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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Show Fejer kernel on the real line is good, without using trignometric integrals.

This is from page 163 of Stein's Fourier Analysis. Fejer kernel on the real line is defined by $$ \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2$$ When $t=0$, $\mathcal{F}_R(t)=R$...
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One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
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Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)

On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula: $\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$ where $s>1$, $\chi$ is a ...
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Help with an Inverse Fourier transform

Can anybody please guide me how to compute the inverse Fourier Transform of: $$ f(k) = \frac{1}{1+k^2} \frac{\pi}{4}(\rm{sgn}(1-k) + \rm{sgn}(1+k)) $$
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$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
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33 views

Expansion theorem or Poisson Summation Formula? - Basis of eigenfunctions gives rise to a Fourier series

Does anyone could explain to me why in the Semiclassical's answer on the question Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula, the basis gives a series of the form $\...
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A funtion and its fourier transformation cannot both be compactly supported unless f=0

Problem : Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$. Hint : Assume $f$ is supported in [0,1/2]. Expand $f$ in a ...
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If $f$ is continuous and moderate decreasing, then Fourier transform of $f$ is continuous.

If $f$ is continuous and of moderate decrease, show that $\hat{f}$ is continuous. My attempt: $$ \hat{f}(\omega+h)-\hat{f}(\omega) = \int_{-\infty}^\infty f(x)e^{-2\pi ix\omega}(e^{-2\pi ix h} - 1) ...
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67 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: $f(x)=\frac{1}{\pi}\...
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24 views

A function and its fourier transfrom cannot both be compactly supported unless f=0 [duplicate]

Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$. I assume that $\hat f$ is compactly supported function. Then, $\exists N$ ...
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22 views

Fourier part series, missing one piece

$$F(x)=\left\{ \begin{array}{rl} ax,&0<x<\pi,\\ bx,&-\pi<x<0, \end{array} \right.$$ So, far i've got: $$a_0 = - \frac{b\pi}{2} + \frac{a\pi}{2}$$ $$bn = \frac{1}{\pi} \frac{(-1)^{...
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13 views

Characterizing functions with controlled Fourier coefficiens

It's a well known fact that an infinite dimensional Banach space $E$ is not locally compact. One may consider, at which point, is this property lost, i.e. what kind of compact sets $K \subset E$ exist....
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Control the value of a function at a point by the norm of its fourier transformation and itself

$n\leq 3$ ,$\Delta$ is the Laplacian on $L^{2}(R^{n})$, $Dom(\Delta) = \{\phi\in L^{2}(R^{n})|\Delta\phi\in L^{2}(R^{n})\}$. Please show that:for any $\phi\in Dom(\Delta)$,there exists constants $c_{1}...
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39 views

Fourier Tranform of $\frac {1}{(1+k \sin(t))^{3}}$

I'm stuck with some Fourier transforms that I'm not being able to solve and Mathematica is not helping: $\frac{1}{(1+k \sin(t))^{3}}$ (and the same one for sinh), where $k$ is a constant. Any ...
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18 views

Windowing effect and Fourier Transform

I understand how windowing effect helps to improve side lobes, of transformed signal in fourier spectrum. One way to explain this, is by pointing out that the sampled signal is considered periodic, ...
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What is the fastest way to show that FT Convolution theorem holds also in the case of a weighted sum?

Given $Y=X+Z$, with $X, Z $ r.v. such that $X \sim f(x)$ and $Z \sim g(x)$, the Fourier Transform Convolution property gets me the result: $$\mathcal{F}[(f \otimes g)(x)] = \hat{f}(\xi) \hat{g}(\xi) $...
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Existence of operator

I want to show that for $ s> \frac{1}{2} $ there is a bounded linear operator $ T: H^s(\mathbb{R}^n) \to H^{s-\frac{1}{2}}(\mathbb{R}^{n-1})$ following the below steps: Consider that $ u \in ...
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64 views

Fourier Transform Dirac Delta

I have recently learnt about tempered distributions, and how one can define the Fourier transform of a tempered distribution $v$ to be $\hat v$ so that $$\langle\hat v,\varphi\rangle=\langle v,\hat \...
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Fourier transform without using Lebesgue measure

Let $\mathbb{L}^p(\mu)$ be a space such that $$ \mathbb{L}^p(\mu) = \left\{f:\mathbb{R}\to \mathbb{R} \mbox{ measurable}: \|f\|_{L^p(\mu)} = \left(\int_0^{+\infty} \big|f(x)\big|^pd\mu(x)\right)^{1/...
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Why equals the z-Transform $c^n * u(-n-1)$? according to Matlab/WolframAlpha?

$x[n] = c^n * u[-n-1]$ Where u[n] is Heaviside step function. According to Matlab and WolframAlpha this equals 0. However if I compute the sum according to the z-Transform definition I got (sum from ...
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43 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...
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How to compute the ifft of a vector?

In the following post Concrete polynomial implementation it is said that the final step before obtaining the product of two polynomials is to compute the ifft of a vector. How to compute the ifft of ...
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Why don't we use unit impulse to find the fourier transform of unit step signal?

I have read that we can't find the fourier transform of unit step as it is not absolutely integrable. So we use signum function to find its transform .But why don't we use unit impulse function to ...
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34 views

Pointwise evaluation of $L_2$ Fourier Transform

We know, that the $L_2$-Fourier Transform of a function $f\in L_2$ is defined as a limit of $L_2$ functions (e.g. $\ \mathcal{F} f=\lim_{n\to \infty} \int_{-n}^{n} f\cdot \chi_{(-n,n)}\ d\lambda $,...
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$f \in L_2$ bandlimited implies $f$ equal to continous function a.e. (without using Parley-Wiener)

I was wondering, if my proof is right as I didn't find any similar statements in books or the internet without using the Parley-Wiener-Theorem: If we have $f \in L_2(\mathbb{R})$, bandlimited (i.e. ...
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26 views

Cross-correlation, Fourier transform and Laplace transform: measure of how much signal are alike?

I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still ...
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Convergent Fourier series of continuous function

Let $f$ be a continuous function. It is known that its Fourier series is convergent almost everywhere to $f$ and it may fail to converge on some measure zero set. However I would like to know whether ...
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42 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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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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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: $$F(\omega)=\frac{1}{\...
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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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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 I ...
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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 $\tilde{...
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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: \begin{...
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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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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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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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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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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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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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About the Fourier-Legendre series of $f(x)=e^{-x}$

So for the function $f(x) = \exp(-x)$ I have the formula for the coefficients of $$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$ which is(by using Rodrigues formula) $$a_n = \frac{2n+1}{2} \int_{-1}^{1}\...
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$L^1$ 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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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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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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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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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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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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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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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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17 views

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 $\mathbb{...