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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trigonometric series

It is known that the eigenvalues of Sturm liouville problem: $$ u''(x)+\lambda u(x)=0 \\ u(0)=u'(\pi)=0 $$ are $\sin\left(\left(\frac{1}{2}+n\right)x\right)$ for $n=0,1...$ If for example we expand ...
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Bit operations to count longest string of 1s in a binary number - connections to FFT?

I found this rather applied question on another forum. How to calculate size of largest string of consecutive 1s in a binary number. However the other forum had neither much of a focus on applied ...
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Can the DTFT be Pi-periodic?

Given is the following property of the DTFT of a time-discrete signal x[n]: $X(e^{j*\theta}) = X(e^{j*(\theta-\pi)})$ In my opinion this DTFT has a period of $\pi$ and not $2\pi$, as the definition ...
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about the property of Fourier transform??

It is said that: $$F[\frac{df(x)}{dx}] = i\omega F(\omega)$$. This expression depends on the initial definition of Fourier transform, yes? if I define Fourier transform as: ...
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Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
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Fourier transform of $f(x+h)$?

Show that $f(x+h)\to \hat{f}(w)e^{2\pi i h w}$ Let $g(y) = f(x+h)$, then $\hat{g}(w) = \int_{-\infty}^\infty g(y) e^{-2\pi i y w} dy = \int_{-\infty}^\infty f(x+h) e^{-2\pi i (x+h) w} dx$, then ...
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How does one find the Fourier Series for a non-periodic function on an arbitrary interval $[-\frac{L}{2},\frac{L}{2}]$ using the complex exponential?

I was given three functions, and told to find the coefficients of their Fourier Series using $\tilde{f_k} = \frac{1}{\sqrt{L}}\int_{-\frac{L}{2}}^{\frac{L}{2}} f(x) e^{i2\pi kx/L}dx$ where ...
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Fourier transform of integral related to zeta function

In this MO question here, I asked about the Fourier transform of the zeta function. The second answer lists the following as a representation for $\zeta(s)$, with $E(x)$ as the floor function: ...
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Extracting Information From Fourier Transforms

I have some data that I believe can be analyzed best by taking the Fourier Transform and analyzing the resulting data in the new domain. In particular, I'm trying to determine if a frequency of ...
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Proof verification : regarding pointwise and norm convergence of a fourier sequence of $L^2$ function

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 $f$. could you verify the proof? ...
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Sampling a sinusoidal signal

Consider the signal $g(t)=\cos(2\pi \lambda t+\phi)$ that is sampled with a frequency $\tau$. Let $g_k$ denote the values of $g$ at the times $t_k=\frac{k}{\tau}$, $k \in \mathbb{N}$. (a) Show that ...
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Finding explicit solution to $-\Delta u + u =f$ using Fourier Transform

This is a question from a previous year's qualifying exam, so it's possible we haven't covered all the material this year in order to solve this problem (we did not discuss PDEs in the class so far, ...
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Fourier transform of function defining half an ellipse

I'm trying to determine the expression for the Fourier transform of a function defining half an ellipse. It's been awhile since I've done Fourier transforms by hand. Obviously I can plug the ...
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Approximating Fourier transform for range of output frequencies

(This may be an elementary question, I am new to Fourier analysis.) I am working on a visualization tool. I have a real function $f(x)$, given by N samples on some interval, and vanishing outside ...
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Is there a Plancherel-type identity for generalized Fourier Transforms?

Let $S$ be in $\mathcal{T}$, the set of tempered distributions, and $\mathcal{F}S$ be its Fourier Transform. Is there some relationship for such distributions, analogous to the Plancherel Theorem for ...
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About the Fourier-Legendre series of $f(x)=e^{-x}$

So for the function $f(x) = \exp(-x)$ I have the formula for the coefficients of $$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$ which is(by using Rodrigues formula) $$a_n = \frac{2n+1}{2} ...
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$L^1$ functions approximated by non-decreasing continuous sequences

Actually the origin problem is: Suppose $f \in L^1([0,1])$, prove that there are two non-decreasing sequences of continuous functions ${g_k},{h_k}$ which are $a.e.$ bounded, and $$f(x)=\lim_{k \to ...
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Fourier Transform pdes

I have an exam next week and I was hoping someone might be able to help me out with this question. Show that the Fourier transform of the function $f(t+a)$ is $e^{iwa}\hat{f}(w)$ . There is a list of ...
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Confusion with fourier coeffients

Consider $f(t) = \frac{\pi - t}{2}$, $t \in [0, 2\pi]$ The complex fourier coefficients are $c_n = \frac{1}{2\pi}\int_0^{2\pi}\frac{\pi - t}{2}e^{-int}dt$ Which turns out to be $-\frac{i}{2n}$ if im ...
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When the fourier series equal to the original function?

Let $f\in L^2([-1/2,1/2])$. Define $a_n=\int_{[-1/2,1/2]} f(x) e^{-2\pi i n x} dx$ for each $n\in\mathbb{Z}$. Define $S_N(x)=\sum_{n=-N}^N a_n e^{2\pi i n x}$ for each $N\in \mathbb{Z}^+$ and $x\in ...
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Recommend resources for understanding Phase spectrum

I am learning Fourier transform. if we apply Fourier transform on a signal, we get magnitude spectrum and phase spectrum. I want to learn phase spectrum part in detail. So can anyone recommend any ...
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Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
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Find the Fourier transform of $\sin x^2$.

I've tried it by applying integratrion by parts, but I'm not getting the answer correct. Its answer is $$\frac{1}{\sqrt{2}}\,\sin\left(\frac{k^2}{4} +\frac{\pi}{4}\right).$$ Please help in this.
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How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
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Fourier Transform of a radial function in $L^1(\mathbb{R}^2)$ [duplicate]

Let $f \in L^1(\mathbb{R}^2)$ be radial, i.e. there exists $g: [0,\infty) \rightarrow \mathbb{R}$ such that $f(x) = g(|x|)$. Prove that $f$ is also radial. (Note that this result is true for ...
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Fourier Transform of operator

Let $A = (a_{jk})$ be a real $n \times n$ matrix. Let $u \in C^2(\mathbb{R}^n)$ and define $L_Au \in C(\mathbb{R}^n)$ by: $$L_Au = -\sum_{j,k=1}^n a_{jk}\partial^2_{x_jx_j}u$$ Assume $u(\infty) = ...
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Understanding Theorem $7$.$23$ of Rudin's Functional Analysis

The Theorem states the following: (a) If $ u\in D'(\mathbb{R^n})$ has its support in $rB$, if $u$ has order $N$ and if $f(z)=u(e_{-z})$ where $z \in \mathbb{C^n}$, then $f$ is entire, the resriction ...
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Discrete Fourier transform implementation giving results that are order of magnitude off

I tried implementing a Discrete Fourier transform in Matlab, but I found my results an order of magnitude off. I used next definition of DFT: $$ F(u) = \frac{1}{2N} \sum^{N-1}_{x=-N} f(x) e^{- \pi i ...
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how can i applay fourier transform to a fourier series+reves+fillter respons

hellow and thenks for helping. i got a tesk in class. we should design a low-pass filter (butter worth fillter) to a given band and stop band freq. i need to cunculate the respont of a periodic rect ...
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Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - ...
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Given a finite number of Fourier coefficients, can we construct a corresponding intergrable function?

Let $\xi_1,\dotsc,\xi_n$ and $\eta_1,\dotsc,\eta_n$ be real numbers. Is there a complex valued function $f\in L^1(\mathbb{R})$ such that: $\int f(x)e^{2\pi i\xi_k x}dx=1$ for every $1\leq k\leq n$. ...
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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 ...
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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 $$ ...
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Definition of the convolution with tempered distributions and Schwartz function

If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$, where we place ...
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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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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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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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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 ...
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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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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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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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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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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. $$ ...
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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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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 ...