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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Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + ...
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How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
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44 views

fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
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2answers
21 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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2answers
62 views

Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t ...
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20 views

Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...
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1answer
22 views

What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
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28 views

Fourier series - Why does $\hat f(0) \ne 0$?

Let $f\in C^1$, $2\pi$-periodic, and let's assume $\int_{-\pi}^\pi |f'|^2 \le 1$. Prove: $$\sum_{n\in\mathbb{Z}} |\hat f(n)|^2 \le \frac{1}{2\pi}$$ There's a $c\in\mathbb{C}$ such that: ...
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27 views

Reading the properties of Discrete Fourier Transformation from the given figure.

Can anybody please help me read the properties of Discrete Fourier Transformation from the given figure. Here is the image link Thank you guys, appreciate you help.
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30 views

Worked out FFT example per Hand

Can anyone please show me the worked out example of FFT. Suppose for the signal x = (1/2, 1/4, 0, 1/4 , 1/2, 1/4, 0, 1/4). What I think I know: First I need to do the bit reversing, I get x = ...
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89 views

Show that $f_n\to f$ uniformly on $\mathbb{R}$

Let $$P_n(x) = \frac{n}{1+n^2x^2}$$. First, I had to prove that $$\int_{-\infty}^\infty P_n(x)\ dx = \pi$$ And that for any $\delta > 0$: $$\lim_{n\to\infty} \int_\delta^\infty P_n(x)\ dx = ...
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1answer
50 views

Prove that $\frac{t}{t^2-1}$ is a tempered distribution

I want to compute the Fourier transform of $\frac{t}{t^2-1}$, and in order to do so I need to prove in which space is the function. Clearly the function is not $L^1(\mathbb{R})$ neither ...
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2answers
24 views

Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
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1answer
39 views

(Distributional) Fourier transform

I need to calculate the (distributional) Fourier transform of $$ f(x) = \frac{x^2}{x^2+1}. $$ I unsuccessfully tried to find a differential equation for $f$ in order to solve the Fourier-transformed ...
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2answers
43 views

Dirac delta function and its property

I am trying to find a formal proof for the following property of Dirac delta function: for any function $f$ : $$\int_{-\infty}^{+\infty} \delta(x)f(x)dx=f(0),$$ where $\delta$ is Dirac delta ...
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37 views

Good reference for Fourier Analysis

Would you please indicate a good reference about Fourier analysis (Fourier series, convergence theorems: pointwise, uniform convergence, $L^2-$convergence...etc)? It should concern the organisation of ...
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37 views

proof translation property of transform fourier

I really need your help here: Think of $f(x,y)$ as an image where the dimensions are: $NxM$. I can't prove this property of 2D transform fourier: if the transform of $f(x,y)$ is $F(u,v)$ so the ...
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30 views

Fourier transforms of Gaussians with rational functions

Consider functions of the form $$ f(x)= e^{- x^2/\sigma^2} \frac{a_0 + a_1 x + \cdots +a_n x^n }{b_0+ b_1 x + \cdots + b_m x^m } $$ Under which circumstances do these functions have closed forms for ...
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29 views

Finding the zeroth Fourier coefficient using limit

The $\text{n:th}$ Fourier coefficient (for the $\cos(nx)$ part) is defined by $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(\theta) \cos(n\theta)d\theta.$$ Inserting $n=0$, we get $$a_0 = \frac{1}{\pi} ...
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14 views

system being absolute continous

The classical Rudin--Shapiro sequence $r_n\in\{-1,1\}^{\mathbb{N}}$ satisfy the following inequality. $(\int_{0}^{1}\left|\sum_{n<N}r_ne^{2\pi in(\theta)}\right|^{2}d\theta)^{\frac{1}{2}}\leq ...
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16 views

Fourier transform of shifted function

I would like to know how to deal with: $$\mathcal{F} g(t-a(t))=??$$ Cause I know that: $$\mathcal{F} g(t-b)=e^{-i2\pi f b}G(f)$$ where $b$ is a real constant and $G$ represents the Fourier ...
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1answer
37 views

Which way does the Fourier Transform go?

This might be a silly question, but I'm really confused by the way Fourier Transform was taught in my algorithms class, and everything else I found on the internet. The way we defined FT is first ...
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1answer
31 views

Why is the Fourier transform of $1/|x|^\alpha$ a function?

I would like to prove that the Fourier transform of $f(x) = 1/|x|^\alpha$ is a function, where $x\in \mathbb{R}^n$ and $0<\alpha <n$. It's clear for me that it is a tempered distribution. I need ...
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16 views

How to show that $\chi(\omega) = \int_{-\infty}^{\infty}{\chi(t)e^{-i\omega t}dt}$ is complex valued if $\chi(t) = 0$ for $t<0$

The question is physics related, but my issue stems from the math so I figured this was an appropriate place. I'm looking at the (electric) susceptibility $\chi$, in both the time $t$ and frequency ...
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2answers
168 views

Book on applied mathematics

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
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1answer
35 views

Non-compactness of support of linear KdV equation solution

The last question in Linares and Ponce's 'Introduction to Nonlinear Dispersive Equations's first chapter asks the reader to prove that, if the following IVP is given: $$\begin{cases} \partial_t u + ...
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1answer
35 views

How to prove that $f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda$ where $f_{\lambda} $ is the eigenfunctions of $\Delta$

On Euclidean space $\mathbb R^n$, how to prove that $$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$ where $\Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) $, whith $\Delta$ is the ...
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Discrete Fourier Transform of a vector

Find the Discrete Fourier Transform of the vector $[1,0,-1,0]^T$ We have not covered this in class. What is it that I'm trying to accomplish and how do I report an answer?
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39 views

Applying Fourier transform to equation

I would like to know why we divide the $(i*ω)^2$ to the equation. When I asked my supervisor, he said I need to learn Fourier Transform: more specifically I need to understand relationship between ...
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1answer
51 views

Parseval's theorem from PMA Rudin

This is Parseval's theorem from book PMA Rudin. Let $\{\phi_n(x)\}$ is orthonormal system on $[a,b]$. Rudin defines Fourier coefficients as $c_n=\int_{a}^{b}f(t)\overline{\phi_n(t)}dt$. And to to ...
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1answer
2k views

Calculating an integral derived from the convolution of two Fourier transforms

Let $\sigma>0$ , $1<\alpha\leq 2$, and $-1\leq \beta \leq 1$. I am looking for a closed-form solution (or something near) for the following integral. $$\frac{1}{2 \pi } \text{PV}\int_{-\infty ...
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60 views

Questions about delta function.

Let $z \neq 1$ be a complex number. Then \begin{align} \frac{1}{1-z} = \sum_{n=0}^{\infty} z^n. \end{align} We have \begin{align} \frac{z^{-1}}{1-z^{-1}} = \sum_{n=1}^{\infty} z^{-n}. \end{align} ...
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2answers
64 views

Fourier transform of a Lévy density $\frac{1}{\sqrt{2\pi }}\int_{0}^{\infty} e^{ikx-\frac{1}{2x}}x^{-\frac{3}{2}}dx$

A Lévy density is defined as $$q(x;1/2,1)=\frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2x}}x^{-\frac{3}{2}}$$ for $x>0$ I am looking for it's Fourier transform: $$g(k;1/2,1)=\frac{1}{\sqrt{2\pi ...
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82 views

Fourier transform of $1/|x|^{\alpha}$.

My problem is to prove the following identity: $$C_{\alpha}\int_{\mathbb R^n} \frac{1}{|x|^\alpha} \phi(x) dx = C_{n-\alpha}\int_{\mathbb R^n} \frac{1}{|x|^{n-\alpha}} \widehat{\phi}(x) dx$$ where ...
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1answer
31 views

Application of Residue Theorem to inverse Fourier transform

I'm reading through a derivation in a book and am having trouble understanding a step. Here's a screenshot 3.46 is the equation in $(k,\omega)$ space. They're doing an inverse Fourier transform back ...
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1answer
29 views

If $\Delta u(x)=\delta(x)$ then $u(x)=C |x|^{2-d}$ via Fourier transform?

Can the behaviour $$ \text{constant}\times|x|^{2-d} $$ be obtained for the solution of the distributional equation in $\mathbb R^d$, for $d\ge 3$, $$ \Delta u(x)=\delta(x) $$ via Fourier transform ...
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31 views

Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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18 views

Wiener's Tauberian theorem for positive-value functions

The elementary Wiener's Tauberian theorem goes as follows: Theorem (Rudin, Functional Analysis Th. 9.5): Suppose $K \in L^1(\mathbb{R}^n)$ and $Y$ is the smallest closed translation-invariant ...
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39 views

Completeness of a set of functions

We know Fourier basis $e^{i k \xi \cdot x}$ forms a $L^2$ complete basis, $|\xi| = 1$ and $k\in \mathbb{N}$. I would like to know if $k$ belongs to a open interval, do we still have the completeness. ...
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36 views

Bound on the following integral using integration by parts

Let $W: \mathbb{R}^s \rightarrow [0,1]$ be a smooth function supported on $[0,1]^s$ that satisfies $$ \left| \frac{\partial^k }{\partial x_{i_1} \cdots \partial x_{i_k}} W(\mathbf{x}) \right| \leq ...
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2answers
59 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
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1answer
35 views

Fast Fourier Transform as Matrix Factorization

I'm given a vector of length 4 and three matrices that correspond to a Fast Fourier Transform, I'm not exactly sure which one, but I guess it's supposed to be the Cooley-Tukey algorithm. Here is the ...
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17 views

How to compute a complex-valued function specifying its amplitude and the ampltiude of its Fourier transform

I want to set the amplitude of a complex function (independent of its phase), and also the amplitude of its Fourier transform (again independent of its phase). Given these two functions (amplitude of ...
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25 views

Equivalence of $H^{1/2}(S^{1})$ norm with integral

I've been attempting to show that the Sobolev space norm $$\|f\|^{2}_{H^{1/2}} := |\hat{f}(0)|^{2} + \sum_{n \neq 0} n |\hat{f}(n)|^{2}, $$ for $f$ on the circle, is equivalent to the integral $$I(f) ...
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22 views

Fourier sine series

Compute the Fourier sine series of $f(t)=t$ over the interval $[1,3]$. The question I have is that over $[-L,L]$, the cosine series is $0$ but does this still apply over the interval $[1,3]$? So ...
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45 views

Solving a simple Schrodinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occuring in the simplest Schrodinger equation possible: $\partial_t \psi ...
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1answer
42 views

Some questions Regarding Fourier Transform

I have some questions regarding Fourier Transform. I have studied a lot of books for Fourier Transform and currently, I am confused weather I should use Riemann integral or Lebesgue integral for the ...
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12 views

Find appropriate sine and cosine terms and coefficient formulas to approximate a function on [1,3]

This is probably a ridiculous/simple question but I am still having trouble! Find appropriate sine and cosine terms and coefficient formulas to approximate a function on [1,3]. I understand how to ...
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13 views

How to create obtain aliased version of $f(t)$ by upsampling whenever $f(t)$ at every $t$ is available

Suppose there is original complex-valued $f(t)$ with $t$ ranging from $-\infty$ to $\infty$. It is possible obtain samples from original $f(t)$ at every $t$ with some negligible error. If one ...
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3answers
110 views

Evaluating the integral $\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$

I want to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$$ for all $t \in \mathbb{R}$. I would preferably do it using the tools of complex analysis, but since I ...