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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12
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4answers
516 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 $f(x) = |f(x)|e^{i\phi(x)}$ ...
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1answer
10k views

Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann

I am trying to compute the Fourier transform of $\frac1{|\mathbf{x}|^2+1}$ where $\mathbf{x}\in\mathbb{R}^3$. Just writing out the integral: $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\...
12
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2answers
4k views

Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| \int_{\mathbb{...
12
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1answer
5k views

Fourier transform in $L^p$

Let the $f$ be a function in $L^s$ where $s \in [1,\infty) $. For which $r$ Fourier transform $\hat{f}$ belongs to $L^r$? I'd be grateful for any kind of help including providing a literature or ...
12
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2answers
1k views

Reconciling two intuitions about convolution

There are two intuitive things convolution does. In the time domain, it represents the distribution of the sum of two independent random variables. In the frequency domain, it's just multiplication. ...
12
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3answers
447 views

Fourier transform of $\Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) /\sqrt{ \cosh(p/2)}$

Is it possible to compute the following Fourier transform analytically? $$\psi(x) = \frac{1}{\sqrt{4 \pi}}\int \Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) \frac{e^{i p x}}{\sqrt{ \cosh(p/2)}} ...
12
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1answer
311 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 ...
12
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1answer
646 views

Deriving Fourier inversion formula from Fourier series

Let $g\in C_0^{\infty}(\mathbb{R})$ (infinitely differentiable with compact support), and let $$\hat{g}(y)=\int_{-\infty}^\infty g(x)e^{-ixy}dx$$ Assume that $\hat{g}$ is in the Schwartz class. Prove ...
12
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1answer
947 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
11
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2answers
28k views

Calculate the Fourier transform of $b(x) =\frac{1}{x^2 +a^2}$ [closed]

I need help to calculate the Fourier transform of this funcion $$b(x) =\frac{1}{x^2 +a^2}\,,\qquad a > 0$$ Thanks.
11
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6answers
1k views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
11
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1answer
47k views

What is the Fourier transform of $f(x)=e^{-x^2}$?

I remember there is a special rule for this kind of function, but I can't remember what it was. Does anyone know?
11
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3answers
2k views

Rigorous derivation/explanation of delta function representation?

I am interested in a derivation of the following representation for the Dirac delta function: $$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$ It is clear to me how the property $\...
11
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2answers
3k views

Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?

It should not be a good argument but rather a short one and one that convinces a physicist ( so no need for mathematical rigor ) that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$ ...
11
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5answers
9k views

Extracting exact frequencies from FFT output

Say I pass 512 samples into my FFT My microphone spits out data at 10KHz, so this represents 1/20s. (So the lowest frequency FFT would pick up would be 40Hz). The FFT will return an array of 512 ...
11
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2answers
966 views

A Fourier transform of a continuous $L^1$ function

I know that for every function $f\in L^1(R)$ its Fourier transform has the following properties: it is continuous, it's bounded, and we have the limit $$ \lim_{\omega\to\pm\infty} \hat f (\omega) = 0 $...
11
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5answers
19k views

How to interpret Fourier Transform result?

Can anybody tell me what result of discrete fourier transform means? I know all theoretical stuff and pretty graphs, that it is a change of domain from time to frequency and so on. But I want to ...
11
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2answers
37k views

Fourier Series of $f(x) = x$

I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = x^2$...
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3answers
32k views

Fourier transform of a triangular pulse

I've been practicing some fourier transform questions and stumbled on this one; To start off, I defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as shown ...
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2answers
2k 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 \infty} ...
11
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1answer
2k views

How do we know the complex exponentials “span” the set of all real functions?

So, we know if $L^2 (0,2\pi)$ is the space of all $2\pi$ periodic square-integrable functions, ie all functions that have finite energy: $$ \int_0^{2\pi} |f(x)|^2dx < \infty $$ Then those ...
11
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3answers
24k views

Fourier transform of even/odd function

How can I show that the Fourier transform of an even integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is even real-valued function? And the Fourier transform of an odd integrable function $f\...
11
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3answers
398 views

The boundedness of an integral

Is there a constant $C$ which is independent of real numbers $a,b,N$, such that $$\left| {\int_{-N}^N \dfrac{e^{i(ax^2+bx)}-1}{x}dx} \right| \le C?$$
11
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4answers
6k views

Numerical Approximation of the Continuous Fourier Transform

Given a function $F(k)$ in frequency space (sufficiently nice enough, eg. a Gaussian), I would like to compute its Fourier inverse \begin{equation}f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}F(...
11
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1answer
1k views

Isoperimetric inequality implies Wirtinger's inequality

Let $C: x=x(t), y=y(t), a\le t\le b$ be a $C^1$ closed curve (not necessarily simple).The isoperimetric inequality says that $$ A\le \frac{\ell^2}{4\pi},$$ where $$A=\left|\int_C y(t)x'(t) dt\...
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2answers
349 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 ...
11
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1answer
687 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is $C^{\infty}(\...
11
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1answer
1k views

Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
11
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1answer
213 views

Is $\int_0^\infty \left (\int_0^\infty f(k) k \sin kr \, \mathrm dk \right) \mathrm dr = \int_0^\infty f(k) \, \mathrm dk$ correct?

I am a physicist, and as a physicist I have proved the following equality: $$ \int_0^\infty \left (\int_0^\infty f(k) k \sin kr \,dk \right) dr = \int_0^\infty f(k) dk, $$ where $f$ is a rapidly ...
11
votes
1answer
376 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 $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ Show ...
11
votes
1answer
153 views

Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: “...
11
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0answers
195 views

The number $\pi$ in an unexpected context

[This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.] Drawing shapes by some predefined Fourier series I found this square ...
11
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0answers
206 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
11
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1answer
394 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, h\rangle_{L^2[...
10
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4answers
591 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}} \int_0^...
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5answers
4k views

Does the phrase “instantaneous frequency” make sense?

I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a ...
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5answers
2k 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 ...
10
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4answers
37k views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = 0$$...
10
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3answers
15k views

Sum of Sinusoids with Same Frequency = Sinusoid (proof)

I am studying Fourier analysis on my own, I realised that probably the first thing you want to proof in Fourier transform is that the sum of 2 sinuoids (namely a sine and cosine) with the same ...
10
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3answers
15k views

The Fourier transform of a “comb function” is a comb function?

Let $f(x) = \sum_{n=-\infty}^{\infty} \delta(x - n)$, where $\delta$ is the Dirac delta function. This function $f$ (a "comb function") is important in signal processing because evenly sampling a ...
10
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2answers
4k views

What will be the support of the convolution of two test functions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
10
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2answers
2k views

How to find the sum $1+\frac{1}{2}-\frac{1}{4}-\frac{1}{5}+\frac{1}{7}+\frac{1}{8}-\frac{1}{10}-\frac{1}{11}+\cdots =\ ?$

Let $\phi(x)=\begin{cases}0, & 0\lt x\lt 1\\ 1, & 1\lt x\lt3 \end{cases}$ We have that the Fourier cosine series is given by $$\phi(x)=\begin{cases}0, & 0\lt x\lt1\\ \frac{4}{3}+\...
10
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2answers
439 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 ...
10
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2answers
16k views

What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
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2answers
2k views

Fourier Transform of $\ln(f(t))$

I want to compute Fourier transform of $\ln(f(t))$ maybe in a sense of distributions? Where we can assume that: $f(t) > 0$ $f(t) \in L^1$ $f(t)$ is continuous $\lim_{t \to \infty} f(t)=0$ and $\...
10
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1answer
937 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 ...
10
votes
1answer
211 views

Finding the period of a nonlinear ODE

I am studying a periodic physical system with a nonlinear ODE $$x''=f(x)+g(x)x'^2$$ I think the periodicity comes from the $x'^2$ term because this provides two possible numbers to give a same right ...
10
votes
3answers
1k views

Calculate the Fourier transform of $\log |x| $

How can one prove that the Fourier transform of $\log |x|$ is $$-\pi \mathrm{pf} \frac{1}{|\xi|} +C \delta,$$ where $\mathrm{pf}\frac{1}{|x|} = D(\mathrm{sign}(x)\log|x|)$ (in the sense of ...
10
votes
2answers
3k views

Is there a way to relate prime numbers and the fourier transform

According to what I know about Fourier transforms, any continuous periodic signal can be represented as a combination of sine and cosine functions. To me, this looks analogous to the "Fundamental ...
10
votes
2answers
4k views

Which function's Fourier transform is the function itself?

We know that the Fourier transform of a Gaussian function is Gaussian function itself. Can anyone give one or more functions which have themselves as Fourier transform?