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 sine and cosine function

For the sine function we can do the following formal computation: $$\mathcal{F} (\sin(2\pi kt))(x) = \int_{-\infty}^{\infty} e^{-2\pi i xt} \frac{e^{2\pi i kt}-e^{-2\pi i kt}}{2i}dt= \frac{i}{2} ...
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Seeking better understanding of Fourier transform?

I'm quite confused on the one part of the Fourier transform. I don't understand what is the term $\left(u*x + v*y \right)$ mean. I mean $u$ and $v$ are the axis for frequency domain and $x$, $y$ are ...
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Find occurrences of sequences in sound wave

I have quite basic knowledge of mathematics. Consider two axis, time and dB (sound levels). My question is if it's possible, using fourier or any other method, to find how many occurrences of ...
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Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
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Does $f(0) = +\infty$ when $\hat f \geq 0$ and $\int \hat f (s) \ ds = +\infty$?

Throughout, $f \in L^1(\mathbb{R})$ and $\hat f \in C_0(\mathbb{R})$ is its Fourier transform $s \mapsto \int e^{its} f(t) \ dt$. Motivation: If $\hat f \in L^1(\mathbb{R})$ too, then, by Fourier ...
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115 views

Convolution theorem for other transforms

The Fourier transform is an integral transform with turns any function into a superposition of sinusoidal waves. The convolution theorem states the astonishing property that if you convolve two ...
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Fourier Transform of $\frac{1}{(1+x^2)^2}$

I need to find the Fourier Transform of $f(x) = \frac{1}{(1+x^2)^2}$ Where the Fourier Tranform is of $f$ is denoted as $\hat{f}$, where $\hat{f}$ is defined as ...
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Relationship between Fourier coefficients of $f\left(x\right)$ and $f^{-1}\left(x\right)$

Say I have a function $f\left(x\right)$, which can be expressed as a Fourier Series: $$f\left(x\right)=\sum_{k=-\infty}^{\infty} c_k e^{ikx}$$ Define the inverse of $f\left(x\right)$ as, ...
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311 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
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Does the Fourier matrix $F_n$ represent a (tensor) multiplicative function?

At "Complex Hadamard Matrices", I found that, two Kronecker (tensor) products of Fourier matrices $k_1$ and $k_2$ are equivalent, if and only if $k_2$ can be obtained from $k_1$ by a combination of an ...
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86 views

Discrete Fourier Transform on a shifted frequency grid

I use the discrete Fourier transform in 3D to solve my model partially in real space and partially in Fourier space. The DFT pair is defined as \begin{equation} ...
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197 views

Parseval's theorem, DFT to FT

So I am working with DFTs for the first time and I have some problems. I have a discrete signal to this signal i want to apply a DFT, then I want to use the output in an integral. So $S(t)$ is my ...
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78 views

symmetric window for discrete windowed Fourier transform

The discrete windowed Fourier transform of a signal $f$ of period $N$ is given by $$ Sf[m,l]=\sum_{n=0}^{N-1}f[n]g[n-m]\exp\left(\frac{-i2\pi l n}{N}\right). $$ Why is it that the window $g$ must be ...
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64 views

Fourier Series and Summation

$\sum_{n=1}^\infty \frac{1}{n^2}$ can be computed in straight-forward way by computing the Fourier co-efficients of $f(x)=x$ and applying Parseval's identity. Likewise, $\sum_{n=1}^\infty ...
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95 views

Convergence of convolution

I have a vector $x = (..., x_{-1}, x_0, x_1, ...)$ and a vector $w = (..., 0, 0, 1, 1, .. , 1, 1, 0, 0, ..)$ (with $2M + 1$ components equals to 1) such that $y = x \cdot w = (0, 0, .., x_{-M},x_{-M ...
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If $f \in L^1(\mathbb{R})$ and $\hat f \geq 0$, is $f$ continuous?

Suppose $f \in L^1(\mathbb{R})$. I am wondering what conditions on $\hat f = \left[ s \mapsto \int e^{its} f(t) \ dt \right] \in C_0(\mathbb{R})$ suffice to make $f$ continuous (or, more accurately, ...
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32 views

Fourier and differentiation operators

For a function $f:\mathbb{R}\rightarrow\mathbb{R}$ in the Schwartz class, define $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ We can show that $T^2f(y)=f(-y)$, and $T^4f(y)=f(y)$. ...
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77 views

Relation between Schwartz space and Sobolev space $H_{1}$

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |x^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ and $S'(\mathbb R) ...
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How do you compute the Fourier Transform of this Unit-Impulse Function?

I have been given this problem from a textbook (not homework, trying to study for an exam. The goal is to find the Fourier transform of this function. $\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$ ...
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58 views

Inverse Fourier transform to get convolution

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $u(x,0)=f(x).$ Let ...
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Finding Fourier transform of initial condition

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
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557 views

Fourier transform on Hermite polynomial

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Define a transformation $T$ as $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ How can I find the ...
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128 views

Differential equation for heat equation

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
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estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
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247 views

Computing Fourier transform for $L^2$ function

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
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306 views

Deriving Fourier inversion formula from Fourier series

Let $g\in C_0^{\infty}(\mathbb{R})$ (infinitely differentiable with compact support), and let $$\hat{g}(y)=\int_{-\infty}^\infty g(x)e^{-ixy}dx$$ Assume that $\hat{g}$ is in the Schwartz class. ...
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Repeated transformation of function yields identity

For a function $f:\mathbb{R}\rightarrow\mathbb{R}$ in the Schwartz class, define $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ I want to show that $T^4f(y)=f(y)$. But plugging in the ...
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147 views

Uniqueness of Fourier transform in $L^1$

The Fourier transform of an $L^1$ function is defined by $$\hat{f}(y)=\int_\mathbb{R}f(x)e^{-ixy}dx$$ Is it true that for functions $f,g\in L^1$, if $\hat{f}=\hat{g}$, then $f=g$?
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Fourier coefficients converging

I'm thinking about this question, which has no answer yet despite being on a bounty and having 100+ views. Maybe it would be easier to start by asking this: Let $g\in C_0^{\infty}(\mathbb{R})$ ...
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Real Periodic Fourier Transforms

Suppose we have some input data (only real numbers) and the data is periodic. Is it true that the Fourier coefficients will also be only real numbers?
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Derive Fourier transform of sinc function

We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the ...
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Inverse of Short-time Fourier transform

The Gabor transform (i.e. Short-time Fourier transform with some Gaussian window) can be defined by (see http://en.wikipedia.org/wiki/Gabor_transform) : $G_x(t_0,\xi) = \int_{-\infty}^\infty ...
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Solving roots of a sum of sinusoids

Suppose I have a sinusoid with fundamental frequency $f_0$ and $N$ harmonics (all with distinct amplitudes $a_k$. Each harmonic also has it's own corresponding phase $\phi_k$ and offset $c_k$. $y(t) ...
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44 views

Cauchy sequence from Fourier coefficients

Let $f\in L^2(\mathbb{R})$, and let $f_1,f_2,\ldots \in L^1(\mathbb{R}),L^2(\mathbb{R})$ be such that $\|f_n-f\|_2\rightarrow 0$ as $n\rightarrow\infty$. Is it true that ...
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When can we interchange Fourier transform and countable sum?

When does $\mathcal{F}\left ( \sum_{n=1}^{\infty} f_n (x)\right ) = \sum_{n=1}^{\infty} \mathcal{F}(f_n(x))$ where $\mathcal{F}$ the Fourier transform operator.
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Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
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Basis For Non-Compact Functions

I am not a mathematician, so excuse me for my incorrect use of terminology. What I would like to ask is if there are basis functions for non compact functions? For example Fourier expansion uses ...
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81 views

Finding an ideal low pass filter convolution kernel

Let $f \in L^2[-\pi,\pi] $ and let: $$f = \sum_{k=-\infty}^{\infty}\hat{f}(k)e^{ikx}$$ the Fourier expansion of $f$. I want to find a convoultion kernel $g_N$ so that: ...
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Can the Fourier transform of a compactly supported function wind around a point?

Suppose $f \in C_c(\mathbb{R})$. That is, $f$ is a continuous, compactly-supported function $\mathbb{R} \to \mathbb{C}$. Its Fourier transform $$ \hat f(s) = \int_\mathbb{R} e^{-its} f(t) \ dt.$$ is ...
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Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.
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Algebraic intuition for Fourier inversion formula

I was thinking about the Fourier inversion formula, which says $$f(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty\left(\int_{-\infty}^\infty f(z)e^{-iyz}dz\right)e^{ixy}dy$$ I know there is an issue about ...
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Fourier transform on $\mathbb R^n$ of Gaussian function

Let $\displaystyle{K(x)= e^{- \pi |x|^2} \quad ,x \in \mathbb R^n}$ be the Gaussian kernel on $\mathbb R^n$. Prove that its Fourier transform is $$ \hat{K} (\xi) = e^{- \pi |\xi|^2} $$ I can ...
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260 views

Schwartz class function convergence in $L^1$ and $L^2$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function in both $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$. I want to show that there exists a sequence of functions $g_1,g_2,\ldots$ in the Schwartz class ...
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Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$ The answer given by Wolfram Alpha is $\sqrt{2\pi}\xi\exp(-\xi^2/2)$. Observe how this is related to the Fourier transform of ...
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Change of variables to derive Fourier series

Let $f\in C^{\infty}(\mathbb{R})$ be a periodic function of period $2L$. Define $$a_n=\dfrac{1}{2L}\int_{-L}^Lf(x)e^{-in\pi x/L}dx$$ Show by change of variables that $$f(x)=\sum_{n=-\infty}^\infty ...
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Fourier Transform over function depend on time and frequency

In my task I need to perform Inverse Fourier Transform from spectrum that depend on time and frequency arguments simultaneously. E.g., I have a discrete spectrum of some function $S(t, f)$ with $2N$ ...
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194 views

Trying to figure out Fourier transform of {(0.5^n)(u(n))

I'm working in a signals class for continuous signals, and we have this problem shown above. I have tried using this function f_1 X f_2 = F_1 * F_2, where I'm ...
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158 views

About coefficients of Fourier integrals

If a $2\pi$-periodic function $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lebesgue integrable in $[-\pi,\pi]$, and the series $\frac{a_0}{2}+\sum_{n=1}^\infty [a_n \cos{nx}+b_n \sin{nx}] $, where ...
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Checking if a function is in the Schwartz space of rapidly decreasing functions.

Is there any neat bi-implication other than the definition that I can use to check this? This question was motivated by a question that asked if $ f(x) = e^{-|x|^3}$ was in S. It isn't infinitely ...
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194 views

Fourier transform of $|x|^{-t}$

In $\mathbb{R}^d$, let $f(x)=|x|^{-t}$, its Fourier tranform $F(f)(ξ):=(2\pi)^{-\frac{d}{2}}∫_{\mathbb{R}^d} e^{ix\cdot ξ}f(x)dx$, is there any fast way to see that this integral converge at $ξ \neq0$ ...