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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Is deconvolution simply division in frequency domain?

Is it correct to say that deconvolution simply division in frequency domain? And that convolution in time domain is multiplication in frequency domain. And is it a convention to notate a function in ...
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74 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
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Finding Fourier coefficients of functions that are defined as integrals with known Fourier coefficients?

Given a continuous periodic function f, with a period of $2\pi$, and Fourier coefficients that are $\hat f(n) = \frac{1}{1+n^2}$ , what are the Fourier coefficients of $g(x) = \int_0^xf(t)dt $? So ...
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Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
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71 views

Is the definition of DTFT using $\omega$ wrong?

I'll briefly explain this problem I faced. Let's take this simple signal: $$x(n)=\cos(\pi n)$$ The signal is not absolutely summable, however we can define its DTFT in terms of distributions. That ...
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51 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
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118 views

Decay of Fourier coefficients sequence

If $f:\Bbb R\to \Bbb R$ is a $2\pi-$ periodic, $C^1$ function, then $k^2a_{k}(f)\to 0$ where $$a_{k}(f)=\frac {1}{\pi}\int_{-\pi}^{\pi}f(x)\cos kx dx$$ are the Fourier coefficients. I ask if this is ...
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45 views

$f(x) =\cos(x-y) -\cos(\delta)$ plotting

Ok, so this is a confusing one. I'm not sure what my teacher is looking for. The problem is: Plot any number $-\pi < y< \pi$ and pick a small number $\delta > 0$ so that the whole interval ...
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18 views

Fourier Operator and roots of Identity operator

I have seen that if Fourier operator is defined by $$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 ...
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75 views

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x.

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x. I keep doing this problem and somehow I keep messing up the constants, and it just jumbles up in my head. $$(1+\cos(x-1))^3$$ $$= ...
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58 views

Basic question about Fourier transform

The problem is: Let $f \in L^{1}(R) \cap C(R)$ . Supose that $f$ is positive. Show that $|\hat{f}(\xi)| < |\hat{f}(0)|$ for all $\xi \neq 0$. My idea: By the definition of the Fourier transform we ...
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74 views

Integrating $\frac{\sin ^2(x)}{x^2}$

Fourier transforming the function: $$f(t) = \left\{ \begin{array}{ll} 1; & \mbox{ } |t| \leq 1 \\ 0; & \mbox{otherwise} \end{array} \right.$$ We get: $$F(y)=2 \frac{\sin y}{y}$$ And ...
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1answer
78 views

How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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38 views

Fourier analysis question

Let $f(t)=\frac 12 -t, t\in(0,1).$ Calculate the Fourier coefficients of the function $f$ and the sum $\sum_{n=1}^{\infty} \frac {1}{n^2}$. Note that $L^2 (\Bbb{T}) \to l^2(\Bbb{Z})$ and ...
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77 views

Relationships between growth rates of a distribution and smoothness of its Fourier transform

Let $f\in \mathcal{S}^\prime(\mathbb{R})$ be a tempered distribution, and $\hat{f}$ be its Fourier transform. It is known that when both $f$ and $\hat{f}$ are $L^2$ functions, there are relationships ...
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155 views

Convergence of the Fourier serie of $f(x)=e^{2\pi i \alpha x}$

I have some difficulties with the last part of an old exam exercise. For the 1-periodic function $f$ defined on $[0, 1[$ by $f(x)=e^{2 \pi i \alpha x}$ with $0<\alpha <1$. I have found that its ...
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69 views

calculate the integral using Fourier transform

I am asked to calculate the integral $$\int_{-\infty}^{\infty}\frac{\sin^2(\omega)}{\omega^2}e^{i\omega}d\omega.$$ I read all the posts on this site about the integral ...
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1answer
52 views

Fourier series using other bases?

The theory of Fourier series, representing a reasonable function by an infinite sum of exponential functions, is very well-developed. In addition to basic functional-analytic results there are things ...
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55 views

Number of roots of sine-and-cosine expression

Is there an easy proof of the following fact? Let $a_0, \ldots, a_n, b_1, \ldots, b_n$ be real numbers, not all zero. Then, the function $$a_0 + a_1 \cos x + b_1 \sin x + a_2\cos 2x+b_2\sin ...
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How to generalise the Fourier transform

The Fourier transform approximates a signal using a bunch of sine and cosine waves. The inverse Fourier transform then reconstructs the original signal from this information. I am told that it's ...
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93 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
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1answer
54 views

Fourier Transform Identity

I'm trying to verify the following: $$ \int_{\mathbb{R}} e^{-z\xi^2} \hat{f}(\xi) \; d\xi = \sqrt{\frac{\pi}{z}} \int_{\mathbb{R}} e^{-\pi^2x^2/z} f(x) dx, $$ for $z = \alpha i$ purely imaginary and ...
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About a function and its Fourier transform being zero at the same time.

If we have $f\in L^{2}(\mathbb{R})$. How can I prove that (if $f$ is not zero), $f$ and its Fourier transform $\cal{F}(f)$ can't be zero out of a bounded interval? I think it involves the inversion ...
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DFT matlab problem

We have signal $\sin(2\pi v_1 t)+\sin(2\pi v_2 t)$ and we know $ν_1\in{700,780,860,940}$ and $ν_2\in{1200,1340,1480}$. Also we have vector here: $$h(k)=\sin(2π ν_1 k Δt)+\sin(2π ν_2 k Δt)$$ where ...
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24 views

Fourier coefficient problem

Calculate fourier coefficient $\hat{s}(-1)$, where 1-periodic signal $s$ :$\Bbb{R}/\Bbb{Z}\to\Bbb{C}$ is defined with equation $s(t)=(2cos(\pi t))^{16}$
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Ques from exam: sequence of functions and improper integrals

$P_n(x):R\rightarrow R$ is a sequence of functions defined by: $$P_n(x)= \frac{n}{1+n^2x^2}$$ f:R->C is continuous and 2pi periodic. We define: $$f_n(x)=\frac{1}{\pi}\int ...
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19 views

Fast Fourier Transform for non-trigoniometric bases

The fast fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other basis, e.g. orthogonal polynomial bases ...
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197 views

Function with product of sine kernel

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Also, $f$ is continuous and goes to $0$ at $\pm \infty$. Let ...
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Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
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76 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

I asked yesterday on math.stackexchange a question and received no answer. Since I'm very interested in an answer, I'm reposting it here: "Let $k$ be a fixed integer, and $\mathcal{F}$ the set of ...
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69 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...
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28 views

question involving integration of fourier transform

I was reading a paper and I came across one equation, in which I had a problem deriving this equation. ...
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37 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

Let $K$ be a fixed integer, and $\mathcal{F}$ the set of trigonometric polynomials with at most $K$ nonzero terms. Let $(f_n)$ be a sequence in $\mathcal{F}$ converging pointwise (on $\mathbb{R}$) to ...
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1answer
64 views

Using the Discrete and Fast Fourier Transform for Polynomial Multiplication

I need to multiply $ f(x) = x^2-3$ by $ g(x) = -2x$ using both Fourier transformations. I think I have found the roots of some equation, and it gives f(x) $$= 1,\frac{-1+i\sqrt3}{2} and ...
2
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proof by induction to fourier problem

So if $h_n (t) = e^{\pi t^2}\frac{d^n}{dt^n}(e^{-2\pi t^2})$. Show proof by induction that $$\widehat{h_n}=(-i)^n h_n$$ Any ideas how to go about with this one? When $n=0 \to \widehat{h_0}=h_0$.
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Fourier on discrete but not sequential data

I have time series data, which is discrete as it is timestamped with microsecond resolution. It is not sequential, as in not every microsecond has a value. How would I go about Fourier in such a case? ...
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1answer
75 views

Frequency result of FFT for data that does not start at t=0

I know there are already a lot of questions about frequency bins in FFT. However I have one that doesn't really fit to the ones I read. I have time dependent data where the time does not start at t=0 ...
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1answer
52 views

Fourier conjugate problem?

Do I need to use conjugation rules to show that $\overline{h(t)}=\hat{g}(t)$ and when $g(t)=\overline{\hat{h}(t)}$? Trying to prove parsevals identity with this one. Edit: Something like this: ...
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Proof of alternate characterization of Schwartz functions

In a book that I am reading called "Integral Geometry and Radon Transforms" by Sigurdur Helgason, Schwartz functions are defined by $f \in \mathcal{S}(\mathbb{R}^n)$ if and only if for every ...
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1answer
31 views

Fourier coefficients of $e^{xe^{it}}$

I am given an arbitrary $x \in \mathbb R$ and the $2 \pi$-periodic function $$f(t) = e^{xe^{it}}.$$ The Fourier coefficients are for any $n$ given by \begin{equation*} 2 \pi c_n ( f) = ...
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simple question about a result of Fourier series

I am studying the proof of this result but i am with a problem in a part of the proof: Result: Let $f \in L^{p}(T) = \{ h : R \rightarrow C , \text{of period 1 such that } \int_{0}^{1}|f|^p < ...
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Maximize a sum of sinusoids with comensurable periods

I'm writing a program that requires finding $$\text{argmax}_\theta\sum_{k=1}^na_k\cos(k\theta+b_k),$$ where $a_k$ and $b_k$ can be any real numbers. How can I do this efficiently?
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Complex Fourier integral

Why is the $\omega$ in the solution for this integral written in absolute value? $$\int_{-\infty}^{\infty} \frac{x e^{i\omega x}}{(x^2+1)^2}dx = \frac{\pi \omega}{2}e^{-|\omega|}$$
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Fourier Series of $f(x) = 0$ from $(-\pi, 0)$, $x$ from $(0,\pi)$

I need to determine the fourier series of the following function, (using trig method, not complex) $$ f(x) = \begin{cases} 0 & \text{if } -\pi < x < 0, \\ x & \text{if } 0 < x < ...
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1answer
48 views

Interesting equation in L^1

Consider $L^{1}(T) = \{ f : R \rightarrow C \text{ with period 1 and } \int_{0}^{1} |f (x)| \ dx < \infty\}$. For $f,g \in L^{1}(T)$ the convolution is given by $(f * g)(x)= ...
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963 views

Fourier transform probability distribution

Suppose I have 1,000 independent random values with a uniform distribution $[+1, -1]$. Now suppose I take the discrete Fourier transform of this data. What the heck is the probability distribution of ...
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Find a sequence of positive functions with non-trivial properties in $L^1([-\pi,\pi])$ and in $L^2([-\pi,\pi])$

I was asked to exhibit a sequence of positive functions $\{f_n\}_{n\in\mathbb{N}}$ belonging to $L^2([-\pi,\pi])$ such that: $\{f_n\}_{n\in\mathbb{N}}$ is strongly converging to $0$ in ...
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179 views

A contour integral for Fourier transform

How does one show the following, preferably with contour integral on the complex plane? $$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$ where $x$ is ...
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47 views

Check answer on given question

I would like to take care on my answer on the following question Fourier transform involving a dirac delta function I have tried to answer this question,of course did not know exact answer,just if ...
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192 views

Given a Poisson-noisy signal, what is the noise distribution of its Fourier transform?

Disclaimer: I'm not a mathematician, but here's my attempt at a mathy version of my question Start with a noiseless, discretely sampled expected signal $I(x_n)$. Construct a Poisson-noisy measurement ...