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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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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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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How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

The Plancherel Theorem for the Fourier transform $\hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt$ on $\mathbb{R}$ states that $$ \int_{-\infty}^{\infty}|\hat{f}(s)|^...
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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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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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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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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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45 views

Prove that $\mathscr{F}[f] \in L^2(\mathbb{R})$

Let $f \in L^2(\mathbb{R})$ (square integrable functions), I'm trying to prove that his Fourier transform also does: $\mathscr{F}[f] \in L^2(\mathbb{R})$. I have tried to bound it \begin{align} \...
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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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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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36 views

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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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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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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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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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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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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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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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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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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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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Computing Hilbert transform and envelope of a function

The following is a function with $\alpha$ being a real constant $$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$ Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope ...
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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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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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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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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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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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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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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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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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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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If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?

if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
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$x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ 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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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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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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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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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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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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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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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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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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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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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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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 ...
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19 views

If D_m is the mth Dirichlet kernel, $||D_m||_1\to\infty$ as $m\to\infty$

I was working on this problem for my own studying but am stuck on how to solve it. Let $D_m$ be the $m$th Dirichlet kernel. Show that $||D_m||_1\to\infty$ as $m\to\infty$. Anything would help. Thanks
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Is $\{(\epsilon + \cos(x))^{2k}\}_{k\in\mathbb{N}}$ a family of good kernels?

Show that for any $0<\delta<\pi$, $$\lim_{k\to \infty} c_k\int_{\delta<|x|<\pi} \left(\epsilon + \cos(x)\right)^{2k} dx = 0 $$ where $\epsilon >0$ is some small number (for example,...
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36 views

Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
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Calculating mean and standard deviation of frequency given time displacement data

For example, using this data set (sample rate = 200 Hz): https://my.mixtape.moe/mtyocd.gz If I take the power DFT of it (either unwindowed or with the Blackman-Harris window) and zoom in, I can see ...
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For fourier series g(x), prove that the fourier series for the integral G(x) can be found by term-by-term integration of g(x)

I want to prove that if I have a fourier series of the form $g(x) = a_0/2 + {\sum_i}^\infty a_icos(ix) + b_isin(ix) $, the fourier series of G(x) $-x*a_0/2$ can be found by simply integrating g(x) ...
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23 views

Fourier series for absolute value of sin functiom

If we take the absolute value for sin function, then it becomes even. However, isn't period of this function pi? To find fourier series, 1.Even 2. period 2 pi. Can we just treat this function as ...