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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Bit operations to count longest string of 1s in a binary number - connections to FFT?

I found this rather applied question on another forum. How to calculate size of largest string of consecutive 1s in a binary number. However the other forum had neither much of a focus on applied ...
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37 views

trigonometric series

It is known that the eigenvalues of Sturm liouville problem: $$ u''(x)+\lambda u(x)=0 \\ u(0)=u'(\pi)=0 $$ are $\sin\left(\left(\frac{1}{2}+n\right)x\right)$ for $n=0,1...$ If for example we expand ...
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Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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Can the DTFT be Pi-periodic?

Given is the following property of the DTFT of a time-discrete signal x[n]: $X(e^{j*\theta}) = X(e^{j*(\theta-\pi)})$ In my opinion this DTFT has a period of $\pi$ and not $2\pi$, as the definition ...
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22 views

about the property of Fourier transform??

It is said that: $$F[\frac{df(x)}{dx}] = i\omega F(\omega)$$. This expression depends on the initial definition of Fourier transform, yes? if I define Fourier transform as: ...
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119 views

Uncertainty principle density argument

I proved the Heisenberg Uncertainty Principle for $f$ in the Schwartz space $ S(\mathbf R)$: $$ \int_{\mathbf R} |\xi \hat{f}(\xi)|^2 \int_{\mathbf R} |xf(x)|^2 dx \geq \frac{1}{(4\pi)^2} ...
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40 views

Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
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20 views

Fourier transform of $f(x+h)$?

Show that $f(x+h)\to \hat{f}(w)e^{2\pi i h w}$ Let $g(y) = f(x+h)$, then $\hat{g}(w) = \int_{-\infty}^\infty g(y) e^{-2\pi i y w} dy = \int_{-\infty}^\infty f(x+h) e^{-2\pi i (x+h) w} dx$, then ...
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How does one find the Fourier Series for a non-periodic function on an arbitrary interval $[-\frac{L}{2},\frac{L}{2}]$ using the complex exponential?

I was given three functions, and told to find the coefficients of their Fourier Series using $\tilde{f_k} = \frac{1}{\sqrt{L}}\int_{-\frac{L}{2}}^{\frac{L}{2}} f(x) e^{i2\pi kx/L}dx$ where ...
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Fourier transform of integral related to zeta function

In this MO question here, I asked about the Fourier transform of the zeta function. The second answer lists the following as a representation for $\zeta(s)$, with $E(x)$ as the floor function: ...
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89 views

Sampling a sinusoidal signal

Consider the signal $g(t)=\cos(2\pi \lambda t+\phi)$ that is sampled with a frequency $\tau$. Let $g_k$ denote the values of $g$ at the times $t_k=\frac{k}{\tau}$, $k \in \mathbb{N}$. (a) Show that ...
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Extracting Information From Fourier Transforms

I have some data that I believe can be analyzed best by taking the Fourier Transform and analyzing the resulting data in the new domain. In particular, I'm trying to determine if a frequency of ...
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35 views

Proof verification : regarding pointwise and norm convergence of a fourier sequence of $L^2$ function

if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. could you verify the proof? ...
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1answer
50 views

If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?

if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
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36 views

Finding explicit solution to $-\Delta u + u =f$ using Fourier Transform

This is a question from a previous year's qualifying exam, so it's possible we haven't covered all the material this year in order to solve this problem (we did not discuss PDEs in the class so far, ...
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1answer
467 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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Fourier transform of function defining half an ellipse

I'm trying to determine the expression for the Fourier transform of a function defining half an ellipse. It's been awhile since I've done Fourier transforms by hand. Obviously I can plug the ...
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17 views

Approximating Fourier transform for range of output frequencies

(This may be an elementary question, I am new to Fourier analysis.) I am working on a visualization tool. I have a real function $f(x)$, given by N samples on some interval, and vanishing outside ...
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43 views

Is there a Plancherel-type identity for generalized Fourier Transforms?

Let $S$ be in $\mathcal{T}$, the set of tempered distributions, and $\mathcal{F}S$ be its Fourier Transform. Is there some relationship for such distributions, analogous to the Plancherel Theorem for ...
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37 views

$L^1$ functions approximated by non-decreasing continuous sequences

Actually the origin problem is: Suppose $f \in L^1([0,1])$, prove that there are two non-decreasing sequences of continuous functions ${g_k},{h_k}$ which are $a.e.$ bounded, and $$f(x)=\lim_{k \to ...
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1answer
18 views

Fourier Transform pdes

I have an exam next week and I was hoping someone might be able to help me out with this question. Show that the Fourier transform of the function $f(t+a)$ is $e^{iwa}\hat{f}(w)$ . There is a list of ...
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47 views

Definition of the convolution with tempered distributions and Schwartz function

If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$, where we place ...
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16 views

Confusion with fourier coeffients

Consider $f(t) = \frac{\pi - t}{2}$, $t \in [0, 2\pi]$ The complex fourier coefficients are $c_n = \frac{1}{2\pi}\int_0^{2\pi}\frac{\pi - t}{2}e^{-int}dt$ Which turns out to be $-\frac{i}{2n}$ if im ...
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522 views

Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n ...
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32 views

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

The Plancherel Theorem for the Fourier transform $\hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt$ on $\mathbb{R}$ states that $$ ...
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1answer
34 views

When the fourier series equal to the original function?

Let $f\in L^2([-1/2,1/2])$. Define $a_n=\int_{[-1/2,1/2]} f(x) e^{-2\pi i n x} dx$ for each $n\in\mathbb{Z}$. Define $S_N(x)=\sum_{n=-N}^N a_n e^{2\pi i n x}$ for each $N\in \mathbb{Z}^+$ and $x\in ...
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60 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
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Recommend resources for understanding Phase spectrum

I am learning Fourier transform. if we apply Fourier transform on a signal, we get magnitude spectrum and phase spectrum. I want to learn phase spectrum part in detail. So can anyone recommend any ...
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Find the Fourier transform of $\sin x^2$.

I've tried it by applying integratrion by parts, but I'm not getting the answer correct. Its answer is $$\frac{1}{\sqrt{2}}\,\sin\left(\frac{k^2}{4} +\frac{\pi}{4}\right).$$ Please help in this.
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Fourier transform of convolution for $L^2$ functions

If $f,g\in L^1(\mathbb{R})$, it is not hard to show by definition that $$(\hat{f\ast g)}(t)=\hat{f}(t)\hat{g}(t).$$ But what about if $f,g\in L^2(\mathbb{R})$? The Fourier transform on ...
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42 views

Extension of the convolution theorem

From the convolution theorem, we know that the multiplication in the frequency domain is equivalent to convolution in the time domain, and vice-versa. I am wondering if there is some kind of ...
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381 views

Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
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convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
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Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
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What will be the support of the convolution of two test functions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
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Convolution of two piecewise functions

$$\phi(t)=\begin{cases}1&t\in[0,1)\\0&\text{otherwise}\end{cases}$$ and $$\psi(t)=\begin{cases}1&t\in[0,1/2)\\-1&t\in[1/2,1)\\0&\text{otherwise}\end{cases}$$ I know that ...
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298 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
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24 views

Computing Hilbert transform and envelope of a function

The following is a function with $\alpha$ being a real constant $$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$ Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope ...
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How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
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Fourier Transform of a radial function in $L^1(\mathbb{R}^2)$ [duplicate]

Let $f \in L^1(\mathbb{R}^2)$ be radial, i.e. there exists $g: [0,\infty) \rightarrow \mathbb{R}$ such that $f(x) = g(|x|)$. Prove that $f$ is also radial. (Note that this result is true for ...
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Discrete Fourier transform implementation giving results that are order of magnitude off

I tried implementing a Discrete Fourier transform in Matlab, but I found my results an order of magnitude off. I used next definition of DFT: $$ F(u) = \frac{1}{2N} \sum^{N-1}_{x=-N} f(x) e^{- \pi i ...
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Fourier Transform of operator

Let $A = (a_{jk})$ be a real $n \times n$ matrix. Let $u \in C^2(\mathbb{R}^n)$ and define $L_Au \in C(\mathbb{R}^n)$ by: $$L_Au = -\sum_{j,k=1}^n a_{jk}\partial^2_{x_jx_j}u$$ Assume $u(\infty) = ...
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Understanding Theorem $7$.$23$ of Rudin's Functional Analysis

The Theorem states the following: (a) If $ u\in D'(\mathbb{R^n})$ has its support in $rB$, if $u$ has order $N$ and if $f(z)=u(e_{-z})$ where $z \in \mathbb{C^n}$, then $f$ is entire, the resriction ...
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518 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
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Can't extract odd function with FFT

I can't correctly extract spectrum from data points of odd function (e.g. $\cos\left(\frac23\pi x\right)$, $16$-points vector $[1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1]$), instead of one function I get a ...
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how can i applay fourier transform to a fourier series+reves+fillter respons

hellow and thenks for helping. i got a tesk in class. we should design a low-pass filter (butter worth fillter) to a given band and stop band freq. i need to cunculate the respont of a periodic rect ...
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48 views

Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - ...
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39 views

Given a finite number of Fourier coefficients, can we construct a corresponding intergrable function?

Let $\xi_1,\dotsc,\xi_n$ and $\eta_1,\dotsc,\eta_n$ be real numbers. Is there a complex valued function $f\in L^1(\mathbb{R})$ such that: $\int f(x)e^{2\pi i\xi_k x}dx=1$ for every $1\leq k\leq n$. ...
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1answer
35 views

Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
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32 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...