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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Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$
As I have heard people did not trust Euler when he first discovered the formula
$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
However, Euler was Euler and he gave other proofs.
I ...
46
votes
10answers
28k views
Fourier transform for dummies
A vague question of Kevin Lin which didn't quite fit at Mathoverflow:
So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)?
...
31
votes
3answers
639 views
Instructive proofs in functional analysis
I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
28
votes
2answers
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Explicitly reconstructing a function from its moments
Let $f$ be an integrable real valued function defined on $[0,\infty)$. Let $$m_n=\int_0^\infty f(x)x^n \mathrm dx$$ be the $n^{th}$ moment, and suppose that all of these integrals converge ...
25
votes
4answers
3k views
Connection between fourier transform and taylor series
Both fourier transform and taylor series are means to represent functions in a different form.
My question:
What is the connection between these two? Is there a way to get from one to the other (and ...
24
votes
1answer
847 views
Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$:
$$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot e^{-\frac{x}{2}})$$
can be plotted with ...
17
votes
1answer
467 views
Accessible proof of Carleson's $L^2$ theorem
Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$).
Some time ago I ...
15
votes
4answers
551 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:
$$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - ...
14
votes
1answer
208 views
Is Fourier transform characterized by its diagonalization properties?
Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space:
$$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$
We then have the following properties:
...
14
votes
1answer
321 views
On Vanishing Riemann Sums and Odd Functions
Let $ f: [-1,1] \to \mathbb{R} $ be a continuous function. Suppose that the $ n $-th midpoint Riemann sum of $ f $ vanishes for all $ n \in \mathbb{N} $. In other words,
$$
\forall n \in ...
13
votes
2answers
140 views
Intuitively, why is the Gaussian the Fourier transform of itself?
It's a standard exercise to find the Fourier transform of the Gaussian $e^{-x^2}$ and show that it is equal to itself. Although it is computationally straightforward, this has always somewhat ...
12
votes
3answers
295 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 ...
12
votes
3answers
652 views
Delta function integrated from zero
I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a ...
11
votes
2answers
188 views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
11
votes
3answers
429 views
learning algebra and harmonic analysis
I've revised my question a bit in response to the (very helpful) advice so far--
I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather ...
11
votes
2answers
242 views
Fundamental role of the Fourier Transform
I am currently learning about the Fourier Transform and the associated Fourier Analysis.
So far I realize that it has a number of applications, but more than that it seems to be central to Functional ...
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votes
6answers
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How is the Fourier transform “linear”?
A "linear" function usually means one who's graph is a straight line, or that involves no powers higher than 1. And yet, many sources will tell you that the Fourier transform is a "linear transform".
...
10
votes
4answers
335 views
Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ ...
10
votes
2answers
368 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 ...
10
votes
1answer
364 views
For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?
Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it.
I have an always-nonnegative (on the ...
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votes
1answer
174 views
Regularizing effect of the heat equation
Consider the heat equation on $\mathbb{R}_+\times\mathbb{R}^d$
\begin{align*}
\partial_t u -\Delta_x u &= f, \\
u(0,x)&=u_0(x).
\end{align*}
In the case where $u_0\in L^2(\mathbb{R}^d)$ ...
9
votes
4answers
278 views
Singular asymptotics of Gaussian integrals with periodic perturbations
At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
9
votes
4answers
2k views
Non-power-of-2 FFT's?
If I have a program that can compute FFT's for sizes that are powers of 2, how can I use it to compute FFT's for other sizes?
I have read that I can supposedly zero-pad the original points, but I'm ...
9
votes
5answers
321 views
Notes for Beginner Fourier Analysis?
Are there any good lecture notes or books on basic fourier analysis that authors publish freely online?
It is very difficult to find rigorous mathematical theory of fourier analysis because google is ...
9
votes
1answer
211 views
Applications of Pseudodifferential Operators
I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
9
votes
2answers
534 views
Is a Fourier transform a change of basis, or is it a linear transformation?
I've frequently heard that a Fourier transform is "just a change of basis".
However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in ...
9
votes
3answers
159 views
Dirac Delta or Dirac delta function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
9
votes
1answer
138 views
Laplace transform identity
Is there a function equal to its Laplace transform?
I mean
$$ \int_{0}^{\infty}dt\exp(-st)f(t)= f(s).$$
Of course I know $f(t)=0 $ satisfy the equation.
For the case of the Fourier transform, I ...
9
votes
2answers
291 views
Zeros of Fourier transform of a function in $C[-1,1]$
I am trying to prove the following:
Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros.
I know that $G(z)$ is entire and $\lim_{x \to \pm ...
9
votes
1answer
242 views
Show that f is a polynomial
Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon |
\text{Im}\,z|}$$
...
8
votes
3answers
334 views
A log improper integral
Evaluate :
$$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$
I found it can be simplified to
$$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$
I found the exact value in the ...
8
votes
3answers
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The mathematics of music - why sine waves?
Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal.
But what ...
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votes
8answers
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Conceptual/Graphical understanding of the Fourier Series.
I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
8
votes
1answer
870 views
Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$
I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed.
But ...
8
votes
3answers
293 views
Condition for Fourier series
I read that
Any "well-behaved" function of period $2\pi$ can be expressed as a Fourier series.
What qualifies as "well-behaved"? Any examples of functions that cannot be expressed as a ...
8
votes
1answer
333 views
pointwise convergence of Fourier series
I am a bit confused. I have heard today someone saying that the Fourier series of any continues periodic function $f$, say with period 1 for concreteness, converges pointwise to $f$. Wikipedia here ...
8
votes
4answers
7k views
Derive Fourier transform of sinc function
We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the ...
8
votes
1answer
562 views
How do I compute the eigenfunctions of the Fourier Transform?
In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections ...
8
votes
1answer
320 views
Fourier transform of Schrödinger kernel: how to compute it?
Let
$$K_t(x)=\frac{1}{(4 \pi i t)^{\frac{n}{2}}}e^{i \frac{\lvert x \rvert^2}{4t}}\quad x \in \mathbb{R}^n,\ t \in \mathbb{R},\ t\ne 0.$$
Clearly this is not a $L^1$ or $L^2$ function with respect ...
8
votes
1answer
186 views
Expectation of a Random Subset of the Roots of Unity.
Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...
8
votes
1answer
180 views
Ergodic flow in tori
Let $\mathbb{T}^n = { (z_1,\ldots,z_n) \in \mathbb{C}^n : |z_l| = 1, \; 1 \leq l \leq n }$ denote the $n$-torus, and let $t_1, \ldots, t_n$ be arbitrary real numbers. Then it can be shown that the ...
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votes
2answers
780 views
Compactly supported function whose Fourier transform decays exponentially?
It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential ...
7
votes
4answers
329 views
Computing the Gaussian integral with Fourier methods?
There are many proofs that
$$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx = \sqrt{\pi}.$$
For example, using a change to polar coordinates, differentiation under the integral sign, and the theory ...
7
votes
2answers
221 views
Sobolev space is an algebra
How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
7
votes
1answer
142 views
$L_{p}$ distance between a function and its translation
I'm working through a proof and one of the comments is that for a function $f\in L_p (\mathbb{T})$:
$$\lim_{t\to \infty}\;\|f(\cdot + t) - f\|_p = 0.$$
Should this read as $t\to 0$?
If so, how do ...
7
votes
1answer
172 views
Fourier transform of function in $L^{4/3}$
Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f)$. Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by
...
7
votes
2answers
164 views
Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?
Assume that the functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are integrable and equal to zero on $(-\infty,0)$, (i.e $f,g \in L^+$). Then by Titchmarsh's theorem:
$f*g$ is zero almost everywhere ...
7
votes
3answers
450 views
Decomposing a discrete signal into a sum of rectangle functions
Hello math@stackexchange community !
I have a simple question that seems to have a non trivial answer.
Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
7
votes
2answers
255 views
Fourier transform of fourier transform?
I have the definition of Fourier transform $$\hat f(\lambda) = \int_{\infty}^\infty f(t) \exp(- i \lambda t) dt$$ and have proved the following lemmas:
$\hat E(x) = \sqrt{2 \pi} E(x)$ where $E(x) = ...
7
votes
1answer
400 views
Meaning of polynomially bounded
I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
