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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Normaliztion in$L^{p}$ and $L^{q}$

Given a function f in $L^{p}$ and $L^{q}$ where $0<p,q<\infty$ Is f can always be normalized s.t. $\left\Vert f \right\Vert_p=\left\Vert f \right\Vert_q=1$
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Even or odd function. Fourrier coefficients

This is probably a very easy question, but I can't find the answer to it.. I'm working on Fourier coefficients and whether or not the integrals become zero. As far as i'm concerned this integral ...
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34 views

Positivity of the Fourier transform of a certain function

I am trying to show that the Fourier transform of $\cosh(x)^{-\nu}$ is positive for every $\nu>1$. I know that such a function has even Fourier transform and... that's about it. Could you suggets ...
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48 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
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118 views

Laplace transform of $y'' + 3y' + 2y = f(t), \; y(0) = y'(0) = 0$ [closed]

Can you do this? This is part of my final year EE work. I need to solve this in order to figure out how my sensor is behaving. $$y'' + 3y' + 2y = f(t), \; y(0) = y'(0) = 0$$ where $f(t)$ is a ...
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Example of Parseval's Theorem

In the textbook "Mathematics for Physics" of Stone and Goldbart the following example for an illustration of Parseval's Theorem is given: Until 2.42 I understand everything but I don't understand ...
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12 views

Using FFT to compute DFT of a polynomial

i currently studying about FFT and DFT and we were given simple question: Use the recursive FFT to compute the DFT of this polynomial of '3' degree: $$-1\:+\:4x\:+\:3x^2$$. So, i go to this ...
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Questions about the Fourier transform as a unitary transform

As far as I know, the Fourier transform is a (linear) unitary transform: $T: \textbf{L}^2(-\infty, +\infty) \rightarrow \textbf{L}^2(-\infty, +\infty)$ where the basis functions {$e^{i \omega x} | ...
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14 views

Equality with fourier transform

I have problem with the following equality where the Fourier transform appears: Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, ...
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Why do Fourier Series work?

I would like to have an intuitive understanding of Fourier Series. I mean, I know the formulas: $$ f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL) $$ And ...
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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fft phase plot of pure sine function, why so messy?

I am plotting the phase plot of $sin(2*pi*60*x)$ in the frequency domain. Ideally, we should only see two peaks. How come this is not the case in matlab? ...
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9 views

Phase difference of two signal of different frequency

Currently, I have two signals, the main components of both signals are 60Hz, but both also have weaker response at 180Hz + small amount of noise. As shown in the photo below, I want to find the phase ...
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3answers
52 views

Why is the integral from 0 to 1 of $\sin(2\pi nt) \sin(2\pi mt)$ equal to 0,5 if $m$ and $n$ are equal?

I am interested in this result because I am studing Fourier Series. By the way, although I have studied Mathematical Analysis, my background is not so good. Could you please explain why the integral ...
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Prove $ | \int_{\nu}^{\delta} \frac{\sin{((2N+1)\pi t)}}{\pi t} \,dt | \leq 2 \sup \limits_{M > 0} | \int_{0}^{M} \frac{\sin{\pi t}}{t} \,dt |$.

I'm trying to understand the proof of Jordan's criterion for the convergence of the Fourier series of a function $f \in L^{1}(\Bbb T)$. At the end of the proof, the following inequalities are used, ...
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Can this integral similar to the Fourier transformation of $\delta$ function be calculated analytically?

I want to calculate the following integral: $$\int_{-\infty}^{+\infty}dk\ \exp\left[i\big(kx-\sqrt{k(k-b)}\big)\right]$$ where $x$ and $b$ are both real. If $b=0$, the integral reduces to the Fourier ...
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40 views

Fourier sine series of $f = \cos x$

Let $f:(0,\pi) \to \mathbb{R}$ defined by $x \mapsto \cos x $ Show that the Fourier sine series of (odd extension) is given by $$\sum\limits_{n=2}^\infty \frac{2n(1+(-1)^n)}{\pi(n^2-1)}$$ So far, ...
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44 views

Is Fourier transform method suitable for solving equation $\int g(x-t)e^{-t^2} dt = e^{-a|x|}$

Is Fourier transform method suitable for to solve the following equation \begin{align*} \int g(x-t)e^{-t^2/2} dt = e^{-a|x|} \end{align*} Suppose we take the Fourier transform of the above ...
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18 views

How to prove these two equations

How to prove: $$x(t)*\delta^{(n)}(t) = \frac{d^n}{dt^n}x(t)$$ and $$x(t)*u(t) = \int_{-\infty}^tx(s)ds$$ To the first one, I think I could use the following formula: $$ ...
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2answers
47 views

3D Fourier Transform for 2p state Hydrogen Atom

So I have the 2p state Hydrogen wavefunction, $$\psi(r)=Nz\exp(-\frac r{2a})$$, where $N$ is the normalization constant, and $a$, is the Bohr radius. Now I want to calculate the Fourier Transform of ...
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27 views

General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
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an oscillatory integral with two parameters

Consider $$I(a,b)=\int_{\mathbb{R}}e^{i(ax^2+bx)}\psi(x)\,dx$$ where $\psi$ is smooth and supported in $\{x:|x|\in[1/2,2]\}$. How to control $I(a,b)$ in terms of $a$ and $b$? Moreover, is there an ...
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28 views

Laplace-Fourier transform issue

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ we take the generalised Fourier transform $\hat{f}(w)=\int_{-\infty}^{+\infty}e^{iwx}f(x)dx$ where $w\in \mathbb{C}$. Now assume, this transform ...
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Fourier transform of $f(x) = \chi\cos^n(\pi x)$

I ran across an abandon post from 2013 where the OP has no work shown but just a problem statement. The OP was last seen May 2013 so I doubt they will be returning to edit their post with relevant ...
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1answer
2k views

Why fourier transformation use complex number?

I know that the Fourier transform is as follows:$$\hat{f}(\xi)= \int_{-\infty}^{\infty}\exp(-\mathrm ix\xi)f(x)\mathrm{d}x$$ but I couldent understand why should use complex number $i$ in the ...
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The Heisenberg uncertainty principle in the time-frequency plane

The Heisenberg uncertainty principle says that it is impossible to have a signal with finite support on the time axis which is at the same time band limited. Is the following reasoning correct: When ...
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1answer
34 views

Fourier Transform of $|x|^n$

I know that Fourier transform of $x^n \Leftrightarrow2\pi i^n \delta^{(n)}(\omega)$ where $n$ is an integer. I was wondering how is this effect if we want to find Fourier of $|x|^n$? Of course, if $n$ ...
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construction of a partition function

Let $q$ be a large positive integer. How to construct a smooth function $\phi$ with the following properties? i) $\sum_{a\in\mathbb{Z}}\phi(q(x-a/q))=1$ for any $x\in\mathbb{R}$ ii) For any ...
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57 views

Fourier transform of exponent?

Mathematica fails to find a Fourier transform of exponent. Yet according to this page $$\mathcal{F}[e^{2\pi iat}]=\delta(t-a)$$ and via substitution, $$\mathcal{F}[e^{at}]=\delta\left(t-\frac ...
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How to show the Fourier Inverse of the Fourier transform is the identity transformation?

Let $\mathcal{F}\left[f(t)\right](x)$ be the Fourier Transform of $f$, defined regularly as $$\mathcal{F}\left[f(t)\right](x)=\int_{\infty}^{\infty}f(t)e^{-itx}\,dt$$ And let ...
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Computing Fourier Transform of $\frac{1}{t^2+a^2}$

I know this should be relatively simple, but I'm not getting the complete answer correct when I check with Wolframalpha. Here is my attempt. Going straight from the definition, with $x,t,a \in ...
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Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
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Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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Does the convolution property suffice to show $\hat \chi*\hat \chi=\hat \chi$?

Given a compact and sufficiently regular set $\Omega\subset\mathbb{R^n}$ and its characteristic function $\chi=\chi_\Omega$, I would like to conclude that the (inverse) Fourier transform ...
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1answer
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Help with proof of Poisson summation formula

I am trying to understand a proof of the Poisson summation formula and I cannot understand a vital part of it which the author seems to think is obvious, but is not obvious to me. If anyone can fill ...
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Does band-limited imply continuous?

I was hoping someone could point me to an article or text which explores the connection between the continuity of a signal in the time domain and it being band-limited in frequency domain. (Update) ...
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Fourier transform of $\frac{1}{f(t)}$

Suppose we know Fourier transform of $f(t)$ is $F(\omega)$. Can we find Fourier transform of $\frac{1}{f(t)}$. I was thinking we can write $\frac{1}{f(t)}=( f(t))^{-{1}}$. So I guess a more general ...
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1answer
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Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
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inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
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1answer
26 views

Heat flow in 1D bar fourier series problem

I am stuck on this problem: The temperature $T$ in a one-dimensional bar whose sides are perfectly insulated obeys the heat flow equation $$ \frac{\partial T}{\partial t} = \kappa ...
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50 views

Integral Equation and Fourier Analysis

I am trying to solve the following equation \begin{align*} F(\omega) G(\omega)= 2 \pi \delta(\omega)-2\pi \delta^{(2)}(\omega) \end{align*} where $F(\omega)$ and $G(\omega)$ are Fourier transforms ...
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Derivation of Transformation for basic theta function

Given that $\vartheta(x) = \sum_{n = -\infty}^\infty e^{-\pi n^2 x}$, I am trying to finish a derivation that $\vartheta(x) = \frac{1}{\sqrt{x}}\vartheta(1/x)$. I believe that I am very close. I ...
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Good recommendations for solving PDE's by integral transforms

I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't ...
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IIR Filter - Finding Magnitude and Phase

I've been asked to plot the magnitude and phase response of this IIR filter. $H(z)=\frac{0.0534(1+z^{-1})(1-1.0166z^{-1}+z^{-2})}{(1 - 0.683z^{-1})(1-1.4461z^{-1}+0.7957z^{-2})}$ And I've found a ...
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30 views

Is there a trigonometric Fourier transform formula?

I wonder if one can express the Fourier transform in the trigonometric approach like, say, in the case of the Fourier series, where we can write it as: $Sf(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left ...
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304 views

Curve fitting using FFT

In general I want to know how to use FFT to do curve fitting. If I could understand the following example I would be happy: In Matlab I do Y=[10.6534 9.6646 8.7137 8.2863 8.2863 ...
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Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n ...
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What does coefficient before Forier integral and integration limits depends on?

I've read a couple of sources on Fourier transform. All of them give different coefficients and integration limits. Wikipedia: 1, -infinity, +infinity. Russian Wikipedia: 1/sqrt(2*pi), -infinity, ...
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Laplace equation in 2D for a continuous source: is there a continuous solution?

Given the Laplace equation on $\mathbb{R}^2$ $$ \Delta u = f, $$ where $f$ is a continuous function on $\mathbb{R}^2$, can we find a solution $u$ that is in fact also continuous? Given the fact that ...
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Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...