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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Why does the Fourier transform include the base of the natural logarithm, the square root of -1 and $\pi$?

The formula itself, as a vector of summations of products of the original coefficients with some weight, itself a function the original and transformed coefficient indices, is not a hard pattern to ...
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18 views

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin t}{t}J_m(t)\,dt$

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin(t)}{t}J_m(t)\,dt$ We know that $\hat{\chi_{S^{n-1}}}(rx)=(2\pi)^\frac{n}{2}\frac{J_\frac{n-2}{2}(r|x|)}{(r|x|)^\frac{n-2}{2}} \iff ...
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Prove that $\frac{d}{dx}(z^kJ_k(z))=z^kJ_{k-1}(z)$ for every $ k \geq \frac{1}{2}$.

Prove that $\frac{d}{dx}(z^kJ_k(z))=z^kJ_{k-1}(z)$ for every $ k \geq \frac{1}{2}$. Then write an explicit expression for (a) $J_\frac{1}{2}(x)-J_\frac{5}{2}(x)$, and (b) an expression for $J_1+2J_3$ ...
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20 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
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48 views

Finding Fourier transform of $\frac{x}{(x^2 + 4)^2}$

So I have this function $$ f(x) = \frac{x}{(x^2 + 4)^2} $$ and I have to find its Fourier transform. This is however much harder than what I have done before so I don't have a clue where to start. I ...
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28 views

Parseval's theorem.

We consider two signal $h(t)$ and $g(t)$ such that $$\int_{-\infty}^\infty |g(t)|^2dt<+\infty$$ $$\int_{-\infty}^\infty |f(t)|^2dt<+\infty$$ Parseval's theorem states that: ...
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18 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
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46 views

Calculating $\int_{-\pi}^{+\pi} e^{ixt} e^{-i \omega t} dt$

We know that Fourier Transform of $e^{ixt}$, where $x$ is a real parameter, $t\in \mathbb R$ is $$\int_{-\infty}^{+\infty} e^{ixt} e^{-i \omega t} dt=\int_{-\infty}^{+\infty} e^{ixt-i \omega t} ...
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1answer
25 views

Fourier transform of $L^2$ function

Let us the definition of Fourier transform $$\hat f(\lambda) = \int_{-\infty}^\infty f(t) \exp(- i \lambda t) dt$$ How do I change this expression if $f(t)\in L^2[-\pi,\pi]$? $$\hat f(\lambda) = ...
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36 views

Proving pointwise convergence of Fourier series [on hold]

I'm trying to prove the pointwise convergence of a $2\pi$-periodic function $f$ to its Fourier expansion. The proof on my lecture notes stops at this formula: $$f(t)-P_{N,f}=\frac 1 {2 \pi} ...
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11 views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
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1answer
23 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
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1answer
20 views

How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in ...
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1answer
17 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
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1answer
24 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
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25 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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19 views

Which of the functions below define a tempered distribution? The functions in a, b, c are defined in R.

Which of the functions below define a tempered distribution? The functions in a, b, c are defined in R. a) $f(x)=x^3+3x$ $$|T(\phi)| \leq \int_R |(x^3+3x)\phi(x)|dx=\int_R |x(x^2+3)\phi(x)|dx \leq ...
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30 views

In general, let $f \in L^2(-1,1)$ and let $g:R \to R$, $g(x)=f( \{ x \} )$. How are the Fourier transforms of f and g related?

Recall that $\{ x \}$ is a decimal part of a real number $x$. For example, if $x=3.41$, then $\{ x \}=-0.41$. Part A: Let $f(x)=2x$, with $x \in (-1,1)$ and let $g:R \to R$, $g(x)=2 \{ x \}$. Sketch ...
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35 views

Use the Fourier inversion formula to compute h(x) when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$

Use the Fourier inversion formula to compute $h(x)$ when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(x)=\frac{1}{(1+y^2)^2}=\frac{1}{1+y^2} \times \frac{1}{1+y^2}=\widehat{\frac{1}{2}e^{-|x|}} \times ...
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63 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
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11 views

Find the Fourier Transform of a Riesz Basis.

Find the Fourier Transform of a Riesz Basis for $L^2[-\pi,\pi]$: $e^{ix_nt}$, where $x_n$ is a sequence of real numbers, $t\in \mathbb R$. I state that I am not very expert in Riesz Basis and Fourier ...
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2answers
45 views

Find the Fourier Transform of $e^{i x t}$.

Find the Fourier Transform of $e^{ixt}$, where $x$ is a real parameter, $t\in \mathbb R$. I started writing: $$\int_{-\infty}^\infty e^{ixt} e^{-i\omega t}dt$$ but I do not know how to go on!
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33 views

Parseval's theorem to $\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$.

Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's ...
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1answer
17 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
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$\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with ...
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Check: Find $h(x)$ when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$.

Find h(x) when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(y)=\frac{1}{(1+y^2)^2} =\frac{1}{1+y^2} \times \frac{1}{1+y^2} =\hat{\frac{1}{2}e^{-|x|}} \times \hat{\frac{1}{2}e^{-|x|}}$$ Let ...
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Check: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. [on hold]

Part A: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. Part B: Generalize the previous problem and deduce a formula for the Fourier transform of a polynomial of degree m. ...
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1answer
90 views

Is $\hat S$ a function?

Let $S(x)=\sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x)$. Find the Fourier transformation of $S(x)$. Is $\hat{S}$ a function? $$\hat{S}=\int_R \sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x) ...
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25 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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9 views

impulses in domain region and something in frequency region

I am studying about the fourier series and fourier transform. and I am suffering to deal with the picture that I will show you I am sure that the signal upper in the picture has periodicity and ...
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11 views

the multidimensional Hilbert transforms or partial Hilbert transforms

The one-dimensional Hilbert transform can be defined by the convolution $Hf:=f*\frac{1}{\pi x}$, or can be given by Fourier multiplier $(Hf)\hat{\,}(\xi)=-i\mathrm{sgn}(\xi)\hat{f}(\xi)$. ...
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some questions on the soluction of the Dirichlet's problem in the unit disk

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
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52 views

Fractional powers of the operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$, $Bf = f-f^{''}$.

Consider the linear operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$ defined by the following mapping: $Bf = f-f^{\prime\prime} \equiv (I-\Delta)f$, where $\Delta$ is the Laplace operator that ...
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58 views

Showing that Gaussians are eigenfunctions of the Fourier transform

I'm having a bit of trouble on this problem: I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar ...
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39 views

A continuous function which does not converge to its Fourier series. [duplicate]

Where can I find an example (or the theorem) for a continuous function which does not converge pointwise to its Fourier series, as well as its explanation? I would prefer a web page or a free site or ...
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3answers
41 views

Characterization of $\mathscr{S}(\mathbb R^n)$?

Consider the vector space $$\displaystyle\mathscr{S}(\mathbb R^n)=\{f\in C^\infty(\mathbb R^n): \lim_{|x|\to \infty} |x^\alpha \partial^\beta \phi(x)|=0, \forall \alpha, \beta\in\mathbb N_0^n\}.$$ ...
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Under what condition on the constants c and c' are the boundary conditions …? [closed]

Under what condition on the constants $c$ and $c'$ are the boundary conditions $f(b)=c\cdot f(a)$ and $f'(b)=c'\cdot f'(a)$ self-adjoint for the operator $L(f)=(rf')'+pf$ on $[a,b]$? (Assume as usual ...
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91 views

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$. So if I create a number line marking a-b and a+b. If that the integral above has 5 different answers depending on where (-t,t) is located on the number ...
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Eigenvalues of rotation invariant operators on 2-sphere

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore ...
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41 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
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1answer
24 views

Interpreting Fourier transform frequency graph

I've been trying to understand Fourier transform for some time now and I think I've perhaps finally got the idea now. What I would like to do now is to make an example of Fourier transform for ...
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1answer
33 views

Given Tf(x), find the equivalent operator m(k)f^(k) in the Fourier transform sense.

Let $f\in L^2(\mathbb R)$, let $$ g(x) = Tf(x) = \int^{x+1}_{x} f(s)ds $$ Find $m\in L^\infty(\mathbb R)$ such that $\hat g(k)=m(k) \hat f(k)$. Use this to show that $T$ is a bounded linear operator ...
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14 views

Top and bottom power spectral density of a height profile

Imagine I have a simple 1D height profile which is NOT symmetric. Now, what is truly important for me is to know what are the frequency content of the top profile (i.e. a cut profile above the ...
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1answer
56 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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33 views

Estimate involving almost orthogonality

Let $\{x_k\}$ be a $\frac{1}{R}$-separated set of points on $S^{d-1}$. Then $$ \left\|\sum_ka_ke^{ix_k\cdot\xi}\right\|_{L^2\left(B\left(0,R\right)\right)}\lesssim R^{d/2} ...
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12 views

Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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14 views

Nyquist stability criterion with Fourier Transform

I want to ask if I have a system of partial differential delay equations that I have linearized and then found the transfer function in fourier space, is it possible to analyze this using the Nyquist ...
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1answer
28 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
2
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1answer
36 views

To What Extent is the Fourier Inversion Theorem Due to the Self-Adjointedness of the Laplacian

I've tried looking this up (I looked at various spectral theorems) but couldn't find anything that talks about the connection between Fourier transforms and the eigenfunctions of the Laplacian (we may ...
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1answer
26 views

How to interpret the imaginary part of an inverse fourier transform

The fourier transform of arbitrary real data can (usually will) result in complex data. If the real data represents samples in time, then the complex FT data represent frequencies with the magnitudes ...