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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Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
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Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
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Filter on Fourier Series

i have a lowpass filter H(ω) which is $ H(ω) = e^{-jω} $ on -2π≤ω≤2π, and $0$ elsewhere and i have a function in fourier series y(t), i need to find the new signal (z(t)) after the application of the ...
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Inter-neighbor resistance on triangular prism

Given a triangular prism of infinite length along the X direction. A graph is formed with the set of nodes all the points on an edge of the prism with integer values of X, and the with each node ...
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wavelets, fourier-transform [duplicate]

We are given that $f\in C^{p} \ if \int_\mathbb{R}|\hat{f}(\omega)|(1+|\omega|^p)d\omega <+\infty$. Now if $\hat{f}$ has a compact support then how $f\in C^{\infty}?$
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Fourier transform of a 2-D Gaussian on a ring

I need some help obtaining the 2-D Fourier transform of the following function: $$f(r)=e^{-\frac{-2(r-a)^{2}}{w^{2}}}$$ Where $r$ is the polar radius, $a$ and $w$ are positive. So this describes a ...
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Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\circ$ ...
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How can apply the $L^p$ norm in a circle to $L^2$ norm in a square?

Let $f(x_1,x_2)=\frac{1}{(x_1+ix_2)^2} \chi_{ \{ (-1,1) \times (-1,1) \}}$ Part A: Evaluate the $L^p(R^2)$ norm of $f(x_1,x_2)\chi_{ \{ (x_1,x_2): x_1^2+x_2^2 \leq 1 \}}$ for every $1 \leq p \leq ...
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Fourier Transform for option pricing

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and sotck price Brownian motions of call options under stochastic interest rates? So lets ...
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2answers
39 views

Why Fourier transform owns two different signs?

In my book, the defination of Fourier transform is $$F(\lambda)=\int_{-\infty}^{+\infty}f(t)e^{i\lambda t}dt$$ While the reverse one is: ...
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26 views

Use the interpolation theorem to estimate the $L^p$ norm of f(x) when $p>2$.

The maximum of the function $\displaystyle f(x)=\frac{\sin(x)}{x}$ is $1$ and $\displaystyle \int_{-\infty}^\infty(\frac{\sin(x)}{x})^2 dx= \pi$. Use the interpolation theorem to estimate the $L^p$ ...
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1answer
71 views

Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
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28 views

Discrete Time Fourier Transform of a real signal

I want to prove that if we have a real signal x[n] then for the DTFT it is applied that we have an even symmetry: | X(Ω+1/10) | = | X(-(Ω+1/10)) | (I mean the ...
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Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
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1answer
184 views

Can this be simplified?

$$ e^{-i\frac43\pi n} - e^{-i\frac23\pi n}, n\in \mathbb{N} $$ I am trying to simplify this but cant. Any ideas appreciated.
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Existence of Density in Bochner's Thoerem

Bochner theorem for locally compact abelian group, $G$ and a positive definite function $f$ there exist a unique measure $\mu_f$ such that: $$f(x)=\int\limits_{\hat G}(x,\gamma)d\mu_f(\gamma)$$ Where ...
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1answer
42 views

Fourier series of oscillation in form $\cos(2 \pi \frac{k}{T}+\phi)$

I would like to calculate the fourier coefficients of $\cos(2 \pi \frac{k}{T}+\phi)$ where $T \in \mathbb{N}$ is the period and is arbitrary but fixed, $k \in [1, N-1]$ is the number of oscillations ...
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1answer
20 views

Evaluate the $L^p(R^2)$ norm of $f(x_1,x_2) \times \chi_{ \{(x_1,x_2):x_1^2+x_2^2 \leq 1\} }$ [duplicate]

Let $f(x_1,x_2)=\frac{1}{(x_1+ix_2)^2} \times \chi_{(-1,1) \times (-1,1)}(x_1,x_2)$. Evaluate the $L^p(R^2)$ norm of $f(x_1,x_2) \times \chi_{ \{(x_1,x_2):x_1^2+x_2^2 \leq 1\} }$ for every $1 \leq p ...
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1answer
34 views

For which values of $p \geq 1$ is $f \in L^p(\mathrm{R}^2)$?

Let $f(x_1,x_2)=\frac{x_1}{x_2^{1/3}} \times \chi_{\{ [0,1] \times[-1,1]\}}$. Part A: Evaluate $\int_{\mathbb{R}^2} f(x_1,x_2) ~ \mathrm{d}x_1~ \mathrm{d}x_2$ $\int_{\mathbb{R}^2} f(x_1,x_2)~ ...
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77 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
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34 views

Fourier Transform of exp(-a|x-.5|)

So I've been working on the fourier transform of $exp(-a|x-\frac{1}{2}|)$ (with $a>0$) and keep getting: $\left(e^{-\pi i}\right)\left(\frac{2a}{a^+4\pi^2x^2}\right)$. A research partner keeps ...
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39 views

Levy processes, vanilla option and Fourier Transform

The context to this problem is mathematical finance, although the answer does not need specific knowledge of the area. I am trying to work out the expression for the price of a call option using Levy ...
3
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1answer
38 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
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1answer
36 views

How can we represent an image using basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But i don't understand terms like "orthogonal " and "basis ...
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49 views

Fourier transform of a function

I'm struggling with FT, I just can't grasp the concept of it. Can somebody explain it on an example Ex 1: $f(t) = e^{-|t|}$ EX 2: $x(t) = \cos(\pi t/T)$ where it's different from $0$ just on ...
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Are the dominant frequencies preserved under fractional inversion

Let $f(t)$ be a signal that is a function of time. Let $F(f)=\mathcal{F}\{f(t)\}$ be the Fourier transform of $f(t)$. If $F(f)$ is dominated by a sparse set of frequencies $(f_1,f_2,\cdots,f_n)$ (only ...
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Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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Does the integral in the formal 2D Fourier transform of the logarithm converge?

If $k$ is a nonzero vector in $\mathbb R^2$, how to interpret this integral: $$\int_{\mathbb R^2}e^{ik\cdot x}\ln{|x|}dx$$ Does it converge and in what sense? Thanks in advance.
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42 views

Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
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42 views

Why are divergent Fourier series all so 'HARD'?

I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and ...
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1answer
33 views

A function sequence converge to a Fourier series implies point-wise converge?

Assume $f(x)$ is a smooth $2\pi$ periodic function and can be decomposed as $$f(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$ A function sequence $f_m(x)$ satisfies $$\int_0^{2\pi}f_m(x)e^{-inx}dx\to ...
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Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
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1answer
40 views

Proof of Fourier series Theorem (k-continuous derivatives)

Here's the theorem: Theorem: If $f$ is periodic with Fourier coefficients $a_n,b_n$ and if the series $$\sum_{n=1}^\infty (|n^{k}a_n|+|n^{k}b_n|)$$ converges for some integer $k \geq 1$, then f ...
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Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
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Pseudodifferential operators and amplitudes

I am studying psudodifferential operators on $\mathbb{R}^n$. Let $U\subset \mathbb{R}^n$ an open subset. A function is $b\in C^\infty(U\times U\times U \times \mathbb{R}^n)$ is an amplitude of order ...
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1answer
33 views

Discrete Fourier Transform

I am studying DFT and am having trouble with the notation system. The frequency is from $0$ to $2B$ - in DFT the frequency domain does not have negative frequencies. But if this is the case, and we ...
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20 views

Fourier Transform by generalized distribution

I am having one little doubt on the subject. When we are defining Fourier Transform via generalized distribution on test functions, is this mandatory that the pairing has to be defined for all ...
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2answers
26 views

$\int\exp(-jnw_0t)\,dt$ integral calculus.

I seem to forgot these parts of integral calculus. I am trying to determine the Fourier coefficient in complex exponential form. Here, $t$ is the variable being integrated and $n$ is for all ...
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1answer
34 views

How to calculate the value of $\int_{-\infty}^{\infty} y(t)dt$?

For a function $g(t)$, $\int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2}$ for any real value $\omega$. If $y(t)=\int_{-\infty}^{t}g(\tau)d\tau$ then how to calculate the value of ...
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1answer
31 views

Fourier Transform and its Inverse

Could anyone show me how to prove the following results about Fourier Transform, please? It is stated in my book without proof. Thank you. Let $\mathcal F$ denote the Fourier linear operator and $f$ ...
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1answer
19 views

N-point FFT and 2-radix FFT

I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length $2^n$) For example a is a sequence: ...
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Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
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How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
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Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
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What does the Fourier transform of $1/x^2$ mean?

If I ask Mathematica to compute the Fourier transform of $\frac{1}{x^2}$ using the FourierTransform function, it gives me a result of ...
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1answer
38 views

intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
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26 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
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51 views

Fourier transform of exp(cos)

How do I calculate the Fourier transform ($t \rightarrow \omega$) of the following: $\exp(A\cos(\omega_0 t))$ $A$ is a real constant, and $\omega_0$ is a real and positive constant. I know that this ...
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1answer
33 views

a question about Fourier transforms

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
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2D Fourier Transform proof of Similarity Theorem

I have to solve an exercise, but if i could use the following theorem, it would be piece of cake Similarity Theorem if $ \mathscr{F}\{g(x,y)\}= G( f_x,f_y)$ then $ \mathscr{F}\{g(ax,by)\}= \frac ...