The fourier-analysis tag has no wiki summary.

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### History of the Sampling Theorem

In January, 1949, Shannon publishes the paper Communication in the Presence of Noise, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available here, which establishes the Information Theory. In this paper, the ...

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176 views

### Function with Fourier coefficient of order $o(n^{-m})$

Let $(a_n)$, $(b_n)$ be the Fourier coefficients of a periodic, locally integrable function $f: \mathbb{R} \rightarrow \mathbb{R}$. Assume that $n^m a_n, n^m b_n \rightarrow 0$ when $n \rightarrow ...

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189 views

### The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...

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540 views

### karhunen-Loeve expansion of Poisson process

Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda ...

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### Analogues of the Riemann-Roch Theorem

In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...

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1k views

### General Procedure for Inverse of an Integral Transform

Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform) $\int^{a}_{b} f(x) g(x,\xi) dx $ ?
...

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174 views

### Decay of Sobolev fractional norms outside a ball

Suppose you have a function $u$ that lies in the (fractional) Sobolev space $H^s$,
with $0<s<1$.
Take a smooth cut-off $\varphi$ such that $\varphi(x)=0$ if $|x|>R+1$. What can be said ...

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496 views

### Fastest decay of Fourier Transform for Generalized Functions of compact support

What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ...

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88 views

### Characterizing Identification of a Deconvoluted Density Without Independence

Hey, I'm new here and this is my first attempt at a question. My background is in econometrics, so I apologize in advance if I use unfamiliar notation or display ignorance of important work.
Suppose ...

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161 views

### Fourier transform in n-dim Euclidean and Minkowski space

As far as I understood, the Fourier decomposition of a function $\boldsymbol{F}\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ where $\mathbb{R}^{n}$ is endowed with the Euclidean inner product ...

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### Fourier transform

Does anyone know what the Fourier transform (in the sense of distributions) of
$$
f(x) = (x^2 - 1)^{1/2}x, \quad |x|\ge 1,
$$
and $f(x) = 0$ otherwise, is?

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179 views

### Gausian distributions in the Frequency domain

I have read in many texts that the Fourier Transform of a Gaussian is yet another Gaussian, however how does the mean and standard deviation change?
Also if we convolve a Gaussian with itself then ...

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76 views

### Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.)
...

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197 views

### Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...

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397 views

### Simple closed curves and the coefficent of $\exp(i\theta)$ in the associated Fourier series

Given a continuous map $f:S^1\to \mathbb{C}$ from the unit circle to the complex numbers, one can form its Fourier series $\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those $f$ ...

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### Asking for a Fourier inverse transform, which is related to stable laws

Dear friends,
Denote the function
$$
G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;.
$$
It is well known that if $a\in ]0,2]$, ...

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### How does the Laplace Transform work for circuit analysis? [closed]

I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to ...

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**1**answer

110 views

### uniform bit generator

I'm trying this question for days but no luck, if someone can give me a lead or an article that solves this.. it would be great:
A uniform bit generator is a function $f:\{0,1\}^n \times \{0,1\}^n ...

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720 views

### Quick computation of the Pontryagin dual group of torus

I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan ...

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### Peter-Weyl theorem as proven in Cartier's Primer

I'm reading Pierre Cartier's A primer of Hopf algebras to educate myself. In its subsection 3.3 (which doesn't need any Hopf algebra theory), he sketches a proof why compact Lie groups are algebraic. ...

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219 views

### Inequality for the first Fourier level of a Boolean function

In the study of Boolean functions, the hypercontractive inequality enables one to bound from above the norm of $Tf$ by some norm of $f$, where $T$ is the noise operator depending on the noise ...

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### Unitary Representations and Integral formulas

While reading the appendix to 4th chapter of Iwaniec and Kowalski's analytic number theory I came upon a remark relating unitary representations and some integral transforms involving J-Bessel ...

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### Connections of results in Harmonic analysis in the theory of Transcendental Numbers

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$.
A famous result of Polya says if $f$ is an entire function of ...

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170 views

### Spectral gap of tempered distributions

Hi,
Let $\Lambda\subset\mathbb{R}$ be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and $\delta_{\lambda}$
is a unit mass located at the point ...

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281 views

### Basic examples of induction on scales arguments

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth.
Here is a ...

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357 views

### General Isoperimetric Inequality via Representation Theory of SO(n)

Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case?
Specifically, I imagine such a proof would ...

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213 views

### Circlular Convolution for two finite sequences of different length.

Hello.
I just want to know how the circlular Convolution for two finite sequences of different lengths is defined.
As I have found in MIT lectures for the sequences of the same length we have
...

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### Differential equation with switched parameters and boundary conditions in integral form

Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...

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### Wiener Tauberian Theorem for nonunimodular group

Is there a nonunimodular group for which Wiener's Tauberian theorem is true?
Is a locally compact topological group whose volume grows polynomially with radius always unimodular?

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308 views

### Asymptotics of Fourier coefficients of power-type functions

I would like to understand the asymptotic behaviour of the Fourier coefficients of
power type functions
$f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$
I suppose this is a classic ...

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519 views

### An approximate converse of discrete uncertainty principle

Let $f:\mathbb{Z}_n \rightarrow \{0, 1\}$ and let's normalize the Fourier transform $\hat{f}$ so that $\|\hat{f}\|_2 = \|f\|_2$, i.e.
$$\hat{f}(\xi) = \frac{1}{\sqrt{n}}\sum_{x \in ...

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### Fourier transform of a differential operator

I have a differential operator defined by its Fourier transform:
$\left(\alpha k_x^2 + \beta k_y^2 + \gamma k_x k_y \right)^\delta \hspace{20pt} \alpha,\beta,\gamma,\delta \in \mathbb{R}$
I don't ...

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### Restriction of Haar measure to Borel $\sigma$ -algebra

Let $(G,M,\mu)$ be a measure space, where $\mu$ is the Haar measure on topological group $G:=\mathbb R\times\mathbb R_d$, ($\mathbb R$ is the group of reals with the natural topology whereas $\mathbb ...

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### explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

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280 views

### Fourier series/transform of an amplitude-limited sinusoid

I am trying to estimate the amplitude of an original unlimited sine wave from a measurement of the power spectral density (PSD) of an amplitude-limited version. I expect that I may be able to do so ...

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### Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...

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573 views

### A problem concerning $L^2([0,1]\times[0,1])$

Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let ...

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### Rough estimate for the fourier coefficients of a discrete bump function

Setup:
Let $p$ be a large prime number and $F = \{0, \dots, p-1\}$ be the field of order $p$.
Let $I$ denote the discrete interval $I = \{1, \dots, M\}$ for some $M < p$.
Regard both $I$ and $F$ as ...

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### Fourier transforms of functions not in $L^2.$

This is probably something five-year-old physicists know, but here goes: Is there a standard methodology for computing Fourier transforms of things like $\log |x|$? Wolfram Alpha will happily give an ...

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### Constructing an L2-space with a given orthonormal basis

Coming from a completely different world, I am trying to learn some very basic Fourier analysis, and have been scratching my head around this (it may be a very stupid question with an obvious answer):
...

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369 views

### The log kernel and Bochner Theorem

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...

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### bounds on solution of an ODE

I am interested in getting a good bound for solution of the following ODE:
$$ f''(t) + n^2 f(t) = (\sin(\theta t))^n$$
with the boundary condition $f(0) = f'(0) = 0$ and $t \in [0,1]$, where $\theta$ ...

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### A variational problem involving a negative fractional Soboblev norm.

I've run into the problem of trying to evaluate the following:
$\max_{ \xi} \iint_{\partial B \times \partial B} \xi(x) \Phi(|x-y|) \xi(y) dS(x)dS(y)$
subject to $\int_{\partial B} \xi(y)dS(y) = 0$ ...

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### What does Mellin inversion “really mean”?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...

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366 views

### Decomposition of distributions

Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions
$\psi = \psi_1 + \psi_2$
such that $\psi_1$ and the Fourier transform of $\psi_2$ are ...

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### Fourier transform of a real-valued function.

My chemist roommate asked me the following question. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real-valued function and $F$ its Fourier transform. Suppose we know the modulus function $|F| : ...

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### How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times ...

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### What square-summable sequences are “sinc-summable”?

$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by $\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 & \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x & \text{else} ...

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### injectivity of Fourier transform: is there algebraic proof?

Let $L$ be the Banach algebra of $L^1$-functions from $\mathbb{R}$ to $\mathbb{C}$ with $L^1$-norm and convolution as algebra multiplication. Assume that we knew that the homomorphisms from $L$ to ...

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### Fejer's theorem and convergence of Fourier series in measure

Fejer's theorem says that for any continuous function $f \colon S^1 \to \mathbb C$ with Fourier coefficients $a=(a_n)_{n \in \mathbb Z}$ the sequence
$$\sigma_n(a) := \frac1n \sum_{k=1}^n ...