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 is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
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61 views

Create periodic function from combining non-periodic functions

I'm studying recurrent neural networks which often use tanh as an activator function which is not periodic. However in research and papers it's shown that these recurrent neural nets can exhibit ...
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44 views

Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
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15 views

Fractional Sobolev space on an interval

Consider a fractional Sobolev space $H_p^t(I)$ defined on an interval $I\subset \mathbb{R}$. When $I=\mathbb{R}$ the space can be defined via Fourier transform. Is it possible to do it when $I=(-1,1)...
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9 views

Sobolev spaces on non-compact manifolds — independence on charts

Are there some standard references where basic facts about fractional-order (or at least integer-order) Sobolev spaces on non-compact manifolds are treated? More precisely I would like to be able to ...
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31 views

Evaluation of an integral associated with integral kernel of resolvent of Laplacian

I came across evaluating the following sort of integral when I was considering the integral kernel for resolvent of Laplacian $(I-\Delta)^{-1}$: $$ K(x)=\int_0^{\infty}\frac{\exp(-t-\frac{|x|^2}{4t})}{...
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22 views

Fourier decomposition of solutions of the wave equation with respect to the spatial variable

Say I have a wave equation of the form $$\nabla^{2}f(t,\mathbf{x})=\frac{1}{v^{2}}\frac{\partial^{2}f(t,\mathbf{x})}{\partial t^{2}}$$ which is clearly a partial differential equation (PDE) in $\...
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34 views

Théorie de Fourier in Sontag`s book

I was reading Sontag`s In America and she mentions: "La théorie de Fourier sur les douze passions radicales.." What is this theorem about?
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24 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ P(p_i,x)=\...
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12 views

Fourier transform of a continuous periodic spectrum of frequencies

Suppose I have a function of the form $$ f(t) = \exp(i\phi(t)) $$ where $$ \phi(t) = \int_0^t\omega(t) \ dt + \phi_0 $$ is the phase of the function and $\omega$ is the angular frequency, which is ...
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41 views

Discrepancy in Discrete Fourier Transform Algorithm Formula?

I'm having a bit of trouble with a small part of the following formula (taken from this page): $$F_k=\sum_{n=0}^{N-1}f(x_n)e^{-(2\pi i)k\frac{n}{N}}\tag{1}$$ This formula is supposed to ...
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40 views

Evaluate $\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$ where $g(x) = \int_{0}^x f(t) dt$

Let $f$ be a $2\pi$-periodic function such that $\int_{-\pi}^\pi f(t) dt = 0$. Define $g(x) = \int_{0}^x f(t) dt$. Evaluate $$\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$$ I hope the integral is equal ...
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24 views

Show that $\int_{-\infty}^{\infty}f(\xi+i\eta,z_2,\ldots,z_n)e^{i[t_1(\xi+i\eta)+t_2z_2+\cdots+t_nz_n]}d\xi$ is independent of $\eta$

Show that $$\int_{-\infty}^\infty f(\xi+i\eta,z_2,\ldots,z_n) e^{i[t_1(\xi+i\eta) + t_2z_2+\cdots+t_nz_n]} \, d\xi$$ is independent of $\eta$, for arbitrary real $t_1,\cdots,t_n$ and complex $z_1,\...
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12 views

Scale of Oscillations

I'm reading an article which claims the following result : (paragraph 2.2) for a function $f = \sin (N g(x)) h(x) $ where $g$ and $h$ are $C^{\infty}$ scalar functions non oscilattory and $N$ a large ...
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290 views

Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain calculations,...
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20 views

Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
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36 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation $\...
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29 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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8 views

How describe functions with finite bandwidth?

What is the sufficient and necessary conditions for a $f:\mathbb R\to\mathbb R$ has finite bandwidth (Fourier spectrum is non-zero on a bounded interval)? I'm guess this equivalent $f$ is continuous ...
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22 views

An example that does not satisfy the conditions of the Fourier inversion theorem?

Here is the Fourier inversion theorem page in Wikipedia. It states that for every function $f(x)$ that satisfy some conditions ($f(x)$ can be a function such as a Schwartz function, an integrable ...
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14 views

Support of the Fourier transform of $\int_{[0, 1]}f(\xi)e^{-2\pi i(\xi x_1 + \xi^2 x_2)}\, d\xi$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Schwartz function. Let $$F(x_1, x_2) := \int_{[0, 1]}f(\xi)e^{-2\pi i(\xi x_1 + \xi^2 x_2)}\, d\xi.$$ In other words, $F$ is the Fourier transform of $f$...
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20 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) +...
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30 views

$x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$

I am trying to establish the following $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$ The right sight looks the the Fourier expansion of ...
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684 views

heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
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23 views

Controlling $\dot W_{k,1}$ norm of a schwarz function

If $\phi \in \mathscr{S}(\mathbb{R})$ then does it follow that there is $c$ s.t. $||\nabla^k \phi||_{L_1} \leq c^k$? From the definition I have only been able to get that this is true for all $k$ in ...
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44 views

Linearspan of Gaussians dense in Schwartz space

as the title already says I am trying to show that the linear span "A" of the gaussians $e^{\frac{-|x|^2}{2}}$ and their translations/ dilations are dense in the Schwartzspace. This is the space of ...
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1answer
25 views

Does $\sum_{i=1}^n \alpha_ie^{2\pi i\xi_ix}\equiv0$ imply $\forall i,\alpha_i=0$?

Let $\alpha_1,\dotsc,\alpha_n$ be complex numbers. Let $\xi_1,\dotsc,\xi_n$ be distinct real numbers. Define a function $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=\sum_{i=1}^n \alpha_ie^{2\pi i\...
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22 views

Proving an identity involving Fourier coefficient

If $f \sim \sum A_n e^{inx}$ and $g \sim \sum a_n e^{inx}$ and $f,g$ are continuous $2\pi$ periodic functions, show that $$\int_{-\pi}^\pi f(t) \overline{g(t)} dt = \sum_{-\infty}^\infty A_n \...
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15 views

Variant: Bounding Fourier coefficients in terms of supremum norm

This is a variant on this answered question. Let $\alpha_1,\dotsc,\alpha_n$ and $\beta_1,\dotsc,\beta_n$ be real numbers satisfying: $\alpha_i,\beta_i \geq 0$ for every $i$, $\sum_{i=1}^n\alpha_i=1$ ...
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81 views

What is the name of that theorem?

Here is the statement : Let $f:\mathbb{R}\to \mathbb{C}$ a continuous map which is $\mathcal{C}^1$ by pieces and such that $f\in \mathcal{L}^1(\mathbb{R})$. Moreover, $\hat f \equiv 0$ in $\...
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33 views

Bounding Fourier coefficients in terms of supremum norm

Let $\gamma_1,\dotsc,\gamma_n$ be nonnegative real numbers. Let $\eta_1,\dotsc,\eta_n$ be real numbers. Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by: $$f(x)=\sum_{i=1}^n \...
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15 views

Significance of the complex conjugation symmetry of the DFT for real-valued input

For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that $$X_{N-k} = X_k^*$$ where * denotes complex ...
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66 views

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert ...
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39 views

Folland 8.20 (Fourier Analysis)

I'm stuck a bit on this problem from Folland: The first part I can't figure out at all. The second part, I know: $\|Pf(x)\|_1 = |Pf(x)| = |\int f(x,y)dy| \leq \int |f(x,y)|dy$. If the last term is ...
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18 views

Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that $...
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35 views

Schwartz function whose Fourier transform is compactly supported and $\geq 1$ on the unit ball.

I need to construct such a function but the closest I have come to is to take $f(t) = e^{-|t|}, t\in\Bbb{R^d}$. But its Fourier transform is not compactly supported as is $\hat{f}(x) = \frac{2}{1+x^2}$...
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17 views

DFT confusion about complex conjugation in forward process

I learning about the DFT and there's one thing that's sort of confusing me, I hope it's not too dumb of a question! I understand that the DFT involves a process of taking an input signal vector and ...
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1answer
18 views

Computing the Fourier Transform of the square pulse

The function in question is $f(x) = H(a - |x|)$, where the Fourier transform is given by $F(k) = (\frac{2}{k}) \sin(ak)$. Initial attempt: $F(k) = ( H(a - |x|), e^{-ikx} )$ = $- (\delta(a - |x|), -\...
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How is the Fourier transform a generalization to the Fourier series?

I have taken a self-tought course on the subject of Fourier series and Fourier transform and I got the message that the latter is a generalization of the first. I know that the idea that the Fourier ...
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27 views

Coefficients of N-dimensional Chebyshev polynomial using discrete cosine transform

I am using Chebyshev polynomials to interpolate a multidimensional function f(x,y,z). I sample f on the Chebyshev roots grid and want to obtain the coefficients of the interpolation polynomial. I now ...
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555 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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59 views

Fourier transform of distribution

Let $f\in S_{\infty}$ be a Schwartz function and let us define a linear functional,for any $\varphi\in S_{\infty}$, $S_{\infty}\to\mathbb{C}$, $\varphi\mapsto (f,\varphi)$ by$$(f,\varphi):=\int_{\...
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684 views

Does $f(x)\in L^1$ imply that $\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0$?

Suppose that $f(x)$ is $L^1$ and R- integrable function, problem is to resolve if it is possible existence of such a $f(x)$ that: $$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ...
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Fourier transform of translation in $L^2$.

For a function $f : \mathbb{R} \longrightarrow \mathbb{R}$, let $(\tau_y f)(x) = f(x - y)$. If $f \in L^1(\mathbb{R})$, then it is straightforward to show that $\widehat{\tau_y f}(\xi) = e^{-2\pi j \...
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81 views

Fourier sine transform of $\frac{1}{x}$

Fourier sine transform of $\displaystyle \frac{1}{x}$ is ..(fill in the blanks).. My thoughts: By Definition, $\displaystyle F_s(s)=\int_{0}^{\infty}f(x)\sin sxdx $ $\displaystyle F_s(s)=\int_{0}^{\...
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30 views

Finding a Fourier Transform

I need some help with the following question. If f has a Fourier transform F(k), what is the Fourier transform of cos(x)f(2x+1). I have made pretty much no progress on this. This seems ...
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23 views

Function Singularity in a Sobolev Space

For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$? UPD: actually I am ...
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41 views

Integral with Dirac delta function

We are given that: $$u(x,t)=\frac{2}{\pi} \int_0^\infty e^{-k^2t}G_s(k)\sin {kx}\space\text{d}k,$$ where $G_s(k)$ is the Fourier sine transform of $g(x)$. Find the solution $u(x,t)$ when $g(x)=\delta(...
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16 views

Poissions Equation (Laplace)

$$\begin{align} u''_{xx}&+u_{yy}= x, \quad 0<x<1, \quad 0<y<1,\\ \\ u(x,0)&=u(x,1) = 0, \\ u(0,y)&=u(1,y) = 0,\\ \end{align}$$ Having some problems with Poissons Equation. I'...
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16 views

How to solve the following partial differntial equation using fourier transform?

How to solve this equation? $$2\iota n_0k_0 \frac {\partial E_x}{\partial y}=\frac {\partial^2 E_x}{\partial x^2} + \frac{\partial^2 E_x}{\partial z^2} $$ where, $E_x(x,y,z)$, $n_0$ and $k_0$ are ...