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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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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1answer
31 views

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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1answer
77 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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2answers
76 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 ...
3
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1answer
2k views

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 ...
9
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1answer
335 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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2answers
370 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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0answers
67 views

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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1answer
412 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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1answer
1k views

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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2answers
322 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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0answers
94 views

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) = ...
4
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2answers
420 views

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 ...
2
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1answer
179 views

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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0answers
100 views

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} = ...
3
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0answers
147 views

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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1answer
72 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 ...
3
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1answer
61 views

$\mathcal D(\mathbb R^n)$ 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 ...
2
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0answers
116 views

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 ...
2
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0answers
184 views

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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1answer
115 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 ...
2
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1answer
41 views

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 ...
0
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1answer
197 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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2answers
2k views

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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0answers
203 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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1answer
108 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 ...
2
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2answers
136 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 ...
2
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1answer
113 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 ...
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2answers
185 views

Detecting increasing pulse trains

I have a one dimensional point process representing the times of events which is also mixed in with lots of data that I regard as noise. The interval between the events in the point process are ...
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3answers
3k views

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ [duplicate]

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ in the range $ -\pi \leq x \leq \pi $ any help much appreciated I need find out $a_0$ and $a_1$ and $b_1$ I can find $a_0$ which is ...
3
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2answers
244 views

Fourier transform of 3D Sinc function

What is the Fourier trasnform of the function $$\frac{\sin(P|\mathbf{x-y}|)}{|\mathbf{x-y}|}$$ where $P$ is a real parameter and $\mathbf{y}$ is a fixed point in three-dimensional space?
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2answers
748 views

A function is $L^2$-differentiable if and only if $\xi\widehat{f}(\xi) \in L^2$.

In this previous question, I defined $L^p$ derivatives of functions in $L^p(\mathbb{R}^n)$. I've been struggling for a while now to prove the following: If $f \in L^2(\mathbb{R})$, then $f$ is ...
2
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0answers
106 views

What is the set of all functions which can be used as a 'convergence factor' for a Fourier Transform?

At times, I am required to take the Fourier Transform of some function that does not decay quickly enough for the Fourier Transform to converge in the usual sense. For example, $$ ...
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0answers
81 views

Is there an expansion for element-wise scaled convolution?

If $x = a\cdot b$ is used to indicate $x_i = a_i\cdot b_i$, $y = a / b$ denotes $y_i = a_i / b_i$, and $a*b$ denotes convolution, then is there a simplification for this expression: $$ ...
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2answers
96 views

Looking for a source: Fourier inversion of $f \in L^1$

Is there a book where I can find a thorough proof of the following assertion? Let $f \in L^1(\mathbb{R}^d)$ be continuous at zero and $\hat{f}\ge0$. Then $\hat{f} \in L^1(\mathbb{R}^d)$ and ...
3
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0answers
64 views

Inverse Fourier transform of $f|X(f)|$

The inverse Fourier transform of $fX(f)$ is simply given by $$\frac{1}{j2\pi}\frac{dx(t)}{dt}$$. But what is the inverse Fourier transform of the following term? $$f|X(f)|$$ Does the inverse ...
13
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2answers
747 views

Are Fourier Analysis and Harmonic Analysis the same subject?

Are Fourier Analysis and Harmonic Analysis the same subject? I believe that they are not the same. Maybe there is big difference between those subjects but I need to know what is the main difference ...
3
votes
1answer
153 views

Schwartz kernel theorem for induced distributions…

I'm studying periodic pseudo-differential operators on torus and I have a question concearning the Schwartz kernel theorem: If $A:C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n)$ is a ...
2
votes
0answers
151 views

Solution of an implicit Fourier transform equation

How does one solve the following equation ($\hat{a}(k)$ denotes the Fourier transform of $a(x)$ and $q$ is real positive): $$\hat{a}(k)=f(k)\widehat{a^q}(k).$$ This equation appeared in some paper. ...
4
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1answer
128 views

Is there a way to do this with fast convolution?

If you could please offer any advice, this puzzle is driving me mad: I've come across a problem that is trivial to compute in $\mathcal{O}(m^2)$ operations, but which very closely resembles a ...
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1answer
206 views

Fourier Transform and amplitude of waves

Given this definition of the fourier transform: $$f(t) \rightarrow \hat{f}(\omega)=\int\limits_{-\infty}^{+\infty}f(t)\,e^{-i\omega t}\,dt$$ and now ...
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1answer
49 views

Positive functions with negative Fourier tail

As the title indicates, my question is: Question: Does there exist a nonnegative function $f\in L^1(\mathbb R)$ such that the Fourier transform of $f$ satisfies $$\hat f(\xi)<0$$ for all ...
5
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1answer
131 views

Uniform boundedness of Fourier series of indicator functions

Suppose $f\in L^1[0,2\pi]$, denote by $S_n f(x)$ the partial sum of the Fourier series of $f$. I am interested in whether $S_nf(x)$ is uniformly bounded independent of $x$ and $n$, i.e. $$(*)\ \ \ \ ...
3
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2answers
148 views

Question on Rudin's Proof of $\int_{-n}^{n}\hat f(\xi)e^{it\xi}\, d\xi\to f$ in the 2-norm

In Real and Complex Analysis pg. 187, (d) property of Theorem 9.13 says: If $f\in L^2(\mathbb R)$ then: \begin{equation}\int_{-n}^{n}f(t)e^{-it\xi}\, dt\to \hat f\text{ and }\int_{-n}^{n}\hat ...
3
votes
0answers
248 views

Causality in Dirac delta forced harmonic oscillator

If I take the simple forced harmonic oscillator equation, apply the Fourier transform to both sides, and assuming the forcing function is a Dirac delta function (at the origin) I get: $ F(s) = \frac ...
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0answers
178 views

Fourier transform of projection of spherical cap

I am currently trying to derive an analytical expression for the Fourier transform of the projection of a spherical cap of the unit sphere onto the xy-plane. Setting up the integration in cylindrical ...
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1answer
803 views

reducing amplitude of fft spectrum with constant phase

I have a time series that I converted to the frequency domain using fft in matlab. I want to reduce the amplitude at a given frequency range (remove a peak between 1.5 and 2Hz) but keep the phase ...
2
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1answer
199 views

Fourier expansion of sine of cosine function

What is the Fourier expansion of $$\sin\left(A\cos(\omega t)\right),\qquad 0<A<1,$$ in the frequency $\omega$ domain?
4
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2answers
508 views

Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
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1answer
988 views

Decay rate of Fourier coefficients of a continuous function with discontinuous derivative

Can you prove or disprove the following: The Fourier coefficients of a continuous function with discontinuous derivative decay like $ 1/n^2$.