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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Integral (Fourier transform) of Heaviside radial function in 3D

I am trying to calculate the following integral: $ \int \frac{d k_x d k_y d k_z}{(2 \pi)^3} \left[ \exp( - \frac{(k_x^2 + k_y^2 + k_z^2) \sigma^2}{2}) + \frac{1}{2} H(\sqrt{k_x^2 + k_y^2 + k_z^2} - ...
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46 views

Approximation by sinc functions in L2

I wish to find the best approximation in $ L_{2} (\Re )$ of $f(x)=\frac{sin(ax)}{ax}$ for $0<a<\pi$ and for $a>\pi$ , Using the system of sinc functions: $$g_{n}(x)=sinc(\pi x-\pi n) = ...
2
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1answer
39 views

Why this formula doesn't work for $n=1$?

I've been studying Fourier series and in trying to compute the Fourier series for the function $f: (-\pi,\pi)\to \mathbb{R}$ given by $f(x)=|\sin x|$ I've found something quite strange that I'm not ...
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45 views

For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$.

As the title states: For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$ where $C_c^\infty(\mathbb{R})$ is the ...
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1answer
39 views

how to disprove uniform convergence

I've been asked to check the uniform convergence of the following function sequence on the real line: $$ f_{N}(t)=\sum_{n=-N}^{n=N}\sin(n) \,\frac{\sin(\pi t-\pi n)}{\pi t-\pi n} $$ It is asked in a ...
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35 views

Find the Fourier transform of the given memory function in the limit volume $V\rightarrow\infty$

The memory function is given by, \begin{equation} \mu (t)=(8\pi e^{2}/3V)\sum_{\vec{k}}|f_{\vec{k}}|^{2}\cos (ckt) \end{equation} where $V$ is the volume, $f_{\vec{k}}$ is the form factor. In this ...
2
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25 views

Fourier transform of a constant - Is it a function of distance only?

I have the following power spectrum $P(k)$, function of the modulus $k$ of the vector $\vec{k}$ in Fourier space: $$P(k) = \begin{cases} P_0 \exp (-\frac{k^2 \sigma^2}{2}) & \text{for} \; k \leq ...
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0answers
32 views

Another equivalent characterization of Schwartz function?

Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \sup_{x\in\mathbb{R}^n}\left||x|^k\Delta^{p}\psi(x)\right|<\infty $$ for all ...
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21 views

If $\{ \phi_n \} $ is an orthonormal basis for $L^2(c,d)$, then $\{ \phi_n \circ f \}$ is an orthonormal basis for $L^2_{f'}(a,b)$

Suppose $f : [a,b] \to [c,d]$ and $f'(x) > 0$ for $ x \in [a,b]$. Show that if $\{ \phi_n \}$ is an orthonormal basis for $L^2 (c,d)$, then $ \{ \phi_n \circ f \} $ is an orthonormal basis for ...
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52 views

Extending an identity for the Dirac delta function

The identity $$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$$ can easily be derived from the generalized Leibnitz formula for $n$ and $p$ positive integers: $$\int \; x^p ...
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1answer
62 views

About PDE problem

Let $ \, u(x,t) : \Bbb{R}\times(0,\infty) \rightarrow\Bbb{R}\, $ be $ \,C^2 \,$ such that $$ {\partial u\over \partial t} (x,t)-{\partial^2 u \over \partial x^2}(x,t)+x^2 ...
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1answer
27 views

Finding the Fourier series of a function

Let $\hat f(n)$ be the fourier coefficients of a a function $f$. Let $g(x)=\int_{0}^{x}f(t)dt$. I'm asked to find the fourier series of $g$. Is it correct to say that since $g' (x)=f(x)$, and ...
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57 views

confused with the FFT output

I am taking some sensor output and doing fft on it. how to get the exact frequencies from the complex output? my understanding is that bin frequencies and the input frequencies are different. Please ...
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Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
2
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1answer
26 views

fourier transform with radial coordinate

I am wanting to find the transform of this: $$f(r) = \frac{e^{-\alpha r}}{r}$$ where $r$ is the radial coordinate. And then I would like to find $\lim_{\alpha \to \infty}$. I have this: ...
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0answers
45 views

Cross-correlation of Gaussian and Jacobian sums

I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them. Let $F$ be a finite field with $q > 2$ elements ...
2
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1answer
35 views

Fourier Expansion

A periodic function f(x) is defined by: $ f(n) = \begin{cases} {-x^2} & \textrm{ for - π < x ≤ 0} \\ x^2 & \textrm{ for 0 ≤ x < π } \\ \end{cases} \space , ...
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1answer
38 views

What does it mean to “simultaneously localize a function in time and frequency domain”?

What does the statement "simultaneously localize a function in time and frequency domain" mean when it comes to signals and systems? What does it mean to localize?
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60 views
+100

solving 2nd order pde with dirac delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ ...
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2answers
216 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)}} ...
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3answers
158 views

Scaling property of Laplace transform

I am not sure how to do the following problem: Let $$\hat{F}(s)=\mathfrak{L}(f(t))$$ be the Laplace transform of $f(t)$. Show that: $$\mathfrak{L}(f(at))=\frac{1}{a}\hat{F}\left(\frac{s}{a}\right) ...
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1answer
14 views

correspondence between linear functional and function

Any Schwartz or $L^p$ function $g$ can be identified with a linear functional via which way? $T_g(f)=\int gf$ or $T_g(f)=\int g\bar f$ ? I have seen these two different definitions in different ...
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1answer
30 views

Norm triangle inequality for convolutions proof

I'm trying to prove that $$\|f*g\|_{L_1}\le{\|f\|_{L_1}\|g\|_{L_1}}$$ with respect to a Haar measure over a group G. Using Fubini's theorem, I'm up to ...
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1answer
23 views

Poisson kernel approximates to the identity

The question comes from Shakarchi and Stien's real analysis p.111, he said that the Poisson kernel for the disc approximates to the identity without proof. However, I don't see that fact immediately, ...
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0answers
15 views

Good Kernel's Properties

I am recently studying properties about a good kernel, and came across a problem. Definition: A kernel $K_\delta$ is 'good' if they are Lebesgue integrable and satisfy the following conditions for ...
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0answers
29 views

Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = ...
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2answers
53 views

Finding the Fourier Transform with Radial Coordinate

I am not sure how find the Fourier Transform of: $$f(r) = \frac{e^{-\alpha r}}{r}$$ where $r$ is the radial coordinate. And then I would like to find $\lim_{\alpha \to \infty}$.
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49 views

Principal component analysis (PCA) results in sinusoids, what is the underlying cause?

Background I'm analysing a data set of $M$ flow measurements (volume per time). The flows go from zero mL/s gradually to higher values and back to zero again, thus: their shapes ideally look like a ...
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1answer
37 views

find the number of times the function is differentiable

find an upper and lower limit for the number of the times the following functions are differentiable and that their derivative is continious $$f(x) = \sum_1^\infty \frac{e^{inx}}{n^4}$$ $$g(x) = ...
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1answer
86 views

Solve integral using Plancherel's formula

This is from a test in Fourier analysis: Define $$ f(\xi)=\int_0^1 \sqrt{x}\rm{sin}(\xi x) \rm{d}x $$ Calculate $$ \int_{-\infty}^\infty |f(x)|^2 \rm{d}x $$ I started with Plancherel's ...
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2answers
54 views

An inequality concerning the Fourier transform

I want to know that the inequality $$ \left| \int_{-\infty}^\infty\frac{f(\xi) e^{ix\xi}}{\sqrt{1+\xi^2}} d\xi\right| \le C \left| \int_{-\infty}^\infty f(\xi) e^{ix\xi}d\xi\right| $$ holds or not ...
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0answers
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Fourier transform of $e^{-at^2}= \frac{1}{\sqrt{2a}}e^{\frac{-w^2}{4a}}$ [duplicate]

I got this answer from wolfram math. But I need to know how this solution?
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1answer
30 views

Integration By Parts on a Fourier Transform

I'm having trouble with the "An integration by parts in $x$ for the first summand...and the assumption that $\phi$ goes to $0$ as $|x|\to\infty$." I tried the integration by parts but ended up with ...
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A question on the conjugate property of Fourier transform

I'm wondering that I can say $$ \mathscr{F}^{-1}(1+|\xi|^2)^{1/2}\mathscr{F} \overline\phi = \overline{\mathscr{F}^{-1}(1+|\xi|^2)^{1/2}\mathscr{F}\phi} $$ where $\mathscr F$ means the Fourier ...
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1answer
52 views

How to understand this notation on Fourier transformation?

For a function $f$, recall the Fourier trnasformation $\widehat{f}(u)=\int_R e^{iux}f(x)dx$ (Maybe someone call it Fourier inversion, but it doesn't matter). Now let $T$ be a bounded self-adjoint ...
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Fourier transform of an almost periodic function.

An almost periodic function $f(x)$ is a tempered distribution. Is its (distributional) Fourier transform a signed measure, $\hat{f}(\xi) = \sum_n c_n \delta(\xi-\xi_n)$, where the $\xi_n$ are the ...
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0answers
48 views

How to show that $\int_{-\infty}^{\infty} \mathrm{d}^3 \textbf{k} \frac {e^{i \textbf{k x}}} {(2 \pi)^3} = \delta^3(x)$ in spherical coordinates?

Recently I had to deal with Fourier transformations and delta functions, and I was wondering how about that. I know, that its trivial to show in cartesian coordinates, but i couldn't do it in ...
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1answer
53 views

Why is the Fourier transform of $f(x)=x$ on $[0,1]$, $0$ otherwise, apparently not square integrable?

Let the function $f(x)=x$ if $x\in[0,1]$ and $f(x)=0$ otherwise. The function is in $L_2$. Unless mistaken, the Fourier transform of f is $\hat f(\xi)=\int_0^1{xe^{ix\xi}dx}=e^{i\xi}(1/\xi^2-i/\xi) - ...
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Fourier transform of $f(x)=x$ if $0<x\leq 1$ and $f(x)=0$ otherwise

What is the Fourier transform of the function defined by $f(x)=x$ on $[0,1]$ and $f(x)=0$ otherwise, i.e., $\hat f(\xi) = \int_\mathbb{R} { e^{-iu\xi} f(u) du }$? Is there a closed-form? Else, how ...
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0answers
41 views

About an inequality concerning the Fourier transform

I want to show that the inequality $$ \sup_{x\in \mathbb R}\left|\int_{\mathbb R} e^{ix\xi} |\xi| f(\xi)\,d\xi \right| \le C \sup_{x\in \mathbb R}\left|\int_{\mathbb R} e^{ix \xi} (1+\xi^2 )^{1/2} ...
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0answers
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Is $X(T) = A \sin(\omega_0 t + \Phi)$ mean ergodic?

This is an example of a tutorial but I think has not been solved properly. Please help me! $X(T) = A \sin(\omega_0 t + \Phi)$ $A$ and $\phi$ are independent $A$ is uniformly distributed over ...
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2answers
35 views

how they prove this Fourier Transform of unit impulse function

In my text book , to prove $\ \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{j\omega t}d\omega $ behaves like unit impulse function they evaluate the integral : $\ ...
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1answer
40 views

Fourier transform of $1-\frac{|\tau|}{2T}$

So far I have tried the following: $$\begin{align} \mathscr{F}(f)&=\mathscr{F}\{1-\frac{|\tau|}{2T}\}\\ &=\int_{-\infty}^{+\infty}(1-\frac{|\tau|}{2T})e^{-i\omega\tau}d\tau\\ ...
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1answer
38 views

Fourier series expansion of $x(t) = \sum\nolimits_{z \in \mathbb{Z}} (-1)^z \delta(t - 2z)$

Find the Fourier series expansion of $x(t) = \sum\nolimits_{z \in \mathbb{Z}} (-1)^z \delta(t - 2z)$, where $\delta(\cdot)$ denotes the Dirac delta function (unit impulse). I can infer that the ...
2
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2answers
61 views

Fourier transform to find an harmonic function (Strauss)

I am trying to solve one of the problems of section 12.4 of the book "Partial Differential Equations" by Strauss. The problem says: Use the Fourier transfor in the $x$ variable to find the harmonic ...
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2answers
83 views

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: ...
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42 views

Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
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1answer
36 views

Product of an $L^\infty$ function and an $L^1$ function is integrable

For every $f \in L^1(\mathbb{R^n})$ let $$ \hat{f} : \mathbb{R}^n \to \mathbb{C}, \quad \hat{f}(\xi) := \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) \exp{(-i \langle x, \xi \rangle)} ...
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0answers
35 views

What functions stem from Fourier Series with rational-only coefficients?

Given the Fourier series $$f(z) = \sum_{k=-\infty}^\infty c_k e^{ikz}$$ but with $c_k\in(\mathbb Q+ i\mathbb Q)$ instead of $\mathbb C$ (or even purely real), are the functions obtained this way in ...
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1answer
16 views

Can an arbitrary real function be written in terms of quadratures of an arbitrary frequency with time dependent coefficients?

Given a real function $f$, and a frequency $\Omega$, is it the case that there exist two other real functions $I$ and $Q$ such that $f$ can be written as $$f(t) = I(t) \cos(\Omega t) - Q(t) ...