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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FFT for Fourier Integrals of analytic Functions

So, I'm trying to implement a fourier-transform of an analytic function through DFT or FFT as I'm only interested in a certain frequency range. So far I've tested Numerical Recipes' FFT algorithm ...
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106 views

Fourier Transform of $ g(x) = (1 + x^{1/a} )^a $

Given a function $g(x) = (1 + x^{1/a} )^a \times 1_{\{x > 0\}}$ that is bounded that is bounded by zero, can I compute the fourier transform? Thanks!
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An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
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158 views

Sum of Fourier Series

I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by $$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$ in the interval $]-\pi,\pi[$ ...
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63 views

Periodic Fuctions - Signals -

If, in the periods, the two half's signal periodic have the same form and opposite phases, the periodic signal has symmetry of half wave. If the periodic signal $g(t)$, of period $T_0$, satisfy the ...
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202 views

For a real valued function $g(t)$, how to prove that $G^{*}(f) = G(-f)$ , where $G(f)$ is the fourier transform of $g(t)$?

Suppose real function g(t) has corresponding fourier transform G(f). In one text book I saw that the complex conjugate of G(f) equals G(-f). How to prove this? ie for a real valued function $g(t)$, ...
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These questions are all about Fourier analysis.

Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: ...
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90 views

I have an equality which I need to prove

This question apppear in the chapter of fourier series. Suppose $f$ is Riemann integrable function in $[a,b]$, and $g$ is a periodic function in $\Bbb R$. The period of $g$ is $T$, and $g$ is ...
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question about 1D & 2D Fourier transformation

I have 100 1D signals (each corresponds to a 1D array of 38684 elements). Those signals are all independent and I need to find the spectrum of each of them. In this case, I have to perform 1D Fourier ...
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156 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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101 views

Is there a link between divergent series and discontinuities in Fourier integrals?

Take the derivative of the function $$\frac{d}{dx}\frac{e^x-1}{e^x+1}= \frac{d}{dx} \left(1- \frac{2}{e^x+1} \right) = \frac{2e^x}{(e^x+1)^2}$$ At $x=0$, this is equal to $\frac{1}{2}$. However, ...
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304 views

Trigonometric interpolation

From http://en.wikipedia.org/wiki/Trigonometric_interpolation trigonometric interpolation can be calculated as follows: Now assume we have 6 data points (0, 0.1), (1, 0.3), (2, 0.4), (5, 0.3), (6, ...
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170 views

Fourier Coefficients : Frequency Shifting

The FS coefficients of a signal $x(t)$ is given by $$x(t) \longrightarrow C_k$$ The frequency shift property says that: if we multiply a signal $x(t)$ by $e^{j m\omega_0 t}$ the fourier series ...
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174 views

Will spectral analysis help me understand digital signal processing better?

I am learning Fourier transforms, Z transforms etc. in Digital Signal Processing and I can work easily with integrals. However, I don't understand how a Fourier transform converts time domain signals ...
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93 views

Integration details

How to do this integral details? \begin{equation*} \int_0^\infty a_0 \sqrt{r} e^{-r^2/R^2} J_\frac{1}{2}(kr)r \, \mathrm{d}r. \end{equation*} It can be done easily by Mathematica: ...
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Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
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Definition of $L^p(\mathbb T)$ with $\mathbb T$ the unit circle

I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation ...
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116 views

Bessel function to $\sin(kr)$

$J_{\frac{1}{2}}(kr)=\frac{\sqrt{\frac{2}{\pi }} \text{Sin}[\text{kr}]}{\sqrt{\text{kr}}})$ This can be easily obtained by Mathematica, How to do the details?
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162 views

Lower bound of Fourier transform

We know the Fourier transform of the Gauss-function: $\displaystyle\int_{\xi\in\mathbb{R}^d}e^{-\pi\, C\,|\xi|^2}e^{2\pi i \xi\cdot X}d\xi=C^{-d/2}e^{-\, \pi\, |X|^2/2}$ for any $C>0$. Then ...
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34 views

How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where ...
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800 views

Fourier transform of the indicator of the unit ball

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.
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Accuracy of Real FFTs

I'm not sure if this is the best place to ask, please advice if not. I'm performing the following test, to check the output of real-FFTs, and I'm getting surprisingly high errors. So a real-FFT only ...
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792 views

Rigorous derivation/explanation of delta function representation?

I am interested in a derivation of the following representation for the Dirac delta function: $$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$ It is clear to me how the property ...
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How to show $\|Pf\|_{L^1(\mathbb T^n)}\leq \|f\|_{L^1(\mathbb T^n)}$:

I need some help with the following problem: Let $P:S(\mathbb R^n)\rightarrow C^\infty(\mathbb T^n)$ be the operator given by $f\mapsto Pf$ where, $$Pf(x)=\sum_{k\in\mathbb Z^n} f(x+k).$$ How can I ...
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73 views

Norm of convolution of $n$ Gaussians

If $$f(x)=e^{-(\pi x)^2}$$ and $$\psi_n(x)=(f* f*\dots*f)(x)$$ ($n$ times convolution). Show that $$\lVert \psi_n(x)\rVert = 1$$ (norm in $L^1(\mathbb{R})$). I've tried using the Fourier ...
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74 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
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Is there an elementary proof of the convolution theorem?

Is there a way, without using much extra theory (other than the basic ideas used in textbooks deriving the Fourier transform for the first time, and ideally just using general theorems about ...
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289 views

Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the ...
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334 views

Proof for Gaussian integrals of form $\int_{-\infty}^{\infty} {\alpha\over\sqrt{\pi}} e^{-\alpha^2 x^2} f(x) dx$?

I have two equations in question here, and I'm looking for a method of solution that doesn't involve the error function $\mathrm {erf}(z)$, and a proof that the answers given are correct. Both these ...
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Moments of Fourier transform

Fix a smooth $\mathcal{C}^{\infty}$ compactly supported function $f$ with the support of $f$ being the unit interval $(-1,1)$ and with $\hat{f} \geq 0$. Is it true that as $k$ goes to infinity $$ ...
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332 views

Parseval's Theorem

The Fourier transform $\tilde f(k)$ of a function $f(x)$ is defined as $$\tilde f(k)=\int_{-\infty}^{\infty}e^{-ikx}f(x)dx $$ and the correlation $h(x)$ between two functions $f(x)$ and $g(x)$ is ...
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Fourier transform of an image: how to interpret the graphs?

Below are graphs (frequency spectrum) of the fourier transform of some simple images. How should I interpret these graphs? I know the center stands for the low frequency of the original image (the ...
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203 views

Fourier transform graph, what are the “negative” frequencies?

I would like to know why the graphs below are symmetrical around x = 0. How can signals exist at a negative point in time or frequencies be negative? Does this have something to do with the complex ...
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Function supported on [-1,1] with arbitrary prescribed sub-exponential Fourier decay?

Given $g:[1,\infty) \rightarrow (0,\infty)$ with $g(t) = o(t)$, does there exist $f:\mathbb{R} \rightarrow [0,\infty)$ with support contained in $[-1,1]$ such that $$ \widehat{f}(y) = ...
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Derivative of Integral (in Fourier transform)

I've taken some analysis, but somehow Fourier transforms were never brought up until they were assumed to be familiar. Fun. Anyway, in a class example (showing the integral of a Gaussian is again a ...
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Determining the inverse discrete cosine transform II (IDCT-II)

I am preparing myself for my upcoming math exam and one of the preparation exercises includes (i) showing that the DCT-II is invertible (ii) determining the formula of the inverse DCT-II The ...
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Fourier transform of $(x^2+b^2)^{-1}$

The question asks to find the Fourier transform of $(x^2 + b^2)^{-1}$ given that $F[e^{-b\vert x\vert}](k) = \frac{1}{\sqrt{2\pi}}\frac{2b}{k^2+b^2} $ $$\sqrt{\frac{\pi}{2b^2}}e^{-b\vert x\vert} = ...
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Fourier coefficient of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x})$ for $\nu \in (0,\frac{1}{2})$.

In Zygmund's Trigonometric Series, vol I, on page 19 section 2.22 they write that Riemann showed that the Fourier coeff of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x}))$ for $\nu \in (0,\frac{1}{2})$ ...
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60 views

To partition the unity by translating a single function

I am trying to show that there exists a (real or complex-valued) function $\psi \in C^\infty(\mathbb{R}^n)$ having the following properties: The support of $\psi$ is contained in the unit ball ...
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$\mathcal D(\mathbb R^n)$ is contained in $\mathcal S(\mathbb R^n)$

The set of smooth compactly supported functions are contained in the Schwartz space. It is somewhat OK to understand the steps in the proof that $\mathcal D$ is dense in $\mathcal S$ but I do even ...
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A Fourier-type integral of a piecewise function

consider the integral : $$f(t)=\frac{t}{\pi}\int_{1}^{\infty}\left(\lambda+ \sum_{n=1}^{\infty}\frac{\left \{ x^{1/n} \right \}}{n} \right)\frac{\cos(t\log x)}{x}dx$$ Where $\left\{ x\right\}$ is the ...
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Closed formula Fourier transform of complex exponential

Is there a closed formula for the Fourier transform of the function \begin{equation} f(t) = e^{2\pi i \sqrt{1-t^2}}, \end{equation} where the square root for $|t|>1$ is $i \sqrt{t^2-1}$. This ...
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101 views

Given a signal in the time domain, is there a way to determine a function that produces that signal?

Disclaimer: I'm by no means an expert in any of this, and I'm just wondering whether a solution to this problem already exists. Using a raw audio waveform as an example, let's say you have a 1:00m ...
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The conservation of a critical non-linear dispersion equation.

Consider the non-linear problem $$ \frac{1}{i}\frac{\partial{u}}{\partial{t}}-\frac{d^2u}{dx^2}=\sigma|u|^{\lambda-1}u$$ $$u(x.0)=f(x)$$ Suppose that $u$ is a smooth solution that decays ...
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1answer
173 views

Inverse Fourier transform to find out $\hat c_1$

If we have an integration which is need to solve inversely $$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$ If I transform the $\sin(kr)$, then we get imaginary part. Please ...
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Relation between Heaviside step function to Dirac Delta function

I understand that "delta function" is a distribution, not a function, as in it acts on another integrand, picking out the value of that integrand at a specific point. The discontinuous function is ...
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145 views

Inverse Fourier-cosine transfrom

Suppose we have a function $F(x)$ given by the integral: $$F(x)=\int_{0}^{\infty}f(t)\frac{\cos(t\log x)}{t}dt\;\;\;\;\;(x>1)$$ This looks tantalizingly like a Fourier-cosine transform of ...
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100 views

Integration of the cosine function in Fourier transform

How do I integrate the following: $$\frac{1}{2\sqrt{2\pi}}\int_{-\infty}^\infty(e^{iax}+e^{-iax})e^{-ikx-b|x|}dx $$ The answer is ...
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122 views

Residue theorem, $\int_{-\infty}^{\infty} e^{-ikx}(1-ika^2)^{-m} dk$ with integer $m$

I am trying to solve this integral $\int e^{-ikx}(1-ika^2)^{-m} dk$ using the residue theorem, but I cannot find the residue of the function. $$\frac{1}{(1-ika^2)^{-m}}=\sum (ika^2)^n(-1)^n ...
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91 views

FFT on circular data?

You may have seen the demos of computing heart rate by looking at video, where they detect the face, compute a mean on the red channel from the face, submit the resulting mean-per-frame to an FFT ...