381 views

### What is the Fourier transform of this function?

Consider the function
$$
f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.
$$
It is known that $f(x_1,x_2)\in L^2(\...

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98 views

### Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...

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50 views

### How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...

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44 views

### Subquadratic multiplication of probability mass functions (with log-convolution?)

We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$.
That is, we are given two finite input vectors $x$ ...

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154 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...

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109 views

### What are the spaces for which the Fourier transform is an automorphism? [closed]

this is well-known that the Fourier transform is an automorphism of $L^2(\mathbb R)$ and also of $\mathcal S(\mathbb R)$ (Schwartz space). Is there any other spaces of functions of one real variable ...

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### Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand
$K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed
form or some kernel like structure. If I try to take derivative, ...

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486 views

### Why does Fourier transform of a function give the frequency spectrum?

Coefficients of the complex Fourier series give spikes at discrete frequencies. I'd like to understand why F.T. gives us the continuous frequency spectrum?

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308 views

### Why decompose a function with eigenvectors of Laplace operator? [closed]

On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied ...

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### Fourier transform in Lorentz spaces

Denote by $E$ the class of nonnegative, even functions on $\mathbb{R}$, monotone
decreasing to 0 on $\mathbb{R}_+$.
In here, Y. Sagher stated the following proposition about Fourier transform of ...

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105 views

### If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4.
However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...

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### Can we do better than zero padding of FFT?

My background is in signal processing, and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...

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### Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$:
$T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial y}-y\frac{\partial}{\...

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### Conceptual explanation for Poisson summation formula

The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have
$$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf Z^...

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### What is the actual meaning of a fractional derivative?

We're all use to seeing differential operators of the form $\frac{d}{dx}^n$ where $n\in\mathbb{Z}$. But it has come to my attention that this generalises to all complex numbers, forming a field called ...

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### Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible:
$$\partial_t \...

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352 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 ...

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### Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it.
Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that
$$
...

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268 views

### G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by $O(\log n)$ ...

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385 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 ...

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301 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: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...

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### Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...

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### Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...

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### Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...

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### A case of nested central limits

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. We are interested in $N$ large and prime. The Fourier transform $\hat{S}...

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### Statistical independence and the Fourier transform

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. With the Fourier transform we can define $N$ random walk variables
$$
\...

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### Help with notations from 2D to 3D FFT representations as 1D FFT

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other.
I need some help and clarifications for my ...

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### 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 ...

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251 views

### Proof of Biot-Savart theorem in 3D using distributions and fourier transform

While studying the book Vorticity and Incompressible Flow by A. Majda and A. Bertozzi I came across the following Biot-Savart theorem in 3D:
The solution to:
$$div \ v=0$$
$$curl(v)=\omega$$
is
$$v=\...

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386 views

### Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
...

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571 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 ...

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### Simplifying an expression using tools from Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help:
$f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{ \omega r^{-\gamma}}} \...

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### Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...

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### Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta q}dx)...

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### Fourier series and transform related to Epicycles

Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation.
1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and ...

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577 views

### Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function:
$$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$
Newton series expansion of a function:
$$f(x)...

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### How are the infinity norm of Fourier transforms of sign vectors distributed?

This is a follow up to an earlier resolved question. Define the $n$-dimensional discrete Fourier transform via the matrix
$$
D_{s,t} := \omega^{st},
$$
where $\omega=\exp(-2\pi i/n)$. Notice that $D$ ...