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2
votes
1answer
175 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 ...
4
votes
0answers
188 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 ...
3
votes
1answer
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 ...
7
votes
2answers
614 views

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 ...
1
vote
4answers
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 $ ? ...
0
votes
1answer
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 ...
6
votes
1answer
494 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 ...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
6
votes
1answer
428 views

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?
1
vote
1answer
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 ...
2
votes
0answers
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.) ...
2
votes
0answers
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 ...
6
votes
3answers
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 $f$ ...
3
votes
3answers
263 views

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 $a\in ]0,2]$, ...
4
votes
2answers
2k views

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 ...
2
votes
1answer
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 ...
3
votes
4answers
717 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 ...
3
votes
3answers
597 views

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. ...
3
votes
1answer
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 ...
2
votes
0answers
211 views

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 ...
10
votes
0answers
444 views

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 ...
2
votes
0answers
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 ...
6
votes
0answers
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 ...
11
votes
1answer
356 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 ...
0
votes
0answers
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 ...
0
votes
0answers
99 views

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. ...
3
votes
2answers
274 views

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?
1
vote
2answers
307 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 ...
4
votes
1answer
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 ...
0
votes
1answer
1k views

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 ...
0
votes
1answer
383 views

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 ...
19
votes
6answers
3k views

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 ...
1
vote
0answers
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 ...
8
votes
2answers
678 views

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 ...
10
votes
1answer
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 ...
1
vote
0answers
165 views

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 ...
10
votes
3answers
1k views

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 ...
2
votes
1answer
472 views

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): ...
6
votes
0answers
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 ...
1
vote
2answers
192 views

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 $\theta$ ...
1
vote
1answer
181 views

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 $\int_{\partial B} \xi(y)dS(y) = 0$ ...
23
votes
6answers
4k views

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 ...
3
votes
1answer
364 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 ...
6
votes
2answers
935 views

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| : ...
0
votes
3answers
703 views

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 ...
2
votes
0answers
465 views

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} ...
6
votes
0answers
616 views

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 ...
4
votes
1answer
603 views

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 ...
4
votes
2answers
481 views

$L^1$ norm of the Fourier transform of a truncated Gaussian

I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes: Consider the Gaussian $G(x):=e^{-x^2}$ ...