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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The fourier transform of $f$ in $L^2$

Let $f$ is continuous and piecewise smooth, $f\in L^2$ and $f'\in L^2$. Show that $\hat{f}$ (the fourier transform of $f$) is on $L^1$.
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Suppose $f$ fulfils a unity condition. Prove $\hat{f}(0)=1$

Suppose that $ f \in L^1(\Bbb{R})\cap L^2(\Bbb{R})$ satisfies the following unity condition $$\sum_{k\in \Bbb{Z}} f(x-k) = 1\ \ \ \ , \forall x\in\Bbb{R}$$ Prove that $\hat{f}(0)=1$ Here $\hat{f}$is ...
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Approximation by trigonometric polynomials in two dimensions

Let $f(x,y)$ be continuous function on $[0,1] \times [0,1]$. Prove that there exists polynomial in form $$P_{n}(x,y)=\sum_{|k_1|,|k_2|<N}e^{i(k_1x+k_2y)}$$ that we have: $$\sup_{x,y ...
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14 views

Fourier Transform of $n$ functions

I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$? ...
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28 views

A trajectory for shortened k-space data acquisition MRI

Given a real function $f:\mathbb{R}^n \to \mathbb{R}$, denote by $\hat{f}$ its Fourier Transform. I have shown that $\hat{f}(\vec \omega)=(\hat{f}(-\vec \omega))^*$ where $^*$ denotes complex ...
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1answer
16 views

How to derive the discrete fourier transform of $(n+2)a^nu[n]$ where $|a| < 1$?

This is a rather simple question, but I'm stuck on one step. Here's what I've done: 1) $x[n] = (n+2)(\frac{1}{2})^nu[n] = n(\frac{1}{2})^nu[n]+2(\frac{1}{2})^nu[n]$ The discrete fourier transform is ...
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36 views

Relating Fourier transform theory on two distinct subspaces

In Fourier transform theory (on $\mathbb{R}$), three vector spaces play a very important role: $L^1(\Bbb R)$, $L^2(\Bbb R)$ and the Schwartz space $\mathcal{S}(\Bbb R)$. Arguably the nicer spaces of ...
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1answer
28 views

Fourier transform ID

I'm assuming $\frac{d}{dw}$ should be written as $\frac{\partial}{\partial \omega}$ I'm a bit confused by the part highlighted in green. I'm think i'm right in saying that when we integrate wrt one ...
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4answers
211 views

Fourier transform of the wave equation

On the LHS side of the highlighted expression should it not read: $\displaystyle \frac {d^2 \hat{u}}{dt^2}$ as the Leibniz Integral rule requires you to transform the partial derivative to a ...
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27 views

Bound on partial derivatives when the total derivative is bounded.

Let $f\in\mathbb{Z}[x,y]$ and fix $k\in\mathbb{N}$. Suppose that for all $v=(v_1,v_2)\in S^1$, $$ \left|\frac{d^k}{dt^k}f(tv)\right|_{t=0} $$ is bounded by $C$. (The value of $C$ that I use comes ...
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49 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
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56 views

Conditions for a Fourier transform of a PDE

I understand that when you are taking the fourier transform of a function you need it to decay at $\pm \infty$ in order to have a chance of having a finite area. However in this case $u $ only has ...
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1answer
45 views

Sobolev spaces and Fourier transform

Does the equation $$\partial^{\alpha} (Ff)(y) = (-i)^{|\alpha|} F(x^{\alpha} f)$$ still hold for $|\alpha| \le m$ and $f \in H^m(\mathbb{R}^n) = W^{m,2}(\mathbb{R}^n)$$? $F$ denotes the Fourier ...
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43 views

one parameter divergence formulation

I was reading this article: http://www.cims.nyu.edu/~trogdon/index_files/publications/periodic.pdf screenshot: And it is not clear to me how to write a pde in a divergence form. How can I transform ...
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62 views

Discrete Fourier Transform question

Let $R_{M\times N}$ be a space of size $M\times N$. Define the 2D Discrete Fourier Transform of $f\in R_{M\times N}$ to be \begin{equation} ...
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1answer
52 views

Odd Inverse of a Characteristic function

I saw this formula today: $$ \mathbb{P} \left[ X > K \right] = \frac{1}{2} + \frac{1}{\pi} \int_0^\infty Re\left( \frac{e^{-iuK}\varphi(u)}{iu} \right) du $$ Where $\varphi(u)$ is the ...
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1answer
28 views

Maximum and minimum of this complex periodic function

I came up with this function by using fourier transform. My only problem is how to get the amplitude of this function. Im planning to get the difference between their maxima and minima. I get its ...
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27 views

Sum of Random Variables and Fourier Transform

I have two dependent Random Variables, having different distribution, and I want to calculate the sum of them. To be more specific, the dependence is given by the formula: Y = a * X, where x,y are RVs ...
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48 views

Which of these is correct for Fourier transform?

Which of these is a correct definition of Fourier transform? $$\mathcal{F}(f(x))= \int\limits_{-\infty}^{\infty} f(x)\, e^{-i k x} \, \mathrm{d} x \ \ \ \ (1)$$ and$$\mathcal{F}(f(x))= ...
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2answers
46 views

What is the intuition behind summability kernel and convolution?

In Fourier analysis, when I look at the theorems and useful results derived using summability kernel and convolution, I get to think "Ok, I guess it works that way. but what is the intuition behind ...
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49 views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
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48 views

Convex function on closed interval: boundary points?

$I=[a,b]$, let the function $f:I\rightarrow\mathbb{R}$ be convex. (1) Is it possible to prove the existence of the limits: $$\lim_{x\rightarrow a^+}f(x) \ \ \ \ \ \lim_{x\rightarrow b^-}f(x)$$ If ...
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$g$ is differentiable and $g'(y)=\int_{\mathbb{R}}ixf(x)e^{iyx}dm(x)$

Let $f \in \mathcal{L}(\mathbb{R},\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ measurable Lebesgue. Asumme that $x\to f(x)$ is measurable. For $y \in \mathbb{R}$ define: ...
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PCA Vs Fourier Transform

As a general rule when trying to deconstruct a noisy signal into its components. When is it better to use Principal Component Analysis and when is it better to use a Fourier Transform?
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If the signal's frequency is multiples of the first harmonic frequency, transform method similar to DFT but use less number of samples?

Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This ...
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1answer
48 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
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2answers
52 views

Fourier coefficient of convex function

On $I = [0, 2π]$ consider the function $f : I → \mathbb{R}$ to be convex. Define: $$a_k\pi := \int_0^{2\pi}f(x) \cos(kx)\,dx$$ Show that the convexity of $f$ implies that $a_k ≥ 0$ when $k ≥ 1$. ...
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About a generalization of the Riemann-Lebesgue lemma

We have that for $f$ $\in$ $L^{1}$$(\mathbb{R}^n)$, $g$ measurable and bounded on $\mathbb{R}^n$ and, for any rectangle $R$, $$ \lim_{m(R)\rightarrow\infty} \frac{1}{m(R)}\int_Rg(x)dx = 0 $$ Then: ...
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1answer
25 views

fourier series sketching (by hand)

I calculated the Fourier Series representation of $f (x) = 1 − |x|$ on $−1 ≤ x ≤ 1$ and now I am asked to sketch the graph of the series on $−3 ≤ x ≤ 3$ by hand. How do I do this? I read through my ...
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1answer
43 views

Book recommendation for wavelet analysis

I am master student doing research in data mining, i read a paper about wavlet analysis for data mining, so i think it may help me in the future. But in my undergraduate degree the last course in ...
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About Fourier multipliers

I need some help with the following question: How can I prove that a Fourier multiplier sequence $\lbrace{m_n\rbrace}_{n=-\infty}^{\infty}$ mapping $L^{\infty}(\mathbb{T})$ into $C(\mathbb{T})$ ...
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Integrability of fourier transform

Let $f\in L^1(\mathbb{R})$ such that there exist $R,\delta >0$ for which $f$ is bounded in $[-\delta, \delta]$ and $\hat{f}(\xi)\geq 0$ for $|\xi|\geq R$. Then $\hat{f}\in L^1(\mathbb{R})$. ...
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23 views

Iterpolating to find the zeros of a complex function

I have an $N\times M$ grid of complex points sampled from some unknown complex function. I would like to interpolate and find the zeros of that function. I believe that this function can be well ...
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Are frequency domain and Fourier space the same thing?

What is the difference between frequency domain and Fourier space? When we perform FFT on an image the result is in "frequency domain", right? How does this relate to Fourier space? I spent some time ...
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26 views

Anti Hermitian Operator

I am required to show that the operator $\partial_t$ is Anti-Hermitian. This operator is defined such that $$\partial_t: s(t) \rightarrow \partial_t s(t) $$ Where the definition of an Anti-Hermitian ...
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How to take dft of irregularly sampled function in k space?

I would like to take inverse dft of irregularly sampled complex function in k-space. I am just summing in a loop over the length of the k vector, but is quite slow. ...
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How to understand the mapping between a periodic function to its Fourier coefficients?

For a periodic function $f(x)=f(x+T)$, its Fourier transform can be written as an infinite sum: $$ f(x)=\sum_{-\infty}^{\infty}c_n e^{2\pi i x/T}. $$ This seems to suggest that the information ...
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1answer
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matlab problem - removing frequencies after FFT, signal processing

I want to stress that this is not a coding problem, my problem is that i don't fully understand the mathematics surrounding the subject and that's why I believe I have a problem. I was given an ...
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Fast Fourier Transform: How is the roots of unity matrix divided?

For an example for input size N=8, how is the roots of unity matrix divided for a divide and conquer approach? My understanding is that it's divided into four quadrants, Ma with J&K evens; Mb ...
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1answer
31 views

Inverse Fourier Transform Proof

I am aware of how Fourier Transformation and Fast Fourier Transformation works, however I do not understand the logic of the inverse of FFT. Could someone explain why the inverse fourier ...
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70 views

Fourier transform of $ \log(x^{2}+a^{2}) $

I would like to evaluate the Fourier cosine transform of $\log(x^{2}+a^{2})$ or the integral $$\int_{0}^{\infty}\cos(ux)\log(x^{2}+a^{2})\,dx$$ for any real $u,a$. However, it seems that this ...
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1answer
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Computing the Fourier transform of $e^{-|x|}$

I'm trying to compute the Fourier Transform of $e^{-|x|}$ however I receive a different answer than wolfram alpha. Wolfram Alpha gets $\sqrt{\frac{2}{\pi}}\frac{1}{1+\varepsilon^2}$ and I get ...
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1answer
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Problem with finding a fourier transform

Help me to find a 2D fourier transform: $$\int dx dy\ \frac{e^{-ik_x x}e^{-ik_y y}}{\sqrt{x^2 + y^2}}.$$ All I've done so far is $$\int dx dy\ \frac{e^{-ik_x x}e^{-ik_y y}}{\sqrt{x^2 + y^2}} ...
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Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
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Several questions about integral operators.

I have been fumbling with expressions of the form \begin{equation} A\{f\}(s) = \int A(s,t)f(t)\operatorname{dt} \tag{$\star$} \end{equation} as a generalization of the matrix product. When looking ...
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1answer
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convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
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How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

Let $m$ be a positive integer and define $$ Lf = -\frac{d}{dx}(1-x^{2})\frac{df}{dx}+\frac{m^{2}}{1-x^{2}}f $$ on the domain $\mathcal{D}(L)\subset L^{2}(-1,1)$ consisting of all twice ...
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Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain ...
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Intuition behind the proof of the Inverse Fourier Transform?

I am interested in the proof of the Inverse Fourier Transform for absolutely integrable real valued functions. The proof I have read asks you to consider an auxiliary function $g_{a}(x)$ defined as ...
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1answer
60 views

Compact set of measure zero and sequence of Harmonic Functions with nice properties.

I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem: Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there ...