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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2answers
665 views

How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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933 views

Fundamental solution to the Poisson equation by Fourier transform

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ u(x)=\begin{...
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1answer
306 views

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} \sqrt{1-k^...
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959 views

How to justify the solution of this problem?

Assume $\mathbf{x} \in \mathbb R_+^N$ with support $P=\{p_1,p_2,\cdots,p_K\}$ ($P$ is unknown). We already know that $$f_1(\mathbf{x}) = f_2(\mathbf{x}) = \cdots = f_{N-1}(\mathbf{x})$$ where $$f_l(\...
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235 views

Show that $\int_a^b |f(x)|^2 \,\mathrm dx \le \frac{(b-a)^2}{\pi^2}\int_a^b |f'(x)|^2 \,\mathrm dx$

$\def\d{\mathrm{d}}$Show that if $f \in C^1[a,b]$ and $f(a)=f(b)=0$, then $$\int_a^b |f(x)|^2 \,\d x \le \frac{(b-a)^2}{\pi^2}\int_a^b |f'(x)|^2 \,\d x.$$ By a change of variable, it suffices to ...
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0answers
557 views

Discrete Fourier Transform: Shift by Fraction of a Sample

I use spectral Fourier methods to numerically solve PDEs and these methods make heavy use of the discrete Fourier transform (DFT). With respect to the DFT I have some issues understanding the discrete ...
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1answer
260 views

Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
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Prove: Fourier series of $e^{\cos x} \sin (\sin x)$ is $\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$

I'd love your help with proving that the following series $$\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$$ is the Fourier series of $e^{\cos x} \sin (\sin x)$. I tried to find $\hat f(n)$ using ...
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7k views

Why is $8 \times 8$ matrix chosen for Discrete Cosine Transform?

In JPEG and MPEG, why is $8 \times 8$ matrix chosen for Discrete Cosine Transform? Why not any other, say $64 \times 64$?
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5answers
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What does it mean for two functions to be orthogonal? [duplicate]

When two finite dimensional vectors are orthogonal, i.e. perpendicular, their dot product is exactly zero, e.g. $$\mathbf{a}\cdot\mathbf{b}=a_1b_1+\cdots+a_nb_n=0.\tag{1}$$ When I studied functional ...
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How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
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3answers
2k views

Fourier transform of a compactly supported function

In which space does the Fourier transform of a smooth compactly supported function $\phi$ lie? I would not say it lies in $\mathcal{S}$, heuristically as one can approximate the step function which is ...
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1answer
4k views

Does rapid decay of Fourier coefficients imply smoothness?

Under the isomorphism of Hilbert spaces $L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n$, smooth functions on the circle are mapped to rapidly decaying sequences (see wikipedia). Is the ...
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4answers
9k views

“Every function can be represented as a Fourier series”?

It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series. So this got me thinking about the ...
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19k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
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2answers
4k views

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the ...
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2answers
6k views

Fourier transform of the error function, erf (x)

I define $\text{erf}(x):=\frac{2}{\sqrt{\pi}}\int_0^xe^{-\xi^2}d\xi$. What is its Fourier transform (unitary, ordinary frequency)? That is, simplify $$\frac{2}{\sqrt{\pi}}\int_{-\infty}^\infty\int_0^...
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140 views

Looking for guidance on a Fourier integral

Working with a Fourier transform problem, I've encountered the following integral: $$ \int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx $$ where $a$, $b$, and $c$ are real ...
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Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $$y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$$ I am having some ...
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Uniform convergence of Fourier Series

I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand: Given that we know the series $f(x) = \sum c_k e^{ikx}$ ...
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1answer
936 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 $f\in L_p (\mathbb{T})$: $$\lim_{t\to 0}\;\|f(\cdot + t) - f\|_p = 0.$$ How do I prove it? I think it is intuitively ...
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269 views

Is the Fourier series always the “best” approximation?

In Stein and Sharkachi's Introductory Fourier Analysis book, in Chapter 3, we're given the "Best Approximation Theorem". The theorem says that if $f$ is Riemann integrable on the circle and we ...
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404 views

Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$

Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \...
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How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

Without using Fourier transforms, how do I derive the spectral measure for $A=\frac{1}{i}\frac{d}{dt}$ on the domain $\mathcal{D}(A)$ consisting of absolutely continuous functions $f\in L^{2}(\mathbb{...
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2answers
14k views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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Fourier transform of the Cantor function

Let $f:[0,1] \to [0,1]$ be the Cantor function. Extend $f$ to all of $\mathbb R$ by setting $f(x)=0$ on $\mathbb R \setminus [0,1]$. Calculate the Fourier transform of $f$ $$ \hat f(x)= \int f(t) e^{-...
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1answer
2k views

Comparing/Contrasting Cosine and Fourier Transforms

What are the differences between a (discrete) cosine transform and a (discrete) Fourier transform? I know the former is used in JPEG encoding, while the latter plays a big part in signal and image ...
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For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
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1answer
2k views

Pointwise but not uniform convergence of a Fourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
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199 views

“Bad” Fourier Series derivation

Let $f(\theta)$ $2\pi$-periodic such that $f(\theta)=e^{\theta}$ for $-\pi<0<\pi$, and $$e^{\theta}=\sum_{n=-\infty}^{\infty}c_{n}e^{in\theta}\,\,\, \mathrm{for}\,\, |\theta|<\pi $$ it's ...
8
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1answer
382 views

Fourier transform of meromorphic function

Suppose that I have a function $f(z)$ which is meromorphic on the entire complex plane, meaning holomorphic everywhere except for a discrete set of poles. I then take a vertical slice of this ...
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3answers
385 views

Fourier Transform of $\frac{1}{\sqrt{|x|}}$

I want to find the fourier transform of $\frac{1}{\sqrt{|x|}}$. I checked the table of common fourier transforms in Wikipedia, and I know the answer should be $$\sqrt{\frac{2\pi}{|\omega|}}$$ What I ...
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1answer
1k views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
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4answers
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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 (...
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1answer
118 views

Why is the plot of $f(t)=\frac{\partial}{\partial t}\left\{\sin(\sin(\pi t))\right\}$ so similar to a triangle wave?

I was playing around the other day and I found that the function $$t\to f(t), f(t)=\frac{\partial}{\partial t}\left\{\sin(\sin(\pi t))\right\}$$ seemed to be very close to the triangle wave. Is there ...
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1answer
177 views

Show that $f_n\to f$ uniformly on $\mathbb{R}$

Let $$P_n(x) = \frac{n}{1+n^2x^2}$$. First, I had to prove that $$\int_{-\infty}^\infty P_n(x)\ dx = \pi$$ And that for any $\delta > 0$: $$\lim_{n\to\infty} \int_\delta^\infty P_n(x)\ dx = \...
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What are discrete and fast Fourier transform intuitively?

I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them.
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400 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
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1answer
108 views

Almost-identity: $[\int_0^\infty{\rm d}x-\sum_{x=1}^\infty] \prod_{k=0}^N\text{sinc}\left(\frac{x}{2k+1}\right) = \frac{1}{2}$

Show that the identity $$\int_0^\infty \prod_{k=0}^N \text{sinc}\left(\frac{x}{2k+1}\right)\,{\rm d}x - \sum_{n=1}^\infty \prod_{k=0}^N \text{sinc}\left(\frac{n}{2k+1}\right) = \frac{1}{2}$$ where $\...
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1answer
459 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
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1answer
760 views

Two definitions of Fourier transform for $L^1$ and $L^2$ coincide

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
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2answers
249 views

What are the properties of the fourier transform of a phase-only function?

Given a function of the form: $$ f(x) = e^{i\phi(x)} | \phi(x)\in\Re $$ What are the properties of its Fourier transform? For instance, purely real functions have Fourier transforms with symmetric ...
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1answer
2k views

Show the Fourier transform is continuous in the Schwartz space $\mathcal S(\Bbb R)$

Show the Fourier transform $\mathcal F$ is continuous in the Schwartz space $\mathcal S(\Bbb R)$. Use the standard $\mathcal S$-norms $$ \|f\|_{a,b}=\sup_{x \in \Bbb R} \left| x^af^{(b)}(x)\right|, \, ...
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1answer
760 views

Exercise 22, Chapter 5 of Stein and Shakarchi's Fourier Analysis

I am working through Stein and Shakarchi's Fourier Analysis and am stuck on Exercise 22 of Chapter 5, which I quote below. Preliminary notation: $\mathcal{S}$ is the Schwartz space of functions on $\...
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2answers
494 views

Eigenfunction of the Fourier transform

I want to show that $$ \frac{1}{ \sqrt{ 2 \pi}} \int_{-\infty}^{\infty} \frac{e^{-iwx}}{\cosh{ (x \sqrt{\frac{\pi}{2}}} ) } = \frac{1}{\cosh{ (w \sqrt{\frac{\pi}{2}}} ) } .$$ My attempt is to ...
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3answers
290 views

Is there a name for a function whose square is an involution?

An involution is a function $f:X\to X$ such that $f\circ f=\text{id}$. Is there a name for a function $g:X\to X$ such that $f\equiv g\circ g$ is an involution? An example is multiplication by $\pm i$ ...
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2answers
2k views

Hilbert transform and Fourier transform

Assume the following relationship between the Hilbert and Fourier transforms: $$ \mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)), $$ where $ \displaystyle [\mathcal{H}...
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3answers
272 views

Fourier transform of $e^{-i\lambda\sqrt{1+x^2}}$ - asymptotics for $\lambda$?

As the title says: I want to compute the Fourier transform (in the distributional sense) of $f(x)=e^{-i\sqrt{1+x^2}}$, $x\in \mathbb{R}^n$ - say $n=1$ for the moment. I have no idea how to get it done:...
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1answer
220 views

Is there a continuous waveform that sounds the same as a square wave?

The fourier series $$f(t)=\sum_{n\in\mathbb N\\n\text{ odd}}\frac1n\,\sin(nt)$$ converges to a square wave. Square waves are discontinuous functions. I'm wondering if there's a continuous function ...
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1answer
51 views

Closed epicycloids

Consider a system of $n+1$ celestial bodies - e.g. a sun, a planet, a moon, a satellite of the moon and a satellite of the satellite - which run around each other on circles in the same plane but with ...