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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Evaluate the limit of function presented as a series

This is an additional exercise given in my Fourier analysis course. Define $F(t)=\sum_{n=-\infty}^\infty (-1)^n e^{-2n^2t^2}, \,t>0$, Prove that $\lim_{t\to \infty}F(t)=1$. ...
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38 views

Evaluate integral using Fourier analysis

$\int_0^\infty \frac{\cos (x)}{1+4x^2}\, dx$ $\int_0^\infty \frac{1}{(1+x^2)^2}\, dx$ There is no hint for these two questions. I think for Q2, since it's a square, I can use Plancherel ...
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13 views

Fourier coefficients of three times differentiable functions.

I was wondering, can we determine somehow the decay of the Fourier coefficients of a function $g \in C^3\mathbb{(T)}$ /three times continuously differentiable/ as $|n|\rightarrow \infty$? Any help ...
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11 views

fourier transforms of $e^{(-kw^2+b)t}$

I was solving and PED using fourier transforms and reached this point: $$v(x,t)=f(t)*F^{-1}\big[e^{(-kw^2+b)t}\big]$$ $F^{-1}$ denotes inverse fourier transforms, and $*$ is used for convolution. I ...
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22 views

Continuity of Fourier-Stieltjes transform

I read in Kolmogorov-Fomin's (p. 419 here) that, if $F$ is a function having bounded variation on $\mathbb{R}$ then the Fourier-Stieltjes transform$$g(\lambda):=\int_{-\infty}^\infty e^{-i\lambda ...
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137 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 \| ...
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30 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
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11 views

Solving DFT of this array

I am told to solve the DFT of the following array (5,5,5,5) However, according to the answer sheet, it is suppose to be (5,0,0,0). I tried working it out by hand, and F(0) was correct. But my ...
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1answer
18 views

Convolution with heaviside function, argument of the heaviside carry on to the dirac function?

So I have this equation to demonstrate: $$ x(t)*u(t)= \int_{-\infty}^t x(\tau)d\tau $$ , where $u(t)=\int_{-\infty}^t \delta(\tau)d\tau$ I opened the convolution as $ \int_{-\infty}^\infty ...
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11 views

Inverse Fourier Transform with Duality

I need to find the inverse Fourier transform of the following equation using the duality property: $X(w)=\begin{cases} 2w+2 &\mbox{if} -1<w<0 \\ -2w+2 &\mbox{if }0<w<1 \\ 0 ...
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25 views

Inversion formula for $\int_{\mathbb{R}}f(x)e^{-izx}dx$

Let $f:\mathbb{R}\to\mathbb{C}$ be a measurable function such that$$\forall x\ge 0\quad|f(x)|<Ce^{\gamma_0 x}$$$$\forall x<0\quad f(x)=0$$I must specify that all the integrals I am going to ...
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75 views

Relation between Fourier components of a positive function

Here's a problem that has recently come up in my physics research: Let f be a function on [0, 2 $\pi$], which yields positive real numbers. Let the integral of $\int_0^{2\pi}f(x)= 1$. (Just for the ...
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12 views

Wiener's lemma and Hulanicki's lemma

Let $\mathcal{A}(\mathbf{T})$ be the Banach algebra of continuous complex-valued functions on the unit circle with absolutely convergent Fourier series. Then Wiener's lemma states that if $f \in ...
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14 views

Fourier series, even and odd n properties

I recently started learning about Fourier series, so I'm still kind of shaky on the topic Given $f(t)=f(t+T)$ and $$f(t)=\sum_{n=-\infty}^{\infty}F_ne^{jn \omega_0t}$$ Show that: (a) If ...
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60 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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1answer
27 views

Support of polynomial distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
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61 views

Fourier Transform of $\exp(-t)$

$$f(t)= \begin{cases} e^{-t} & 0<t<1 \\ 0 & \text{otherwise} \end{cases}$$ How can I solve this function's Fourier transform? I am stuck at here: Daniel R - OP \begin{align} ...
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15 views

Existence of certain function in Schwartz space

Suppose a polynomial $P(x_1,\cdots,x_n)$ is given. Does there exist a function $\phi\in\mathcal{S}(\mathbb{R}^n)$ such that supp$(\phi)\subset\mathbb{S}^{n-1}$ and ...
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17 views

One simple question about Fourier transformation of system of PDE's

Let's assume set of equations $$ \tag 1 \frac{\partial \mathbf A}{\partial t} = \Delta \mathbf A + a [\nabla \times \mathbf A] - d\mathbf b_{k} (\mathbf b_{k} \cdot \mathbf A), \quad \mathbf A(0) = 0 ...
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40 views

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$.

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$. Normally, I would do is find a matrix representation and ...
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13 views

Solvability of eigenvalue problem with Schwartz data

Fix $a\in\mathbb{R}$ and define the operator $T$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ by sending $\phi$ to $\Delta\phi-a^2\phi$. Then $T$ is clearly a bounded operator.Question is ...
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77 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
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61 views

Fourier transform inversion formula for $f\in L_1(\mathbb{R}^n)$ and Dini condition

Let us define the Dini condition for a function $f\in L_1(-\infty,\infty)$, i.e. Lebesgue summable on $\mathbb{R}$, as Given an $x\in\mathbb{R}$ there is a $\delta>0$ such that the Lebesgue ...
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4 views

Short-Time-Fourier-Transform: why overlapping the window?

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I ...
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33 views

Orthonormal system

Let $\varphi\in L^2(\mathbb{R})$, prove that $\{e^{2\pi i m x}\varphi(x)\}$ is an orthonormal system iff $$\sum_{n\in\mathbb{Z}}|\varphi(x-n)|^2=1 \ \ a.e \ x$$ How do you prove this. The hint is ...
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31 views

Parseval's Identity holds for all $x\in H$ implies $H$ is a Schauder basis

Prove that any set $\{v_j\}_{j \in \mathbb{Z}}$ for which the Parseval identity $\|x\|^2=\sum_{j=1}^\infty |\langle v_j,x\rangle|^2$ holds for every $x \in H$ is a Schauder basis. I know that a ...
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12 views

integration concening Fourier transfom variable and space variable

We define the short time Fourier transform as follows: $$V_{g}f(x,w)=\int_{\mathbb R} f(t)g(t-x)e^{-2\pi itw} dt, (x,w \in \mathbb R).$$ (We may assume that $f$ and $g$ nice functions so that every ...
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33 views

Voltage Distribution Inside a Cylinder [closed]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
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9 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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16 views

Fourier transform of $|x|^{-s}$

Using the definition of Fourier transform $\hat{f}(p) = (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(x) e^{ix \cdot p} \ dx$ where $u \in \mathbb{R}^n$. What is the fourier transform of $|x|^{-s}$.
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Absolute continuity as a condition for $F[f^{(k)}](\lambda)=(i\lambda)^k F[f](\lambda)$

In read in Kolmogorov-Fomin's (p. 429 here) that if function $f$ is such that $f^{(k-1)}$ is absolutely continuous on any interval and if $f,...,f^{(k)}\in L_1(-\infty,\infty)$, [...] we get ...
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25 views

Fourier rotation theorem in higher dimensions

Let $F(\mu, \nu)$ denote the Fourier transform of $f(x,y)$, then the (2D) Fourier rotation theorem says that the Fourier transform of a rotated function $f(x \cos \theta + y \sin \theta, -x \sin ...
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29 views

Fourier Transform of Newton's Law of Cooling

I am attempting to solve Newton's Law of Cooling differential equation with Fourier Transforms for a high school math report. Can Fourier Transforms be used to solve first-order ODEs? The equation is: ...
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24 views

How to find the Fourier Transform of the form $\frac{cos(2\pi t)}{t^2}$?

I'm having trouble on figuring out how find the Fourier Transform of the following function, and I'm not allowed to use the straight up definition of the Fourier Transform but rather use it's ...
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26 views

If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?

Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}>0: ...
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1answer
39 views

Does the Fourier Transform exist for f(t) = 1/t?

My professor says that the following function has a Fourier Transform: $$f(t) = \frac{1}{\pi t}$$ He said that all I have to do is apply some of the Fourier Transform properties and not the direct ...
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1answer
27 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
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25 views

Fourier transformation of $\cos^2(\pi t)$

I need to fouriertransform the function $$f(t) = \cos^2(\pi t)$$ First question is if there is some kind of theorem to solve it in a very easy way but I don't know what will happen to the ...
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Fourier transform in reconstruction problem

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in ...
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25 views

To bound a heat equation on a real line?

Let $\displaystyle\mathcal{H}_{t}(x)=\frac{1}{(4\pi t)^{1/2}}e^{-x^{2}/4t}$ be the Heat Kernel. The imposed initial condition for the heat equation on a real line is $u(x,0)=f(x)$ a function belongs ...
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81 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
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75 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
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41 views

Finding Fourier cosine series of sine function

I am trying to find Fourier cosine series of following function, but think that I am messing up somewhere. $$ f(x)=\sin \bigg ( \frac{\pi x}{l} \bigg ) $$ Fourier cosine series can be written as $$ ...
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58 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
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24 views

Fourier transforms in combinatorics

I've got something like a "meta-question": In some parts of combinatorics, looking at Fourier transforms can be a very helpful tool. For example, an early proof of Roth's theorem (any sufficiently ...
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16 views

Wavelet Transform of a shift invariant function

I want to calculate the wavelet transform of a shift invariant function. For example Gaussian - $\exp{-\|x-y\|^2_2} $. There is no restriction on the wavelet basis that can be used here. Can anyone ...
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1answer
127 views

Multidimensional Fourier Transform

I'm having difficulty with multidimensional Fourier Transforms. I have the following problem for $u=u(t,x) \in \mathbb{R}$ $$ \frac{\partial u}{\partial t} = \sum_{m,n=1}^d ...
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19 views

Convolution using Fourier Analysis

I need to make the following convolution using Fourier analysis. Evaluate $x(t)*x(t)$ where: a)$x(t)=\frac{1}{T}[u(t+\frac{T}{2})-u(t-\frac{T}{2})]$ where u(t-x) is the heaviside function ...
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1answer
67 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...