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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Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
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95 views

The Continuity of the Discrete Time Fourier Transform of Absolutely Summable Series

I saw on a book to following claim: Given an Absolutely Summable Series $ \sum_{n = -\infty }^{\infty}\left | x\left [ n \right ] \right | \leqslant \infty $, Namely, $ l_1 $ series it is possible to ...
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224 views

Show a Schwartz function vanish at infinity

Let $f$ be in the Shwartz space $\mathcal S(\Bbb R)$. Why does the $\mathcal S$-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)|, \text{ for } a,b \in \Bbb Z_+, $$ implies that $f$ vanish at ...
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83 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
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66 views

A question about Fourier transform

I just don't know how to calculate the the fourier transform of $1/(1+x^2)$.Can you help me guys? Thx
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680 views

Show smooth functions of compact support are dense in the Schwartz space

Show $\mathcal{D}=C_c^\infty(\mathbb R^n)$ is dense in the Schwartz space $\mathcal{S}(\mathbb R^n)$. Use the standard topology on $\mathcal{S}$ $$ \|f\|_{a,b}=\sup_{x \in \Bbb{R}^n}\left| ...
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165 views

Why doesn't repeating a signal give rise to a finer resolution of DFT/FFT?

If x = [1 2 3 4 3 2]; and x1= [x x x x x x x x x]--that a new vector made of duplicating copies of x, then why is it that the FFT of x and x1 are essentially the same. When I plot the FFTs of each ...
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235 views

Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$

Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $$ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $$ But ...
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417 views

Show the usual Schwartz semi-norm is a norm on the Schwartz space

Let $f \in C^\infty(\mathbb R)$. Define the semi-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)| $$ where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$. Show ...
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156 views

Question about Fourier Transform

I am reading a Fourier Transform definition in two places, in the first is $$\int_{-\infty}^{\infty}f(x)\exp(-ijw)dx$$ and another is $$\int_{-\infty}^{\infty}f(x)\exp(-2\pi ijw)dx$$ I want know ...
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142 views

Question on Parseval's theorem

If $\sum_{k=-\infty}^{\infty}|a_k|^2$ is not finite, does Parseval's theorem say that the Fourier transform of $a_k$ is also not finite?
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317 views

Regularizing effect of the heat equation

Consider the heat equation on $\mathbb{R}_+\times\mathbb{R}^d$ \begin{align*} \partial_t u -\Delta_x u &= f, \\ u(0,x)&=u_0(x). \end{align*} In the case where $u_0\in L^2(\mathbb{R}^d)$ ...
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92 views

Suggestion for a project on Harmonic measure and Fourier analysis

I have a course project on harmonic measure and Fourier analysis. The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis. Harmonic measure is a vast ...
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436 views

Inverse fourier transform of $f(x)=\frac{\text{u}(x)}{\sqrt{1-x^2}}$

How to calculate (or derive) the inverse Fourier transform of $$f(x)=\frac{\text{u}(x)}{\sqrt{1-x^2}}$$ where $u(x)$ is the rectangular function? I know that $f(x)$ is the Fourier transform of the ...
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1answer
126 views

Violation of Parseval's theorem?

Can a function $f:G\to\mathbb{C}$ in $L^p,\ p>1, p\neq 2$ have a Fourier transform $F:\hat{G}\to\mathbb{C}$, where $\hat{G}$ is the Pontryagin dual space of $G$? I believe it can be shown that such ...
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249 views

Schwartz space: semi norm estimate on translation

the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$: $$ \|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| ...
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176 views

integral evaluation of an exponential

let be the function $$ e^{-a|x|^{b}} $$ with $ a,b $ positive numbers bigger than zero then how could i evaluate this 2 integrals ? $$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$ here 'c' can ...
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1answer
94 views

Proof for Fourier transform in $L^2$

This question makes me really confused: Let $f$ and $g$ two functions in $L^2$. Show that: $$\int \widehat f\cdot gdx= \int f\cdot\widehat gdx,$$ where $\widehat f$ is the Fourier transform ...
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118 views

$L^p$ space and Fourier Transform

Let be $q>p$ then $L^q(\Omega)\subset L^p(\Omega)$. I will be able to say that all $f \in L^q$, such that $q>1$, have a Fourier Transform?. pdta:I asking this because I am read that exist ...
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350 views

Discrete Fourier transform of shifted N-periodic sequence

I'm trying to show that if f[n] is an N-periodic sequence then the Discrete Fourier Transform of the shifted sequence f[n-m] for some constant $m\in\mathbb{Z}$ is $e^{\frac{-2\pi i mk}{N}} F[k]$ ...
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56 views

So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?

For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation} H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\}, \end{equation} where $\mathcal{S}'$ is the space of ...
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120 views

Fourier Series and Filter in Function

Let $L=1, A=1$ and $f\in L^2([-L/2, L/2])$, with Fourier series $$f^{t}=\sum_{n=-K}^{K}a_n \exp(2j\pi xn/L),$$ truncated at $K$. Has this function, $f^{t}$, any relation with Fourier Inverse ...
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352 views

Proove the Riemann-Lebesgue lemma for the Fourier Transform on $\mathbb R$ using the Riemann-Lebesgue lemma for the Fourier coefficient on the circle

Proove the Riemann-Lebesgue lemma for the Fourier Transform on $\mathbb R$ using the Riemann-Lebesgue lemma for the Fourier coefficient on the circle. Periodize $f \in L^1(\mathbb R)$ and ...
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1answer
108 views

Dual space of the function $f$ in Fourier Transform

Let $f\in L^1{(\mathbb{R})}$. Why the Fourier Transform $\hat{f}\in L^{\infty}{(\mathbb{R})}$. Is it because $(L^1{(\mathbb{R})})^*=L^{\infty}{(\mathbb{R})}$?
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50 views

Proving/disproving an identity on a Hessian.

Let $A=(a_{i,j})$ be a $n$ x $n$ matrix $(n\geq 2)$, where $a_{i,i} = |x|^2-2x_i ^2$ and $a_{i,j} = -2x_i x_j$ for $i\neq j$. Here $|x|^2 = x_1^2+x_2^2+ \cdots + x_n^2$. I'd like to compute the ...
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183 views

Expression in Fourier Transform

Let be $f\in L^1(\mathbb{R})$, I will be able to say that $$ \dfrac{\hat{df(w)}}{dx} = \int_{-\infty}^{\infty}\dfrac{df(x)}{dx}\exp(-2\pi j wx)dx $$? Why?
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124 views

Taking inverse Fourier transform of complicated multipart equation

Define $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} ...
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121 views

Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
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1answer
89 views

Expressing the cantor function on $[0,1]$ as a function on $\text{Ternary}([0,1])$

I would to link the simple function and probabilistic approach for the calculation of the Fourier transform of the Cantor function. Let $f:[0,1] \to [0,1]$ be the Cantor function. In the simple ...
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3answers
501 views

Heat equation, separation of variables and Fourier transform

I have a question about the heat equation $\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}$ with the conditions that $\varphi(x,t=0) = f_0(x)$ and $\lim_{x \rightarrow\pm ...
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Difference between Fourier series and Fourier transformation

Whats the difference between Fourier transformations and Fourier Series? As I've been working with Fourier Series in my maths lectures yet a friend of mine also doing engineering has been working with ...
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215 views

Fourier transform of a Generalized Gaussian

I've got a family of functions called Generalized Gaussians. They're given by: $f(x) = \exp(-ax^{2p})$ Where $p \in \{1,2,3,\ldots\}$ Could anyone tell me how to find their Fourier transforms?
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73 views

Probabilistic calulation of the Fourier transform of the Cantor function

This is on the same theme as in this post, where the Fourier transform was derived using simple function. Let $f:[0,1] \to [0,1]$ be the Cantor function. Then $f$ is the cumulative distribution of ...
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1answer
372 views

Deriving complex form of Fourier series

I have encountered a problem where I get the correct outcome, but I am uncertain as to whether or not my steps are logically justified. I would really appreciate some input regarding this! The ...
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1answer
29 views

how to show that g(t) is continuous?

how to show that g(t) is continuous? $g(t) = piecewise(t \neq 0, 0, t=0, 1)$ $\lim_{t \rightarrow 0} g(t) =1$? So it should be continuous at $t=1$? Well it seems this question doesn't meet you ...
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1answer
437 views

Fourier transform of the Cantor function

Let $f:[0,1] \to [0,1]$ be the Cantor function. Extend $f$ to all of $\mathbb R$ by setting $f(x)=0$ on $\mathbb R \setminus [0,1]$. Calculate the Fourier transform of $f$ $$ \hat f(x)= \int f(t) ...
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458 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: ...
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44 views

The Fourier transform of $x \mapsto |x|^{-1/3}e^{-x^{2}}$ is not in $L^{1}$.

Okay previously my lecturer showed that this is so by proving in the following way: Proof by contradiction. Suppose the transform is in $L^{1}$. Then as $f \in L^1$, we may use Fourier Inversion ...
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$f \in L^1(\mathbb R), f>0$ then $|\hat f(y)| < \hat f(0), y \ne 0$

Suppose $f$ is a strictly positive function in $L^1(\mathbb R)$. Show $$ |\hat f(y)| < \hat f(0) \text{, for all } y \ne 0. $$ Using monotonicity of the integral, I can show $|\hat f(y)| \le \hat ...
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143 views

Multiplicative formula for Fourier transforms

Suppose I have two functions continuous functions $f,g \in \mathcal{S}(\Bbb{R})$, the Schwartz space. Now I know that the following multiplicative formula holds. Namely, if $\hat{g}$ denotes the ...
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370 views

What is the Fourier transformation of a uniform B-Spline?

I'm looking for the Fourier transformation of the (constant) uniform B-Spline $$N_0(x) = \begin{cases}1 & 0 \leqslant x < 1 \\ 0 & otherwise \end{cases}$$ If $N_0(x)$ would also attain ...
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90 views

How to show that for arbitrary( complex) trigonometric polynomials $P$ and $Q$ holds…

How to show that for arbitrary( complex) trigonometric polynomials $P$ and $Q$ holds $$\frac{1}{2\pi} \int_{-\pi}^{\pi} P(t)Q(mt)dt = \frac{1}{2\pi} \int_{-\pi}^{\pi} P(t)dt \frac{1}{2\pi} ...
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87 views

If the Fourier series of $f$ is absolutely convergent does it implies that it converges to f

If $$ \sum_{k=-\infty}^{\infty} |\hat f (k) | < \infty $$ does it implies $$ S_n(t)=\sum_{k=-n}^{n} \hat f (k) e^{ikt} \to f(t) \; ? $$ I know $S_n$ converges for each $t$ to some function $S$. ...
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50 views

ways to prove that $e^{i|\xi|}\phi$ is a fourier transform of $L^1$ function

Let $\phi\in C_{0}^{\infty}(\mathbb{R}^n)$ and equals to 1 near the origin,then show that $e^{i|\xi|}\phi(\frac{|\xi|}{\mu})$ is a fourier transform of an $L^1$ function with any $\mu>0$,and how ...
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135 views

Sobolev-type inequality.

Let $0<\alpha<n$, $1<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq ...
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23 views

How to change variables if you only know the invariant measure

I want to find an equation relating the coefficients of two different harmonic expansions of the same function (and a relation between their respective basis functions). ...
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1answer
96 views

How to formulate arbitrary complex trigonometric polynomial?

How to formulate arbitrary complex trigonometric polynomial? I know that in real form it is $\displaystyle\sum_{n=1}^k a_n\cos(nx)+b_n\sin(nx)$
12
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1answer
440 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
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1answer
81 views

Fourier transform and sampling time

Given a signal $x(t)$ and the $X(\omega)$ obtained from $x(t)$ using a FFT with a sampling time $Ts$, I get a subset of $X(\omega)$: $Y(\omega)$ obtained from $X(\omega)$ taking it between $\omega_0$ ...
2
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291 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...