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 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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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 $L^...
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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} \exp(\frac{-...
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142 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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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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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. $$(1+\cos(x-1))^3$$ $$= (1+\...
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$f(x) =\cos(x-y) -\cos(\delta)$ 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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63 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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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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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 L^...
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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 $\sum_{n\in\...
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construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that $$\frac{1}{m}=\sum_{j=0}^{\...
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182 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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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 $$\int_{-\infty}^{\infty}\frac{...
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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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62 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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109 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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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 2x+\ldots+...
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73 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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56 views

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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695 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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Multidimensional Fourier transform of the laplacian

In my course on electromagnetic field theory we use the Fourier transform to simplify Maxwell's equations, for example: $$\frac{\partial ^2\vec E(\vec r,t)}{\partial t^2} \rightleftharpoons -\omega^2\...
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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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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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29 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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98 views

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 \limits_{-\infty}^{\infty}...
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218 views

Artifacts and low frequencies FFT.

I am working on analyzing a time signal and want to preform a FFT. However I run in to some artifacts at low frequencies. I have managed to reproduce the behavior in a test signal. Given by $S(t) = \...
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77 views

Integrating the Fourier Transform

I am trying to show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d w'.$$ Shouldn't it be $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{w}^{+ \...
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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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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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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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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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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. $$\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{G(\omega-\omega_c)}{j\omega}e^{j\omega(\frac{p-x_0}{...
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72 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 \frac{-1-i\...
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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 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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how to find fourier transform of $\exp(-x^2/2)$

How can we find the fourier transform of $e^\frac{-x^2}{2}$ where -$\infty $ < x < $\infty $. I tried applying the standard formulae but ended up in un defined form..
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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) = \int_{-\pi}^{\...
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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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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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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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47 views

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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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)= \displaystyle\int_{0}^{...
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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 $L^1([-\pi,+\...
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Understanding Fourier transform example in Matlab

I'm studying about Fourier series and transform and I get confused with the following Matlab example of Fourier transformation: ...
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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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319 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 ...
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97 views

Fourier transform involving a dirac delta function

I know that $\int \delta(x-a)f(x) dx =f(a) $ , the fundamental defining property of the delta function. How does this change if we no longer consider $x-a$ but $a^2 -x^2$, such that the integral is ...