# Questions tagged [fourier-transform]

The fourier-transform tag has no usage guidance.

263
questions

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### How to switch the order of limit and integration here?

I asked it on math.stackexchange but didn’t get an answer.
Let $f\in L^1(\mathbb{R}^n)$ and $\phi_\varepsilon(x):=\varepsilon^{-n}e^{-\pi|\varepsilon^{-1}x|^2}$. It is straightforward to see that $\...

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74 views

### Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...

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56 views

### (Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$.
We assume moreover that the square-summable Fourier coefficients of $f$, ...

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37 views

### Fourier transform of a general normal distribution [closed]

How can I calculate the Fourier transform of
this equation? Here is a known equation that may be helpful.

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49 views

### Conditions for absolute continuity in the Bochner-Schwartz theorem

Suppose that $f$ is a positive-definite Schwartz distribution, that is,
$$\langle\phi,f*\phi\rangle\geq0\qquad\text{for every }\phi\in C_0^\infty(\mathbb R^n).$$
By the Bochner-Schwartz theorem, there ...

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178 views

### Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.

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32 views

### eigenvectors of a graph Laplacian VS Fourier basis

Could you please illustrate the following statement:
the eigenvectors of a
graph Laplacian behave similarly to a Fourier basis, motivating
the development of graph-based Fourier analysis theory.

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61 views

### Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$

It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...

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112 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 ...

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66 views

### How to calculate Fourier transformation of eigenstates in CV quantum information [closed]

The position $\hat{q}$ and momentum $\hat{p}$ has $[\hat{q},\hat{p}]=i$.
And we set there eigenstates as $|s\rangle_q$ and $|s\rangle_p$ with eigenvalue s.
In the paper [Phy Rev A. 79, 062318 (2009)],...

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87 views

### How related are Fourier transforms on finite groups and Fourier transforms on graphs?

Here are two generalizations of the notion of a Fourier transform. I am also aware of the Pontryagin Duality generalization for locally compact abelian groups, though I am personally more concerned ...

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92 views

### A close formula for a Fourier transform

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define
$$
I(\alpha, x)=\int_\mathbb R \cos(xt) e^...

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74 views

### Laplacian spectrum

I have found (in lecture notes) a method to calculate the spectrum of the operator
$$
A:D(A)\subset L^2([0,\pi])\longrightarrow L^2([0,\pi])\text{, such that}
$$
$$
Au=\dfrac{\partial^2u}{\partial^2x^...

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230 views

### Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...

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99 views

### A Fourier elliptic vector field on a Riemannian manifold

Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...

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68 views

### Fourier dimension of radial set

In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...

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35 views

### Cumulant of functions of weakly dependent random variables

Suppose $X_1,\dots,X_4$ are Gaussian random variables with mean and variance
$$\mathbf E X_i = 0,\quad \mathbf E X_i^2=1.$$
Furthermore suppose that the random variables have a certain weak ...

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**1**answer

162 views

### Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$

Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \...

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119 views

### Wavelet momentum identity

I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31):
\...

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222 views

### Certain Fourier transforms involving Whittaker function and Bessel functions

I recently meet the following two weird "Fourier transform" questions.
(I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ and $\...

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90 views

### Common eigenvector of the Wigner D matrices for the eigenvalue $\lambda = 1$?

Consider the family of spherical harmonics $Y_{n}^m$ on the sphere $\mathbb{S}^2$, with $n \geq 0$ and $-n \leq m \leq n$. The Wigner matrices are matrices $\mathrm{D}_{\mathrm{R},n} \in \mathbb{R}^{(...

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80 views

### Advantage of fractional Fourier transform over multiscale wavelet

What is the best argument of fractional Fourier transform over multiscale wavelet in data analysis purpose.
Optimization of the good time-frequency domain parameter ? "Good" will be, find the %time-%...

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10k views

### Why is the Fourier transform so ubiquitous?

Many operations and equivalences in mathematics arise as some sort of Fourier transform. By Fourier transform I mean the following:
Let $X$ and $Y$ be two objects of some category with products, and ...

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235 views

### The Hubbard-Stratonovich transformation

is there a way to extend the Hubbard-Stratonovich transformation
$$e^{\frac{1}{2}Ks^2}=\left(\frac{K}{2\pi}\right)^{1/2}\int_{–\infty}^\infty e^{–Kx^2+Ksx}dx$$
to the case $e^{\frac{1}{2}Ks^p}$ for $...

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**1**answer

272 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 ...

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**1**answer

115 views

### Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$

How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?

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66 views

### Which set of functions/measures has range $\mathrm{L}^\infty$ under Fourier transformation

I have a question concerning the Fourier transformation. What I know is that $\mathrm{L}^{\infty}=\{\hat{u}:\ u\in Y\}$ for some space $Y$. Now, I want to specify the space $Y$. The question is, is ...

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165 views

### Fourier transform derivation from Laurent series

Using Laurent Series of a function $f(z)$ around a point $a \in \mathbb{C}$
$$f(z) = \sum^{\infty}_{n=-\infty} c_n(z-a)^n \ \ \ \ (1)$$
where
$$c_n = \frac{1} {2\pi i}\int\limits_{\gamma}\frac {f(z)} {...

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125 views

### Given $\theta$, find $f$ such that $\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0,$ for all $h \in \mathbb{N}$

Let $\theta$ be a $C^{\infty}$ (resp. analytic) real-valued function on $\mathbb{T}=[0,2\pi]/\{0,2\pi\}$.
When can one find $f \neq 0$, $C^{\infty}$ (resp. analytic) real-valued function on $\...

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69 views

### Is harmonic mean of linear functions a Bernstein function?

According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function:
$f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$
is a Bernstein function, ...

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74 views

### Expressing 1-e^{-z} as a Fourier integral

According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...

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### Deriving spectral measure

I am reposting this question from Cross Validated as I have not received any responses.
While reading this book, I got stuck on page 266 where the authors found the spectral measure $F(du)$ of the ...

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**1**answer

53 views

### Hilbert transform of a signal to measure skewness and asymmetry of a sinusoidal wave

Thank you for taking the time to read this. I was hoping to get some assistance in understanding how these equations function:
$$As=\frac{\langle H(\eta)^3\rangle}{\langle\eta^2\rangle^{3/2}},\qquad ...

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### Properties of microlocalization

Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...

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124 views

### What subjects of Fourier analysis have had more effect on machine learning? [closed]

What is the salient uses of Fourier analysis in machine learning? What subjects of Fourier analysis have had more effect on machine learning?
Please mention the references.

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140 views

### Solving an equation of function

How to solve, or at least how to proceed to solve, the following equation for $g(u)$
$$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$
Here $0<\alpha\leq2$ and $-\...

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160 views

### Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$

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24 views

### Lower Bounds on Oscilatory Integral in Restriction Theory

Consider the oscillatory integral that appears in restriction theory:
$$
I(x,y)=\int_{-1}^1 e^{i(x\xi+y\xi^2)}d\xi.
$$
Using basic theory on oscillatory integrals, we may bound
$$
|I(x,y)|\lesssim |x|^...

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175 views

### Is there a practical application of natural integral or differintegral?

The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$...

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133 views

### Translated version of a Caratheodory article

This excellent introduction to Compressive Sensing cites a couple of (seemingly) interesting Caratheodory papers from 1907-1911.
These are:
[46] C. Caratheodory. Uber den Variabilitätsbereich der ...

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429 views

### Process quicker than Fourier for squares of polynomials

FFT is a quick algorithm for multiplying two polynomials, but given it's a square (i.e. multiplying the polynomial with itself) can we find something better?

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194 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\...

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101 views

### Decay of Laplace (or Mellin) transform beyond region of convergence?

Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...

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116 views

### Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...

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**1**answer

136 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 $s \in [0, \tfrac{1}{2}]$. ...

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### Approximate method to extract behavior of a Laplace transform in an intermediate region

In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...

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158 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,...

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121 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 ...

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**1**answer

205 views

### Hörmander-Mikhlin theorem on the torus

Let me first recall a particular case of the classical Hörmander-Mikhlin multiplier theorem: Let $m$ be a bounded function on $\mathbb {R} ^{n}$ which is smooth except possibly at the origin, and ...

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363 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{\...