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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Discrete Fourier Transform index n from 0 to N-1?

I understand that when we compute a DFT, there are $N$ data points and the indices $n$ and $m$ go from zero to $N-1$ where $y_m$ are the data points in the time domain, and $Y_n$ are the amplitudes in ...
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189 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
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Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
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Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
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91 views

What does the Fourier transform of $1/x^2$ mean?

If I ask Mathematica to compute the Fourier transform of $\frac{1}{x^2}$ using the FourierTransform function, it gives me a result of ...
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38 views

intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
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24 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
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50 views

Fourier transform of exp(cos)

How do I calculate the Fourier transform ($t \rightarrow \omega$) of the following: $\exp(A\cos(\omega_0 t))$ $A$ is a real constant, and $\omega_0$ is a real and positive constant. I know that this ...
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33 views

a question about Fourier transforms

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
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1answer
14 views

2D Fourier Transform proof of Similarity Theorem

I have to solve an exercise, but if i could use the following theorem, it would be piece of cake Similarity Theorem if $ \mathscr{F}\{g(x,y)\}= G( f_x,f_y)$ then $ \mathscr{F}\{g(ax,by)\}= \frac ...
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34 views

Is there an equivalent of Plancherel's theorem with wavelets transform?

For the Fourier transform, we know that: $$||f||_2=c||\hat{f}||_2$$ where $c$ depends on the normalization. Is there an equivalent with wavelet transform? Thanks.
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the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
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36 views

Is there any commonality between the use of Parseval's Identity in two different contexts?

In Fourier analysis, Parseval's Identity relates to "the summability of the Fourier series as a function." In inner product space analysis, the "identity" works as a "Pythagorean theorem" relating ...
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14 views

Fourier transform Piecewise function

i have this piecewise function and i need to get the fourier transform $$f(n)=\begin{cases} 0, & t<-2 \\ -1, & -2\le t<-1\\ 1, & -1\le t \le 1\\ -1, & 1<t\le2\\ 0, & ...
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11 views

infinite discrete abelian group

I am not too familiar with Fourier analysis, but I needed to use a certain result. I would appreciate any assistance. I was reading a literature in Foruier analysis and it said something like "Every ...
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46 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
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30 views

When is a function a Fourier transform of an integrable function?

Specifically, in the case $f(\xi)=\frac{1}{(1+\xi ^2)^\epsilon}$ where $0<\epsilon<1$. I wish to prove this is a Fourier transform of a $L_1$ function. Any insight into the manner would be ...
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21 views

How to show something is a convolution operator?

I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't ...
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110 views

Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
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42 views

Proving that $\sum_{n=1}^\infty{[a_ncos(nx)]^{(m)}+[b_nsin(nx)]^{(m)}} \leq \sum_{n=1}^\infty{(|a_nn^m| + |b_nn^m|)}$

Okay so here's the the problem: Let $k \in \mathbb{Z}.$ Then, $\forall$ $m \in [1,k]$, $$f^{(m)}(x) = \sum_{n=1}^\infty{[a_ncos(nx)]^{(m)}+[b_nsin(nx)]^{(m)}} \leq \sum_{n=1}^\infty{(|a_nn^m| + ...
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43 views

Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
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55 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
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How to get the real frequency from an FFT output graph?

I am working in Origin 7.0 and trying to understand how the FFT Analysis works. In an attempt to understand, I decided to run a test of the program using known frequencies. The function I used was: ...
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20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
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Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
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convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
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37 views

Fourier Transform, and identities

I need to calculate the fourier transform of this $ t \cdot sin(t-3)+2 \cdot t \cdot cos(3t) \cdot rect(6t) $ is the following valid, based on the first fourier identity i've read in a book, and ...
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33 views

Three Dimensional Fourier transform of a raidal function

Scaling analysis shows that the three dimensional Fourier transform of the function $f(\mathbf{r})=1/r$ is proportional to $1/k^2$. On the other hand, when working with spherical coordinates ...
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22 views

Fourier transform theorem

I have an expression: $c(s) = \int_{-\infty}^{\infty} d\omega e^{-i*\omega*s}*F(\omega)$ $G(W)=\int_{0}^{\infty} ds e^{i*W*s}*c(s) $ Is $G(W) = F(W)$? What is the relation of G(W) and F(W)? ...
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47 views

Berlekamp Massey and DFT

I was looking into the Berlekamp Massey algortihm, for LFSR, over GF(2) wondering if there was any DFT(alternately FFT), for the above scheme. Also, is there any generalization to Fn, ie, start ...
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24 views

Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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Fourier transforms similar $\Rightarrow$ functions similar?

I am wondering if there is a theorem that states something like the following? If $$\big|\;\tilde f(\omega)-\tilde g(\omega)\,\big| < \varepsilon\qquad \forall\omega$$ then there exists a ...
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Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
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44 views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
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138 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ...
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Development of Hilbert transform relationship

In the development of Hilbert transform relationships, Prof. Oppenheim has chosen \begin{equation} ...
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Exercise - Fourier Transform

I have got another question concerning the Fourier transform. I hope somebody can help me. Let $f \in L^1(\mathbb{R}^n)$. Prove that $\hat{f}$ is continous. (ok, I was able to show it) If ...
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99 views

Schwartz kernel theorem for induced distributions…

I'm studying periodic pseudo-differential operators on torus and I have a question concearning the Schwartz kernel theorem: If $A:C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n)$ is a ...
3
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1answer
111 views

Scale invariance and the Mellin transform?

The Mellin transform of a function is given by: $\mathcal{M}[f](s) = \int_0^{\infty}x^{s-1}f(x)dx$ Supposedly, the magnitude of the Mellin transform is invariant to scaling, analogous to how the ...
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33 views

How to solve this heat equation with fourier method

Solve this via Fourier method: $$u_t-u_{xx}=0 \quad\quad 0< x<\pi, \quad t >0, $$ $$u(0,t)=u_x(\pi,t)=0, \quad\quad t \ge 0$$ $$u(x,0)=2\sin\left(\frac{3x}{2}\right) \, ...
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177 views

Integral through Fourier Transform and Parseval's Identity

$$ \int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,. $$ Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is ...
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Questions about the Fourier series

$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos{(\frac{2 n \pi x}{L})}+b_n \sin{(\frac{2 n \pi x}{L})}) \ \ \ \ \ (*)$$ The symbol $\sim$ has the following meaning: We know that the right ...
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Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
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33 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
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40 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
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30 views

Inverse Fourier transform of $\cot{ a \omega}$?

I am interested to know what is the inverse Fourier transform of $\cot{ a \omega}$, and how to derive it? My attempt was to use contour integration which leads to $$ \sum_{n = -\infty}^{\infty} ...
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40 views

Fourier transform - epicycles.

I have $f(x) = \sin(x)$. I thought that when I do Fourier Transform and construct epicycles, than those epicycles will draw that $\sin(x)$ function (but this is probably not case with $\sin(x)$, cause ...
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62 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
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48 views

Fourier transform of a potential

I need help computing the distributional inverse Fourier transform of the function $1/|x|^2$ in dimension two. The integral makes sense written as \begin{align} 1/2\pi \int_{\mathbb{R}^2} e^{ix\xi} ...