# Tagged Questions

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### Non-Abelian Fourier Analysis

I'm currently thinking to generalize a known result on abelian groups to non-abelian groups. This is the problem. Fix an abelian group $G$. We know that:
$\mathbb{E}_{\chi\in ...

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121 views

### $L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?

Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, ...

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51 views

### Discrete Fourier tranform on $L^2(\mathbb{R})$

I'm studying on the following construction:
For $\lambda\in\mathbb{R}$, denote
$$Y_{\lambda}=\left\{g\in L^{2,loc}(\mathbb{R})\,:\,g(t+2\pi)=e^{2\pi i\lambda}g(t)\right\}$$
We give to $Y_{\lambda}$ ...

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**1**answer

130 views

### Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put,
$$A(\mathbb T):= \{f\in ...

**3**

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**1**answer

123 views

### Real-analytic variant of theorem 4.2.5 of Duistermaat's “FIO”, 1996

Theorem 4.2.5 of Duistermaat's "Fourier Integral Operators", 1996, states:
Let $A \in I^m(X,Y,C)$ be an elliptic Fourier Integral Operator of order $m$, associated to a bijective canonical ...

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vote

**1**answer

112 views

### Absolute convergence of multi-dimensional Fourier series

For a Lipschitz function $f$ defined in $[0,2\pi]^d$ for $d>1$, is that true
that the multi-dimensional Fourier series converges absolutely?
In other words, $\sum_{k\in ...

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118 views

### Decay rate of measures on Cantor set

I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...

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**2**answers

711 views

### A question in Fourier analysis

I recently came across this problem:
Let $T=\mathbb R /2 \pi \mathbb Z$ the circle, with its proabability Haar measure $\mu$. Any integrable function $f : T \rightarrow \mathbb C$
has Fourier ...

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vote

**1**answer

259 views

### Pointwise convergence of double Fourier series

I'm looking for references/theorems that deal with the pointwise convergence of double Fourier series expansions for a particular function.
Let $D \subseteq [-\pi, +\pi]^2$ be an arbitrary set of ...

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**1**answer

102 views

### Almost Everywhere Convergence of Walsh Series of $L^2$ functions

I am currently reading the Hunt's papar (http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0655.0662.ocr.pdf), and am wondering if there is any notes which presents his argument more ...

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**1**answer

998 views

### The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray:
$$
A(t)=\int_0^\infty \omega(E) e^{-iEt}d E,
$$
where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on ...

**2**

votes

**1**answer

221 views

### Fourier transform of a particular function

In order to estimate the fundamental solution of some particular types of differential operators,I need estimates on some kind of oscillatory integrals.For simplicity, consider the Fourier transform ...

**3**

votes

**1**answer

955 views

### History of the Fourier transform

Does anyone know a good book or article on the History of the Fourier transform? It's first appearance (of the transform) and use in particular? Or at least some source with some historical ...

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votes

**1**answer

351 views

### Ask for theory about the weighted L^2(R^d) space.

Dear MOs,
I am now considering the following norm:
$$
||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.
$$
where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...

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738 views

### Non-commutative Fourier transform

What is a good reference for introducing non-commutative fourier transform for Electrical Engineers and Theoretical Computer Scientists in an explicit way?

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**1**answer

2k views

### FFT and Butterfly Diagram

Wikipedia presents butterfly as "a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into ...

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**2**answers

1k views

### How the Fast Fourier Transform got its name

In 1971, T.S. Huang published a paper in IEEE Computer, May-June, pp.15, called How the Fast Fourier Transform Got its Name, available here.
At the bottom of the paper, he wrotes: "The Chinese ...

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**4**answers

715 views

### Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...

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votes

**2**answers

578 views

### Number of integers coprime to l

A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for
$$
\sum_{n \leq x, (n, \ell) = 1} 1
$$
Of course, this is easy to estimate with a trivial error term of ...

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**1**answer

689 views

### Steinmetz, Laplace and Fourier Transforms

I am looking for references on Steinmetz Transform and its relation with Laplace and Fourier Transforms. There is an Italian Wikipedia page about this topic but with no references.

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**1**answer

575 views

### Fourier and Bessel

Oliver Heaviside, on page 387 of Electrical Papers, Vol. I, Macmillan and Co., 1892, available here, writes
$$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots ...

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814 views

### 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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266 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 $a\in ]0,2]$, ...

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**0**answers

324 views

### Fourier theory of characteristic functions

Here is a question which (up to some translation) I have been asked by an electrical engineer. Let $f:\mathbb{R}\to[0,1]$ be a smooth function with $f(x+1)=f(x)$. I would like to approximate $f$ in ...

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1k views

### Tanh version of a Fourier Transform?

I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...

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231 views

### Sparse Eigenvectors for the Discrete Fourier Transform matrix

There are many ways to choose eigenbasis for the Discrete Fourier Transform matrix since it has only $4$ distinct eigenvalues taken from $\{\pm 1,\pm i\}$.
Has there been any refereed work that ...

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votes

**1**answer

548 views

### (Real) algebraic geometry for (real) trigonometric polynomials?

Has somebody developed a comprehensive theory of the algebraic structure of trigonometric polynomials in several variables? If yes, where?
Background:
By a (real) trigonometric polynomial in ...

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4k views

### Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...

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540 views

### Convergence of squares of the moduli of partial sums of Fourier series

Let $\mu$ be a complex measure on the unit circle. The Wiener theorem says that the sequence of the Cesaro means of $|\hat\mu_n|$ has a limit. Define $p_n(z)=\sum_{k=0}^n \hat\mu_k z^k$. Then the Abel ...

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172 views

### Good probability measues on $S^1$ reprented by a kernel

I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...

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**1**answer

2k views

### Universal definition of Fourier transform [closed]

Is there a category theoretic definition for the Fourier transform using only its universality properties? I am not looking for the most general definition -- one that works only in some special ...

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436 views

### How to find the almost period of an exponential polynomial

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...

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3k views

### Are there extensive tables of Fourier transforms available online?

I hope this is suitable for MO... I was wondering if someone can suggest a website (or some online document) containing an $extensive$ table of Fourier transforms? When I try obvious Google searches, ...

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802 views

### Additive combinatorics and large Fourier coefficients

Elon Lindenstrauss explains in his talk at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $\mu$ on the ...

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334 views

### How large (small) can be the measure of a set where a polynomial takes small values ?

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...

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**1**answer

177 views

### Example of a pair which is not weakly annihilating

Let $\mathcal F$ denotes the Fourier transform $\mathcal{F} :L^2(\mathbb R)\rightarrow L^2(\mathbb R)$ and $E, \Sigma$ be two measurable sets in $\mathbb R$.
The pair $(E,\Sigma)$ is called a weakly ...

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**3**answers

1k views

### English translation of Riemann's Habilitation Thesis

Does anyone know where to find an English translation of Riemann's Habilitation Thesis concerning trigonometric series? The German title of the work is "Über die Darstellbarkeit einer Function durch ...

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547 views

### some questions about properties of harmonic measure

The original post
The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...

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**1**answer

480 views

### Salem Inequality

I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" ...