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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Which of these is correct for Fourier transform?

Which of these is a correct definition of Fourier transform? $$\mathcal{F}(f(x))= \int\limits_{-\infty}^{\infty} f(x)\, e^{-i k x} \, \mathrm{d} x \ \ \ \ (1)$$ and$$\mathcal{F}(f(x))= ...
2
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2answers
49 views

What is the intuition behind summability kernel and convolution?

In Fourier analysis, when I look at the theorems and useful results derived using summability kernel and convolution, I get to think "Ok, I guess it works that way. but what is the intuition behind ...
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1answer
57 views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
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1answer
49 views

Convex function on closed interval: boundary points?

$I=[a,b]$, let the function $f:I\rightarrow\mathbb{R}$ be convex. (1) Is it possible to prove the existence of the limits: $$\lim_{x\rightarrow a^+}f(x) \ \ \ \ \ \lim_{x\rightarrow b^-}f(x)$$ If ...
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1answer
18 views

$g$ is differentiable and $g'(y)=\int_{\mathbb{R}}ixf(x)e^{iyx}dm(x)$

Let $f \in \mathcal{L}(\mathbb{R},\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ measurable Lebesgue. Asumme that $x\to f(x)$ is measurable. For $y \in \mathbb{R}$ define: ...
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20 views

PCA Vs Fourier Transform

As a general rule when trying to deconstruct a noisy signal into its components. When is it better to use Principal Component Analysis and when is it better to use a Fourier Transform?
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24 views

If the signal's frequency is multiples of the first harmonic frequency, transform method similar to DFT but use less number of samples?

Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This ...
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1answer
48 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
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2answers
52 views

Fourier coefficient of convex function

On $I = [0, 2π]$ consider the function $f : I → \mathbb{R}$ to be convex. Define: $$a_k\pi := \int_0^{2\pi}f(x) \cos(kx)\,dx$$ Show that the convexity of $f$ implies that $a_k ≥ 0$ when $k ≥ 1$. ...
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34 views

About a generalization of the Riemann-Lebesgue lemma

We have that for $f$ $\in$ $L^{1}$$(\mathbb{R}^n)$, $g$ measurable and bounded on $\mathbb{R}^n$ and, for any rectangle $R$, $$ \lim_{m(R)\rightarrow\infty} \frac{1}{m(R)}\int_Rg(x)dx = 0 $$ Then: ...
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1answer
25 views

fourier series sketching (by hand)

I calculated the Fourier Series representation of $f (x) = 1 − |x|$ on $−1 ≤ x ≤ 1$ and now I am asked to sketch the graph of the series on $−3 ≤ x ≤ 3$ by hand. How do I do this? I read through my ...
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1answer
45 views

Book recommendation for wavelet analysis

I am master student doing research in data mining, i read a paper about wavlet analysis for data mining, so i think it may help me in the future. But in my undergraduate degree the last course in ...
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18 views

About Fourier multipliers

I need some help with the following question: How can I prove that a Fourier multiplier sequence $\lbrace{m_n\rbrace}_{n=-\infty}^{\infty}$ mapping $L^{\infty}(\mathbb{T})$ into $C(\mathbb{T})$ ...
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35 views

Integrability of fourier transform

Let $f\in L^1(\mathbb{R})$ such that there exist $R,\delta >0$ for which $f$ is bounded in $[-\delta, \delta]$ and $\hat{f}(\xi)\geq 0$ for $|\xi|\geq R$. Then $\hat{f}\in L^1(\mathbb{R})$. ...
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23 views

Iterpolating to find the zeros of a complex function

I have an $N\times M$ grid of complex points sampled from some unknown complex function. I would like to interpolate and find the zeros of that function. I believe that this function can be well ...
2
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23 views

Are frequency domain and Fourier space the same thing?

What is the difference between frequency domain and Fourier space? When we perform FFT on an image the result is in "frequency domain", right? How does this relate to Fourier space? I spent some time ...
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1answer
26 views

Anti Hermitian Operator

I am required to show that the operator $\partial_t$ is Anti-Hermitian. This operator is defined such that $$\partial_t: s(t) \rightarrow \partial_t s(t) $$ Where the definition of an Anti-Hermitian ...
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3 views

How to take dft of irregularly sampled function in k space?

I would like to take inverse dft of irregularly sampled complex function in k-space. I am just summing in a loop over the length of the k vector, but is quite slow. ...
3
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25 views

How to understand the mapping between a periodic function to its Fourier coefficients?

For a periodic function $f(x)=f(x+T)$, its Fourier transform can be written as an infinite sum: $$ f(x)=\sum_{-\infty}^{\infty}c_n e^{2\pi i x/T}. $$ This seems to suggest that the information ...
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1answer
46 views

matlab problem - removing frequencies after FFT, signal processing

I want to stress that this is not a coding problem, my problem is that i don't fully understand the mathematics surrounding the subject and that's why I believe I have a problem. I was given an ...
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18 views

Fast Fourier Transform: How is the roots of unity matrix divided?

For an example for input size N=8, how is the roots of unity matrix divided for a divide and conquer approach? My understanding is that it's divided into four quadrants, Ma with J&K evens; Mb ...
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1answer
31 views

Inverse Fourier Transform Proof

I am aware of how Fourier Transformation and Fast Fourier Transformation works, however I do not understand the logic of the inverse of FFT. Could someone explain why the inverse fourier ...
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70 views

Fourier transform of $ \log(x^{2}+a^{2}) $

I would like to evaluate the Fourier cosine transform of $\log(x^{2}+a^{2})$ or the integral $$\int_{0}^{\infty}\cos(ux)\log(x^{2}+a^{2})\,dx$$ for any real $u,a$. However, it seems that this ...
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1answer
48 views

Computing the Fourier transform of $e^{-|x|}$

I'm trying to compute the Fourier Transform of $e^{-|x|}$ however I receive a different answer than wolfram alpha. Wolfram Alpha gets $\sqrt{\frac{2}{\pi}}\frac{1}{1+\varepsilon^2}$ and I get ...
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1answer
32 views

Problem with finding a fourier transform

Help me to find a 2D fourier transform: $$\int dx dy\ \frac{e^{-ik_x x}e^{-ik_y y}}{\sqrt{x^2 + y^2}}.$$ All I've done so far is $$\int dx dy\ \frac{e^{-ik_x x}e^{-ik_y y}}{\sqrt{x^2 + y^2}} ...
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1answer
67 views

Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
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1answer
109 views

Several questions about integral operators.

I have been fumbling with expressions of the form \begin{equation} A\{f\}(s) = \int A(s,t)f(t)\operatorname{dt} \tag{$\star$} \end{equation} as a generalization of the matrix product. When looking ...
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1answer
46 views

convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
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59 views

How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

Let $m$ be a positive integer and define $$ Lf = -\frac{d}{dx}(1-x^{2})\frac{df}{dx}+\frac{m^{2}}{1-x^{2}}f $$ on the domain $\mathcal{D}(L)\subset L^{2}(-1,1)$ consisting of all twice ...
2
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33 views

Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain ...
5
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40 views

Intuition behind the proof of the Inverse Fourier Transform?

I am interested in the proof of the Inverse Fourier Transform for absolutely integrable real valued functions. The proof I have read asks you to consider an auxiliary function $g_{a}(x)$ defined as ...
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1answer
60 views

Compact set of measure zero and sequence of Harmonic Functions with nice properties.

I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem: Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there ...
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1answer
46 views

Partial Sum Fourier Series

Show that the partial sum $$f_N(x)=\frac{4}{π}\sum^N_{n=1}\frac{\sin((2n-1)x)}{2n-1}$$ may be written as $$f_N(x)=\frac{2}{π}\int_0^x\frac{\sin(2Nt)}{\sin(t)}\,dt$$ The original question is 'Sketch ...
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1answer
33 views

Computing inverse Fourier transform

To compute the inverse Fourier transform I need to evaluate the following integral ...
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29 views

If f is continuous and f $\in$ L1 then $\lim_{\tau\to \tau_0} \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $0$?

Where $ \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $\int_\mathbb{T} \vert f(t-\tau)-f(t-\tau_0)\vert \ dt$ I find it easy to see when f is uniformly continuous, since we would have $\vert ...
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1answer
35 views

Using dummy variable to derive Nth partial Fourier sum

I am working on a problem$^{(1)}$ as follow: Using the standard formulas for the Fourier coefficients, show $$F_N(x) = \frac1{2\pi} \int_{-\pi}^{\pi} \left( 1+2 \sum_{n=1}^{N} ...
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0answers
25 views

Calculate Distance between Fourier Transforms

I'm working with signal data (specifically data from accelerators and gyroscopes), and I take their Fourier transforms to get a better idea of the dominant frequencies. I'd like to compare the ...
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2answers
58 views

Fourier Series Representation $e^{ax}$

a) Compute the full Fourier series representation of $f(x) = e^{ax}, −π ≤ x < π.$ b) By using the result of a) or otherwise determine the full Fourier series expansion for the function ...
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1answer
28 views

Help with Homework Problem for Img Processing class

I have this homework about Image Processing: Give the general equation of a complex or real-valued digital image that produces a delta function in the frequency domain. Demonstrate this function for ...
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1answer
20 views

DFT by $n$ samples of a continuous periodic signal with more than $n$ frequencies

It is known that if we only have $n$ samples and take DFT, we only get at most $n$ distinct frequency data. But let's say that there is a continuous periodic signal with more than $n$ frequencies, ...
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1answer
28 views

Fourier transform doubting factorization

I have to find the fourier transform for $$ {1\over 1+16t^4} $$ I guess going there is a better way to solve it than going throug the integral but I'm not even sure if the factorization i made is ...
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9 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...
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1answer
42 views

Sign mistake in Fourier transform of $\frac{x}{1+x^2}$.

I want to calculate the distributional Fourier transform of $u(x) = \frac{x}{1+x^2}$ in one dimension in the distributional sense as $u\notin L^1$. I use the distributional definition of the Fourier ...
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2answers
29 views

If a signal is periodic, can the error of approximation by Discrete Fourier Transform be avoided when using finite number of samples?

As title says, if a signal $f(t)$ is periodic, can approximation errors of approximation by discrete Fourier transform (DFT) be avoided when only finite number of samples are used?
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1answer
25 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
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2answers
41 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
2
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1answer
41 views

Proving that the Gaussian is the minimizer of Heisenberg's uncertainty product via variational calculus techniques

The version of Heisenberg's uncertainty product I am interested in is $$\left(\int_{-\infty}^{\infty} t^2|f(t)|^2\,dt\right)\left(\int_{-\infty}^{\infty}t^2|\mathcal{F}f(t)|^2\,dt\right),\tag{1}$$ ...
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1answer
29 views

Using Discrete Fourier trasform of the samples of a continuous/periodic signal to obtain frequency data similar to FT of the original signal

Suppose we have a continuous and periodic real-valued 1D signal $f(t)$. Let us say we obtain finite number of samples $f(n)$ from $f(t)$. Is there a way to take discrete Fourier transform of $f(n)$ ...
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1answer
45 views

Fourier Series estimation

I know that the Fourier coefficient of $t\mapsto \frac{1}{\sqrt{\vert t\vert}}$ are given by some Fresnel integral, and behave like $O(n^\frac{-1}{2})$. Reciprocally, if I get a Fourier Series whose ...
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1answer
18 views

Discrete fourier transform of random noise interpretation

Let's say i have a real valued random noise $\eta(t)$, for which I took $N$ samples with $N$ even, i have therefore a vector of sampled values. I want to compute the Discrete Fourier transform of the ...