Questions tagged [fourier-analysis]

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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18
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1answer
465 views

Is there a combinatoric identity for the multiplicities of the following set?

Are you ready for some psychedelic pictures? Define the multiset$$S_n=\left\{\sum_{j=1}^n(-1)^{\left\lfloor(k-1)/2^{j-1}\right\rfloor}u_n^j\mbox{ for }1\leq k\leq2^n\right\}$$ where $$u_n^j=\left(\...
17
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3answers
44k views

How to calculate the Fourier transform of a Gaussian function?

I would like to work out the Fourier transform of the Gaussian function $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$ It seems likely that I will need to use differentiation and the shift rule at some ...
17
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3answers
15k views

Why do Fourier transforms use complex numbers?

I know that the Fourier transform is as follows:$$\hat{f}(\xi)= \int_{-\infty}^{\infty}\exp(-\mathrm ix\xi)f(x)\mathrm{d}x$$ but I couldn't understand why we should use the complex number $i$ in the ...
17
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2answers
3k views

Compactly supported function whose Fourier transform decays exponentially?

It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential ...
17
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1answer
3k views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
17
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1answer
5k views

Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
17
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3answers
20k views

What does the Fourier Transform mean in the context of images?

This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is ...
17
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2answers
1k views

Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function $f\...
17
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3answers
9k views

Intuition behind Fourier coefficients

Actually I'm trying to dive into Fourier series and have some trouble understanding the idea behind the Fourier coefficients. Let's have a Fourier series $$f(x) = a_0 + \sum_{n=1}^{\infty}[a_n\cos(\...
17
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5answers
2k views

Interpretation of Poisson Summation Formula

This question arises from a Fourier transform class I took about a year back. The poisson summation formula is: $$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - \infty}^{\...
17
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1answer
851 views

Residue Proof of Fourier's Theorem Dirichlet Conditions

Whittaker gives two proofs of Fourier's theorem, assuming Dirichlet's conditions. One proof is Dirichlet's proof, which involves directly summing the partial sums, is found in many books. The other ...
16
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5answers
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Definition of convolution?

Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it ...
16
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6answers
639 views

Evaluating the improper integral $\int_0^\infty \frac{x\cos x-\sin x}{x^3} \cos(\frac{x}{2}) \mathrm dx $

I've been working through the following integral and am stumped: $$\int_0^\infty \frac{x\cos x-\sin x}{x^3}\cos\left(\frac{x}{2}\right)\mathrm dx$$ Given the questions in my class that have ...
16
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5answers
12k views

How was the Fourier Transform created?

The Fourier Transform is a very useful and ingenious thing. But how was it initiated? How did Joseph Fourier composed the Fourier Transform formula and the idea of a transformation between periodic ...
16
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1answer
958 views

Laplace transform identity

Is there a function equal to its Laplace transform? I mean $$ \int_{0}^{\infty}dt\exp(-st)f(t)= f(s).$$ Of course I know $f(t)=0 $ satisfy the equation. For the case of the Fourier transform, I ...
15
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3answers
9k views

Fourier transform is uniformly continuous

I am trying to prove the following statement: If $f \in L^1$, then $\hat f$ is uniformly continuous. The argument given is as follows : $$|\hat f (\xi +h )-\hat f (\xi)| = \left| \int f(x) (e^{-2 \...
15
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2answers
2k views

Why do we call a Fourier series a series of complex exponentials?

Euler's formula is $$e^{{ix}}=\cos x+i\sin x$$ I have recently learned that a periodic function can be decomposed into an infinite series of cosines and sines. However, in the formulas that I've seen,...
15
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1answer
731 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
15
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3answers
2k views

Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
15
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1answer
10k views

Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ ...
15
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2answers
34k views

Discrete Fourier Transform: Effects of zero-padding compared to time-domain interpolation

While studying the various algorithm implementations available on-line of the Fast Fourier Transform algorithm, I've come to a question related to the way the DFT works in theory. Suppose you have a ...
15
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2answers
9k views

Is deconvolution simply division in frequency domain?

Is it correct to say that deconvolution simply division in frequency domain? And that convolution in time domain is multiplication in frequency domain. And is it a convention to notate a function in ...
15
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7answers
35k views

Derive Fourier transform of sinc function

We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the ...
15
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1answer
237 views

Hilbert-valued Schwartz functions

Let $H$ be a separable complex Hilbert space. We can define Schwartz functions $f\colon\mathbb R^n\to H$ to be the smooth functions for which $$ \sup_{x\in\mathbb R^n}\|(1+|x|^2)^mD^\alpha f(x)\|_H<...
14
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5answers
5k views

What is the difference between a function and a distribution?

I remember there was a tongue-in-cheek rule in mathematical analysis saying that to obtain the Fourier transform of a function $f(t)$, it is enough to get its Laplace transform $F(s)$, and replace $s$ ...
14
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1answer
4k views

Fourier transform of the indicator of the unit ball

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.
14
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4answers
4k views

The mathematics of music - why sine waves?

Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal. But what ...
14
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1answer
1k views

Fourier Transform of Schwartz Space

I am trying to read through Corollary 8.23 in Folland, p. 250, which is a proof that the Fourier transform maps the Schwartz space into itself. I do not see why the following is true $$\|x^\alpha \...
14
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4answers
973 views

What is a good book, or article, that explains the history of fourier analysis?

What is a good book on the history of Fourier Analysis? I'm looking for a book which explains how it came to be and what the mathematicians (or physicists) were thinking when they came up with it. If ...
14
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1answer
2k views

Relation between function discontinuities and Fourier transform at infinity

I have made the following assertion a few times in this space without ever having provided a proof: Let $m$ be the smallest number such that a function $f \in L^2(\mathbb{R})$ has a discontinuity in ...
14
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6answers
2k views

Where is the wild use of the Dirac delta function in physics justfied?

Wikipedia has a wild article about the Dirac delta function. Are the things listed correct? Or is there no proof that they are correct? For my master thesis I want to refer to rigorous proofs of these ...
14
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0answers
184 views

Toeplitz matrices question with Fourier coefficients

Denote: $f(e^{i\theta})$ is continuous and strictly positive on the interval $ 0 \le \theta \le 2\pi$ with Fourier coefficients $$ t_j = \frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta} \quad ...
13
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3answers
15k views

Proving commutativity of convolution $(f \ast g)(x) = (g \ast f)(x)$

From any textbook on fourier analysis: "It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = \int_{-\...
13
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2answers
2k views

A bound on the Fourier coefficients of an $\alpha$-Lipschitz function

I am asked to show that if $0 < \alpha < 1$, and if $f \in \Lambda^\alpha(\mathbb{T})$, then we have for $k\neq 0$, $$|\widehat{f}(k)| \leq \pi^\alpha \frac{\|f\|_{\Lambda^1}}{k^\alpha}$$ I ...
13
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3answers
3k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
13
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2answers
7k views

DFT - Why are the definitions for inverse and forward commonly switched?

Sometimes the forward DFT is defined with a negative sign in the exponent, sometimes with a positive one and occasionally with a $1/N$ coefficient. I see this all over the place online. I don’t see ...
13
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1answer
390 views

Solving Poisson's equation for $\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) \, \delta(y)$

Problem statement I took an exam, where I had the following task: Determine the electrostatic potential for the charge distribution $$\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) \,...
13
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6answers
17k views

What are some good Fourier analysis books?

I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
13
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1answer
700 views

On Vanishing Riemann Sums and Odd Functions

Let $ f: [-1,1] \to \mathbb{R} $ be a continuous function. Suppose that the $ n $-th midpoint Riemann sum of $ f $ vanishes for all $ n \in \mathbb{N} $. In other words, $$ \forall n \in \mathbb{N}: ...
13
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2answers
608 views

Simple proof that Fourier transform is an isomorphism between $L^p$ spaces for $p \neq 2$?

It is known that the Fourier transform $\mathcal F$ maps $L^2 \to L^2$ as an (isometric) isomorphism and $L^1 \to L^\infty$ as bounded operator. Via Riesz-Thorin this result can be extended to give ...
13
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0answers
207 views

Lower Bound on Oscillatory Integral

Let $p,y\in\mathbb{R}^d\setminus\{0\},\beta>0$ be given and fixed and define for all $\alpha>0$, $$I(\alpha) := \int_{x\in\mathbb{R}^d}\exp(\mathrm{i}\alpha p\cdot x-\alpha\beta \|x-y\|^2)f(x)\...
12
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3answers
12k views

Obtaining the $\frac{1}{2\pi}$ factor in the Fourier transform

This MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ ...
12
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3answers
5k views

How does FFT work?

For five years I tried to understand how Fourier transform works. Read a lot of articles, but nobody could explain it in simple terms. Two weeks ago I stumbled upon the video about a 100 years old ...
12
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6answers
6k views

Fourier Analysis textbook recommendation

I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the point....
12
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4answers
2k views

Computing the Gaussian integral with Fourier methods?

There are many proofs that $$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx = \sqrt{\pi}.$$ For example, using a change to polar coordinates, differentiation under the integral sign, and the theory ...
12
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4answers
2k views

Why would you expand a square wave in a Fourier series?

The periodic square wave $$ f(x) = \cases{ 1 \text{ if } 0 \le x \le \pi \\ 0 \text { if } -\pi \le x < 0} $$ seems easy enough to work with. Why transform it into a series of sines and cosines? I'...
12
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2answers
3k views

Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where $\...
12
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3answers
711 views

Can $\log(1-U)-\log(U)+W$ be normally distributed, with $U$ uniform on $(0,1)$ and $W$ independent of $U$?

Assume that $U$ and $V$ are independent random variables with values in $(0,1)$ and that $U$ is uniformly distributed. Can it happen that $$L=\log\left(\frac{(1-U)V}{U(1-V)}\right)$$ is normally ...
12
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6answers
15k views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
12
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1answer
2k views

Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed. But I'...