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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Fourier transform inversion formula for $f\in L_1(\mathbb{R}^n)$ and Dini condition

Let us define the Dini condition for a function $f\in L_1(-\infty,\infty)$, i.e. Lebesgue summable on $\mathbb{R}$, as Given an $x\in\mathbb{R}$ there is a $\delta>0$ such that the Lebesgue ...
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212 views

How do you invert a characteristic function, when integral does not converge?

I need to find the probability density of some distribution with characteristic function given by: $$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$ I know the formula for inverting a ...
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1answer
632 views

Sum of Sinusoids with Same Frequency = Sinusoid (proof)

I am studying Fourier analysis on my own, I realised that probably the first thing you want to proof in Fourier transform is that the sum of 2 sinuoids (namely a sine and cosine) with the same ...
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4 views

Short-Time-Fourier-Transform: why overlapping the window?

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I ...
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38 views

question regarding to study Sobolev space by Fourier transform

I am reading Sobolev space by using Fourier transform approach. Here I have some questions that treated to be "obvious" by textbook but I can not understand it. We define operator ...
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33 views

Orthonormal system

Let $\varphi\in L^2(\mathbb{R})$, prove that $\{e^{2\pi i m x}\varphi(x)\}$ is an orthonormal system iff $$\sum_{n\in\mathbb{Z}}|\varphi(x-n)|^2=1 \ \ a.e \ x$$ How do you prove this. The hint is ...
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31 views

Parseval's Identity holds for all $x\in H$ implies $H$ is a Schauder basis

Prove that any set $\{v_j\}_{j \in \mathbb{Z}}$ for which the Parseval identity $\|x\|^2=\sum_{j=1}^\infty |\langle v_j,x\rangle|^2$ holds for every $x \in H$ is a Schauder basis. I know that a ...
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67 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
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12 views

integration concening Fourier transfom variable and space variable

We define the short time Fourier transform as follows: $$V_{g}f(x,w)=\int_{\mathbb R} f(t)g(t-x)e^{-2\pi itw} dt, (x,w \in \mathbb R).$$ (We may assume that $f$ and $g$ nice functions so that every ...
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33 views

Voltage Distribution Inside a Cylinder [closed]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
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53 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
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9 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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39 views

Is $ \frac{2}{1+e^{t^2}} $ a characteristic function?

I'm trying to establish whether the following is a characteristic function of some random variable: $$ \phi(t) = \frac{2}{1+ e^{t^2}} .$$ It satisfies all basic characteristic function properties, ...
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16 views

Fourier transform of $|x|^{-s}$

Using the definition of Fourier transform $\hat{f}(p) = (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(x) e^{ix \cdot p} \ dx$ where $u \in \mathbb{R}^n$. What is the fourier transform of $|x|^{-s}$.
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36 views

Absolute continuity as a condition for $F[f^{(k)}](\lambda)=(i\lambda)^k F[f](\lambda)$

In read in Kolmogorov-Fomin's (p. 429 here) that if function $f$ is such that $f^{(k-1)}$ is absolutely continuous on any interval and if $f,...,f^{(k)}\in L_1(-\infty,\infty)$, [...] we get ...
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29 views

Fourier Transform of Newton's Law of Cooling

I am attempting to solve Newton's Law of Cooling differential equation with Fourier Transforms for a high school math report. Can Fourier Transforms be used to solve first-order ODEs? The equation is: ...
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1answer
24 views

How to find the Fourier Transform of the form $\frac{cos(2\pi t)}{t^2}$?

I'm having trouble on figuring out how find the Fourier Transform of the following function, and I'm not allowed to use the straight up definition of the Fourier Transform but rather use it's ...
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1answer
26 views

If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?

Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}>0: ...
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39 views

Does the Fourier Transform exist for f(t) = 1/t?

My professor says that the following function has a Fourier Transform: $$f(t) = \frac{1}{\pi t}$$ He said that all I have to do is apply some of the Fourier Transform properties and not the direct ...
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27 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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67 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...
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25 views

Fourier transformation of $\cos^2(\pi t)$

I need to fouriertransform the function $$f(t) = \cos^2(\pi t)$$ First question is if there is some kind of theorem to solve it in a very easy way but I don't know what will happen to the ...
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1answer
56 views

Show that Fourier transformation is differentiable if $\int|xf(x)|\,d\lambda<\infty$

Let $f\in\mathcal{L}^1(\mathbb{R},\mathcal{M},\lambda)$. Then we define the Fourier transform of $f$, denoted $\hat{f}$, by ...
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18 views

Fourier transform in reconstruction problem

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in ...
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25 views

To bound a heat equation on a real line?

Let $\displaystyle\mathcal{H}_{t}(x)=\frac{1}{(4\pi t)^{1/2}}e^{-x^{2}/4t}$ be the Heat Kernel. The imposed initial condition for the heat equation on a real line is $u(x,0)=f(x)$ a function belongs ...
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41 views

Finding Fourier cosine series of sine function

I am trying to find Fourier cosine series of following function, but think that I am messing up somewhere. $$ f(x)=\sin \bigg ( \frac{\pi x}{l} \bigg ) $$ Fourier cosine series can be written as $$ ...
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1answer
75 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
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81 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
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$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
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24 views

Fourier transforms in combinatorics

I've got something like a "meta-question": In some parts of combinatorics, looking at Fourier transforms can be a very helpful tool. For example, an early proof of Roth's theorem (any sufficiently ...
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14 views

Difference between the Rectangular “Window” Function and the Rectangle Function

I'm getting ahead in my differential equations textbook (Fundamentals of Differential Equations by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function ...
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15 views

Converting sum of complex exponential to sum of cosine

So I am trying to convert the equation $$\sum_{k=-2}^2 \alpha_k e^{i \frac{2 \pi}{T_0} kt}$$ Where $\alpha_0 = 1$, $\alpha_1 = 2 \angle30^\circ$, $\alpha_{-1} = 2 \angle{-30^\circ}$, $\alpha_2 = 1 ...
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16 views

Wavelet Transform of a shift invariant function

I want to calculate the wavelet transform of a shift invariant function. For example Gaussian - $\exp{-\|x-y\|^2_2} $. There is no restriction on the wavelet basis that can be used here. Can anyone ...
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19 views

Convolution using Fourier Analysis

I need to make the following convolution using Fourier analysis. Evaluate $x(t)*x(t)$ where: a)$x(t)=\frac{1}{T}[u(t+\frac{T}{2})-u(t-\frac{T}{2})]$ where u(t-x) is the heaviside function ...
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62 views

Using Fourier analysis to show a function is positive

Let $$f(x)= \sum_{n=-\infty}^{\infty} \frac{\mathrm{e}^{i nx}}{n^2+1}$$ on $[-\pi, \pi]$. Prove that $f(x)>0$ for any $x \in [-\pi, \pi]$. How to use Fourier analysis to show that function is ...
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33 views

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded?

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded? I don't even know where to start proving or disproving, ...
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1answer
28 views

Asymptotic behaviour of Fourier transform: $|F[f]|=|\lambda^{-k}F[f^{(k)}]|$ for absolutely continuous $f$

I read in Kolmogorov-Fomin's (p. 429 here) that if function $f:\mathbb{R}\to\mathbb{C}$ is such that $f^{(k-1)}$ [the $(k-1)$-th order derivative] on any finite interval and if $f,...,f^{(k)}\in ...
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33 views

the Fourier transform of a constant

How to calculate the Fourier transform of a constant without the aid of duality property? In other words, how do I calculate $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$
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Laplace equation on infinite strip

I'm trying to solve the following problem using the Fourier transform: $$u_{xx}+u_{yy}=0$$ on the domain $\;0\lt y\lt b$ , $-\infty\lt x \lt \infty \;$ with the following conditions: $$ u(x,0)= ...
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43 views

$\frac{\varphi(x+h)-\varphi(x)}{h}\to\varphi'(x)$ uniformly as $h\to 0$?

Let $\varphi$ be a bounded, differentiable function on $\mathbb{R}$ such that $\varphi'$ is bounded and uniformly continuous on $\mathbb{R}$. We want to prove that ...
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1answer
18 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
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26 views

Almost a Dirac delta

With some mindless playing with Mathematica, I found out that $$ \int_{-\infty}^{\infty} \text{d}x \frac{e^{ikx}}{\alpha+x^n}=\frac{\delta(k)}{\alpha }+\dots, $$ where the dots symbolize "other ...
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1answer
22 views

convergences in $\mathcal {S'}$

strong textLet $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$ My Question is: ...
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53 views

Problem with (classical) Fourier transform

Problem Find the Fourier transformation of $u(x) = \frac{1}{1+x^2}$ I want $\int_\mathbb R e^{-itx} \frac{1}{1+x^2} dx$. Let $f(z) = e^{-itz} \frac{1}{1+z^2}$, $z \in \mathbb C$, let's integrate ...
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57 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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18 views

Compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $

I want to compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $, where $A$ is a constant and $f$ is frequency. In the case of $ e^{-Af} $, I tried to solve it from the Fourier ...
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17 views

How to change the fundamental frequency of a sample signal?

So I am dealing with a 60Hz signal that is sampled at 1kHz. This 60Hz signal has many other harmonics (eg, 120 Hz, 180Hz..... and more). For some reason, we would like it to be 50Hz. Could we ...
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64 views

Fourier transform of Gaussian - sin/cos/Heaviside step function.

I would like to drive the Fourier transform of the following equations: $f_1(x)=e^{\frac{x^2}{2σ^2}}\cos(nx)$, $f_2(x)=e^{\frac{x^2}{2σ^2}}\sin(nx)$, where $n=2πf$ ...
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1answer
18 views

Fourier Transform Inverse of 1 / (jw - a)

I want to find the inverse fourier transform of $$ \frac 1 {j \omega - 1} $$ The fourier transform of $$ e^{-at} u(t) $$ is $$ \frac {1}{j \omega + a} $$ This result if true ONLY if a > 0. If a ...
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13 views

Discrete Fourier Transform actual frequency of pure tone function

Given that a signal s is amplified at $200Hz$ for $10$ seconds, yielding a sequence $s_t$ for $t = 0, 1, ..., 1999$. We have $s_f = \frac{1}{\sqrt{2000}} \sum_{t=0}^{\infty} ...