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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Motivation for proof of Berry-Esséen Theorem

The proof of the Berry-Esseen theorem found in Terence Tao's notes (https://terrytao.wordpress.com/2010/01/05/254a-notes-2-the-central-limit-theorem/, Theorem 37) starts by "smoothing" the cumulative ...
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Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since $$\sin{...
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What is the Fourier transform of spherical harmonics?

What is the definition (or some sources) of the Fourier transform of spherical harmonics?
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A list of proofs of Fourier inversion formula

The reason for this question is to make a list of the known proofs (or proof ideas) of Fourier inversion formula for functions $f\in L^1(\mathbb{R})$ (obviously adding appropriate hypothesis to get a ...
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$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : \|(\langle{\xi}\...
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Why is the Fourier transform self-inverse?

I've seen the standard proof that the Fourier transform is self-inverse (up to an overall factor determined by conventions), which is essentially equivalent to $\int_{-\infty}^\infty e^{2 \pi i \...
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average of maximal function is less than its infimum?

Let M be the dyadic Hardy-Littlewood maximal operator. Prove the following: there is a constant $C$ such that for any $f$, $$ \inf_{x\in I}Mf(x)\le C 2^k\inf_{x\in J} Mf(x) $$ where $I$ and $J$ are ...
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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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Solving the heat equation with Fourier Transformations

Can anyone help me with this IVP heat equation problem? I have $$u_t-u_{xx}=g(x,t)$$ where $x \in \mathbb{R}$, $t>0$, $u(x,0)=0$ So i've found by taking a Fourier transformation that $$\hat{u_t}...
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How to perform a Fourier transform in spherical coordinates?

For a function $f(r, \vartheta, \varphi)$ given in spherical coordinates, how can the Fourier transform be calculated best? Possible ideas: express $(r,\vartheta,\varphi)$ in cartesian coordinates, ...
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What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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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 $u_0\in L^2(\mathbb{R}^d)$ ...
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Why is this integral is super-exponentially small?

Consider the integral $$I_n^{(a,b)} = \int_{-1}^1 (1-x)^a\,(1+x)^b\, P_n(x)\, dx,$$ where $P_n(x)$ is the $n$-th Legendre polynomial. Here's a plot of $|I_n^{(50,20)}|$ for $n=0,\dots,70$: (I just ...
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Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
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Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so $$2I=\int_{-\...
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Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being $\frac{...
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$\int_{-\infty}^\infty e^{ikx}dx$ equals what?

What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? I saw this in the Fourier transform, and am ...
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Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
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Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
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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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Characteristic function of the normal distribution

The standard normal distribution $$f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}},$$ has the characteristic function $$\int_{-\infty}^\infty f(x) e^{itx} dx = e^{-\frac{t^2}{2}}$$ and this can be ...
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Why $\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iw(t-x)}dw$ is the Dirac delta function?

I'm reading a book that says: $$f(x) = \int_{-\infty}^{\infty}f(t)\left\{\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iw(t-x)}dw\right\}dt$$ and then says that the term in curly brackets can be seen as ...
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Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
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Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
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Convolution of a function with itself

Function $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I ...
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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 ...
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For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
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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 ...
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What is the best approach when things seem hopeless?

I'm finding that I am getting to the point of being hopelessly behind in one of my courses. What is the best thing to do when it feels impossible to get caught up in the literal sense. Being "caught ...
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Fourier Transform of $\frac{1}{(1+x^2)^2}$

I need to find the Fourier Transform of $f(x) = \frac{1}{(1+x^2)^2}$ Where the Fourier Tranform is of $f$ is denoted as $\hat{f}$, where $\hat{f}$ is defined as $$\hat{f}(y)=\int_\mathbb{R}f(x)e^{-...
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Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
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Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$.     &...
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Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
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What Is Spectral Leakage?

Can someone, please, explain in simple words what spectral leakage is ? I am interested in the case where a window is used to reduce the truncation error of a Fourier transform of a finite signal. ...
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On the root of $\cos (a_1x) + \cdots + \cos (a_nx) = 0$

This is a problem I was trying to solve for a while with no succeed. Show that the equation $\cos (a_1x) + \cdots + \cos (a_nx) = 0$ has at least one solution in $[0,\frac {\pi}{a_1}]$, where $0 < ...
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A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here $a,b\...
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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 $\...
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Are the Euler-Maclaurin formula and the Poisson summation formula related?

The Euler-Maclaurin formula, beautifully explained here by Justin Rheinstadter is expressed as: $$\sum_{i=m}^{n}f(i)=\int_{m}^nf(x)dx\;-\frac{1}{2}\left(f(n) - f(m)\right)\;+\sum_{k=1}^{p}\frac{B_{2k}...
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How to prove that inverse Fourier transform of “1” is delta funstion?

$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.
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A function is $L^2$-differentiable if and only if $\xi\widehat{f}(\xi) \in L^2$.

In this previous question, I defined $L^p$ derivatives of functions in $L^p(\mathbb{R}^n)$. I've been struggling for a while now to prove the following: If $f \in L^2(\mathbb{R})$, then $f$ is $L^2$...
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What's the Fourier transform of these functions?

The Fourier transform of $|x|^{\alpha}$. This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; ...
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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 ...
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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 ...
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$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
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$\|\hat{f} \|_{\infty} = \lim _ {n \rightarrow \infty} (\|f^{(n)}\|_1)^{1/n}$

Let $f \in L^2 \cap L^1$ on the Real line, and define $f^{(n)}$ to be the $n$-fold convolution $f \circ f ... \circ f $. I want to show that $||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||...
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Absolute convergence of fourier series

Let's say I have a piecewise continuous function which has the fourier series $\sum_n\ c_{n}e^{inx}$ and I assume that $ \sum_n\ n|c_{n}|$ converges, then I know the following holds: The fourier ...
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Existence of a function

I need some help: I am thinking about this problem. Any advice would be appreciated. Let's fix $\epsilon>0$. Does there exists some $f\in C^0([0,\pi])$ such that: $f\mid_{[\epsilon,\pi-\epsilon]}...
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Why does a fourier series have a 1/2 in front of the a_0 coefficient

I am reading up on the fourier series, and I keep seeing it as being defined as: $$ f(\theta)= \frac{1}{2}a_0 + \sum_{n=1}^{\infty}(a_n \cos(n\theta) + b_n \sin(n\theta)) $$ where $$ a_n = \frac{1}{...
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A mathematical way to represent an image kernel?

How to represent the calculation in this image mathematically? For example: With the discrete convolution and Fourier Transform. It tries to do a calculation on the original image (image $A$/input) ...
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Recommended books/links for Fourier Transform beginners?

I am a student taking engineering course and wish to learn more about Fourier Transforms. It seems very useful. Would highly appreciate it if anyone could advise me where to start.