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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Can an arbitrary real function be written in terms of quadratures of an arbitrary frequency with time dependent coefficients?

Given a real function $f$, and a frequency $\Omega$, is it the case that there exist two other real functions $I$ and $Q$ such that $f$ can be written as $$f(t) = I(t) \cos(\Omega t) - Q(t) ...
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Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
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23 views

Deconvolution and Polynomial factoring using the FFT

I've been trying to implement a general N dimensional deconvolver for various engineering applications and some math curiosities. For speed and simplicity I've decided to try and do this with help of ...
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14 views

Project a signal $S(t) = \sum_0^{\infty}A(k)e^{if(k)t}$ to 3d domain $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $

Definitions: $\vec{v}e^{\vec{w}x} = \vec{v}\cos(x) + \vec{w}\sin(x)$ $ \psi_0(t) = x\hat{i}t + y\hat{j}t + z\hat{k}t, \ x,y,z\in R$ $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $ ...
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Necessary to assume $f\in C^\infty$ in this Fourier transform problem?

Consider the following problem. Is the hypothesis that $f\in C^\infty$ necessary, or could we weaken it and assume just that $f$ is continuous? Let $\hat f$ denote the Fourier transform of the ...
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Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
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“Every function can be represented as a Fourier series”?

It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series. So this got me thinking about the ...
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DFT of subdomain of periodic domain

$f(t_i,x_j)$ is a solution of stochastic differential equation on grid. $j=[0,N+1]$, $i=[0,\infty]$ and boundary conditions are periodic: $f(t_i,x_0) = f(t_i,x_N)$ and $f(t_i,x_{N+1}) = f(t_i,x_1)$ ...
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Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
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Behaviour of Fourier Inverse Transform after non-linear modulation

Suppose $\phi$ is a continuous nowhere differentiable function. $g$ some function in Schwartz space such that $\hat{g}$ has compact support. Define $f(x) = \int_{-\infty}^\infty ...
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54 views

A question on use of square integrable functions

I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function $f(x)$ satisfies the condition ...
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35 views

A form of Nash's inequality, $\|f\|_2\le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta$

For $f\in \mathcal{S}(\mathbb{R})$ can anyone help me prove the following Nash inequality, $$\|f\|_2 \le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta.$$ I believe $\alpha$ and $\beta$ should be $2/3$ and ...
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Fourier transform of isotropic Laplace distribution (2D)

How would I evaluate the Fourier transform of an isotropic 2D Laplace distribution? $F(\omega_x,\omega_y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-b \sqrt{x^2+y^2})\exp(-j\omega_x ...
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How to prove these inequalities using Fourier analysis methods

I wondered if anyone could help me prove these inequalities for $f\in \mathcal{S}(\mathbb{R})$ and $\lambda>0$: $(1) ...
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conditions on Fourier Transform of derivative

At page 445 of Myint-U's Linear Partial Differential Equations (4th Ed), Fourier Tranform of derivative is defined as: Let $f$ be a continuous and piecewise smooth in $(-\infty, \infty)$. Let $f(x)$ ...
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Use Fourier Transform to Show that $f=0$ a.e.

I was working through an old qualifier on my own when I ran across this following question that I was unable to crack. Here it is verbatim: "Let $f\in L^2(\mathbb{R}, \mathcal{L}, m)$ and suppose ...
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If the Fourier Transform of $f(x)$ is known, can one deduce the Fourier Transform of $|x|f(x)$? [closed]

If the Fourier Transform of $f(x)$ is known, can one deduce the Fourier Transform of $|x|f(x)$ ? I've been trying to find the Fourier Transform of $|x|^{7/6}K_{-1/6}(x)$. I know the transform of ...
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Inverse Fourier Transform gives a complex function when it shouldn't

Last I had an exam and there was the following question: Find $f: \mathbb{R} \rightarrow \mathbb{R}$ such that ...
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What is Fourier transform of space variable? on the similar grounds what is the Laplace transform of the same?

I understand that the transform of time domain is frequency domain and the transformation of time to frequency domain is done by Fourier/Laplace transforms. I am confused about the transformation of ...
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How is the Fourier transform a geneeralization the the Fourier series?

I have taken a self-tought course on the subject of Fourier series and Fourier transform and I got the message the the latter is a genaralization of the first. I know that the idea that the Fourier ...
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Fourier Transform on $L^1(\mathbb{R})$

For $f,g\in L^1(\mathbb{R})$, prove or disprove: $\hat{f}(\xi)+e^{i\pi \xi^2}\hat{g}(\xi) = 0$ for all $\xi\in\mathbb{R}$ implies $\hat{f} = \hat{g} = 0$.
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Fourier series & Fourier transformation [closed]

Tell me where we use Fourier series & transform in real life? Please mention an example problem which helps me to understand easily about Fourier series &Fourier transformation?
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+50

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
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Inverse-Fourier transform of a function after non-linear frequency modulation

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is ...
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Fourier Transform of $|x|^\frac{7}{6} K_{-\frac{1}{6}}(|x|)$ [closed]

What is the Fourier Transform of $|x|^{\frac{7}{6}} K_{-\frac{1}{6}}(|x|)$ with $K_{-\frac{1}{6}}$ the modified bessel function of the second kind?
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Fourier transform of a pyramid

Has anyone calculated the Fourier coefficients for a pyramid function? Let us define the pyramid function as, $z = f(x,y)$. We are looking at 5 planes making up the pyramid. The 4 base points and apex ...
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Fast multiplication times a fixed constant $A$?

Is there a way to speed up integer multiplication of billions of $B_{i}$'s times a fixed $A$? We can configure $A$ to be either small compared to the $B_{i}$'s (e.g. $10^{10}$ compared to $10^{200}$) ...
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How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
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Inverting a differential operator on $C^\infty$ functions using Fourier series

I am working on the following problem: Let $S^1 := \mathbb{R}/2\pi \mathbb{Z}$ and suppose $p(x) = a_0 + a_1x + \cdots + a_kx^k$ is a polynomial such that for all $n \in \mathbb{Z}$ we have $p(in) ...
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Searching for error reharding computing Fourier coefficient

I try to set up the Fourier series of $e^x$ in $[-\pi, \pi) $ By definition: $ a_n = \int\limits_{-\pi}^{\pi} e^x* cos(nx) = {e^x (cos(n x)+n* sin(n x)) \over (1+n^2)}|_{-\pi}^{\pi} $ Now I ...
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The space of arrival for Fourier transform.

If $f\in L^2[-\pi,\pi]$, let $\hat f$ be the Fourier transform of $f$ $$\hat f=\frac{1}{2\pi} \int_{-\pi}^\pi f(t) e^{-ixt} dt, \ \ (-\infty<x<\infty)$$ we can see Fourier transform as an ...
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What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
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108 views

Reconciling two intuitions about convolution

There are two intuitive things convolution does. In the time domain, it represents the distribution of the sum of two independent random variables. In the frequency domain, it's just multiplication. ...
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Nash's equilibrium

I'm struggling with this question and was hoping someone could help. There are 4 parts which I think lead on from one another. We have Nash's inequality for $f\in\mathcal{S}(\mathbb{R})$ of the ...
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1answer
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Problem in computing complex integrals for fourier transform

This is from a problem set of open course 8.02 by MIT OCW. I am not able to understand how the integral was solved. I have basic knowledge of Fourier transformation, and the Dirac delta function ...
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Simplifying an expression with Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help: $$f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{i \omega r^{-\gamma}}} ...
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Detrending sine waves accurately

I am doing some data analysis where I look at electricity demand over the course of a day, but need to separate the intra-day (constant and periodic) components from daily changes (assumed linear). At ...
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Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
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How does this conjecture correspond to Carleson's theorem for the case of $d=1$?

From this source, on page 36 (bottom) there is a conjecture stated and it was said that the case of $d = 1$ corresponds to Carleson's theorem. A picture included here : But when I look at wiki ...
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Dirichlet energy and Fourier transform

Is there a direct relationship between the Dirichlet energy of a function: $$E(f)=\int_{\Omega}\lvert\nabla f(\mathbf{x})\rvert^2\mathrm{d}V$$ and its Fourier transform ...
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Is there a Fourier invariant basis?

There are some functions which are invariant under Fourier transformation up to scaling factors, eg. sech(pi*x), Gaussian function etc.. Is there a set of basis functions, which form an invariant ...
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Asymptotic form of an integral to an power law decaying function

$$ f(x)=\frac{1}{2}+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right| $$ This function is not analytic at $x=1$. The plot is shown: The integral is: $$ I=\int_0^\infty g(x) \sin(2b rx) dx $$ where ...
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Fourier transforms of $f(t)=\frac{\sin{at}}{t}$

I want to derive the following pair of Fourier transforms: First: $$f(t)=\dfrac{\sin{at}}{t}$$ $$F(\lambda)= \begin{cases} \sqrt{\dfrac{\pi}{2}}, & \text{if } |\lambda|<a \\ 0, & ...
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An intuitive definition of the frequency spectrum of a function.

In a PDE book I'm reading, the author introduces the Fourier transform by first introducing the Fourier series, and then the Fourier integral representation of a function. The Fourier integral ...
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Do they have a mistake in this heat equation?

I need to know if there is a mistake in these notes: In the second page we have a representation of a function $f(x)$ as a $\sin$ series. Dont we need to have $f(0)=0=f'(l)$ for such a ...
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Changing integration limits in a Fourier transform

I've been reading these notes about the Fourier transform of Heaviside function, but I don't fully understand the derivation right after formula (5) - why does it hold only for $t>0$? The author ...
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Relation Fourier/Laplace Transform

I have a question about the relation between Fourier and Laplace transforms. I have seen in some places that the transfer functions in the Laplace space are represented as $G(s)$ where $s$ is the ...
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Discrete Fourier Transform - proof that columns of matrix are orthogonal

The DFT matrix of size $n$ has entries $M_{ij} = \omega^{ij}$ where $\omega$ is the $n$th root of 1. My textbook states that the columns of this matrix are orthogonal because their dot product is 0. ...