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57
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10answers
7k views

Is Fourier analysis a special case of representation theory or an analogue?

I'm asking this question because I've been told by some people that Fourier analysis is "just representation theory of $S^1$." I've been introduced to the idea that Fourier analysis is related to ...
52
votes
4answers
3k views

How Does My Radio Work?

Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical. I live in an area with $n$ AM radio stations and $m$ FM radio stations. AM station ...
43
votes
5answers
4k views

Quasicrystals and the Riemann Hypothesis

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line: $$ \zeta(\frac{1}{2} + i k_j) = 0 $$ Let $f$ be the Fourier transform of the sum ...
32
votes
6answers
5k views

How does one use the Poisson summation formula?

While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. ...
30
votes
1answer
680 views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ...
30
votes
3answers
3k views

Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" of ...
30
votes
0answers
1k views

Fourier transform on the discrete cube

Notation: identify an element of $\{-1,1\}^n$ with the set $S \subseteq \{1, \ldots, n\}$ on which it takes the value $-1$. The following is an asymptotic question. "Close to one" means "more than ...
29
votes
1answer
2k views

Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...
28
votes
7answers
2k views

How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
26
votes
10answers
17k views

Fourier transform for dummies [closed]

So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)? (Answers at any level of sophistication are welcome.)
25
votes
5answers
3k views

Why is Fourier analysis so handy for proving the isoperimetric inequality?

I have just completed an introductory course on analysis, and have been looking over my notes for the year. For me, although it was certainly not the most powerful or important theorem which we ...
25
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 ...
24
votes
3answers
2k views

What are fixed points of the Fourier Transform

The obvious ones are 0 and $e^{-x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
22
votes
6answers
5k views

Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
22
votes
3answers
3k views

Can Hölder's Inequality be strengthened for smooth functions?

Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals, $$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$ Of course, we ...
22
votes
1answer
2k views

Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle

Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely: Question: (Furstenberg) Let $\mu$ be a ...
21
votes
9answers
12k views

Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where? In the beginning, this question might look strange. But by restricting ...
20
votes
3answers
447 views

In what ways is the standard Fourier basis optimal?

Is there any convincing sense in which the standard trigonometric basis for the space $V$ of square-integrable real-valued functions on $[-\pi,\pi]$ is optimal among all the orthonormal bases? (If ...
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 ...
19
votes
1answer
986 views

Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: ...
18
votes
2answers
1k views

What's the relationship between Gauss sums and the normal distribution?

Let $p$ be an odd prime and $\left( \frac{a}{p} \right)$ the Legendre symbol. The Gauss sum $\displaystyle g_p(a) = \sum_{k=0}^{p-1} \left( \frac{k}{p} \right) \zeta^{ak},$ where $\zeta_p = e^{ ...
18
votes
2answers
2k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
17
votes
3answers
874 views

Is there an L^p tauberian theorem?

From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
17
votes
4answers
5k 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 ...
17
votes
2answers
639 views

Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
17
votes
3answers
2k views

How do I compare the different notions of Fourier transform for sheaves?

There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a ...
17
votes
2answers
672 views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
16
votes
7answers
3k views

Does there exist a continuous function of compact support with Fourier transform outside L^1?

Let f be a complex-valued function of one real variable, continuous and compactly supported. Can it have a Fourier transform that is not Lebesgue integrable?
16
votes
1answer
466 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
15
votes
6answers
3k views

Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication - (f.g)(x) = f(x)g(x), and convolution - (f*g)(x) = ...
15
votes
3answers
484 views

An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
15
votes
3answers
1k views

What is the shortest route to Roth's theorem?

Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the ...
15
votes
1answer
2k views

Convergence of Fourier Series of $L^1$ Functions

I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
15
votes
4answers
2k views

FFTs over finite fields?

I'm trying to understand how to compute a fast Fourier transform over a finite field. This question arose in the analysis of some BCH codes. Consider the finite field $F$ with $2^n$ elements. It is ...
15
votes
1answer
2k views

Fast Fourier Transform for Graph Laplacian?

In the case of a regularly-sampled scalar-valued signal $f$ on the real line, we can construct a discrete linear operator $A$ such that $A(f)$ approximates $\partial^2 f / \partial x^2$. One way to ...
14
votes
6answers
2k views

Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question. The ...
14
votes
2answers
720 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 ...
14
votes
2answers
3k views

Range of the Fourier transform on L^1

It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^d)$ into, but not onto, $\overline{C_0^0}(\mathbb{R}^d)$, where the closure is taken in the $L^\infty$ norm. This is a ...
14
votes
2answers
411 views

What geometric information is carried by the Fourier coefficients of the components of a closed curve?

Let $\gamma$ be a smooth closed curve in the plane and let $(x(t), y(t))$ be a parametrization. The functions $x(t)$ and $y(t)$ are smooth and periodic, so each has a uniformly convergent Fourier ...
13
votes
4answers
2k views

complex fourier coefficients, introduced by?

I remember reading somewhere that the complex Fourier coefficients were introduced by an engineer sometime around 1900, but I can't find anymore this information. Does anyone know the name of this ...
13
votes
2answers
266 views

Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?

I asked this in math.stackexchange, but it disappeared from the "main list" almost immediately, so I hope it will be appropriate as a separate question in MO. For a given function $f\in C(G)$ on a ...
13
votes
1answer
673 views

On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO. Let $S$ be a finite set of integers. For $P$ a subset of ...
12
votes
2answers
5k views

Truth of the Poisson summation formula

The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable ...
12
votes
5answers
1k views

Why do Littlewood-Paley projections behave like iid random variables

I have read more than once that the Littlewood-Paley (LP) projections of a function (i.e. decomposing a function into parts with frequency localization in different octaves) behave in some sense like ...
12
votes
1answer
391 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 ...
12
votes
1answer
1k views

Let a function f have all moments zero. What conditions force f to be identically zero?

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
12
votes
0answers
358 views

Correlation of Fourier transforms of characteristic functions

Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...
12
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0answers
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 ...
11
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 ...
11
votes
6answers
39k views

Fourier vs Laplace transforms

In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...