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 analysis of an exponential function review

I am working through and reviewing some of the examples presented on Fourier analysis from a Modern Digital and Analog Communication Systems book. In one of the examples, the author goes through the ...
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Misunderstanding with Fourier sine transform…

In my copy of Table of Integrals, Series, and Products (Gradshteyn & Ryzhik) on p.1121, it says that the Fourier sine transform is defined $$F_s(\xi) = \sqrt{\frac{2}{\pi}}\int_0^\infty ...
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Fourier transform of a unity function and of unit step function

Fourier transform of the unity function is the Dirac delta distribution. I think this means: In particular, the Fourier transform of the unity function is the Dirac delta distribution, ...
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252 views

A contour integral for Fourier transform

How does one show the following, preferably with contour integral on the complex plane? $$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$ where $x$ is ...
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Basic Fourier question

Calculate fourier transform for function $h$: $$\hat{h}(\nu)=\int_{-\infty}^\infty e^{-i2\pi\nu t} h(t) dt $$ When $h(t)=1$ and $|t|\le\frac 12$. And when $h(t)=0$ and $|t|\gt\frac 12$ Also does ...
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Simple question involving the Riemann Lebesgue Lemma

If i assume the sentence: $$ \text{If} \ f \in L^{1}(0,1) \ \text{then} \ \displaystyle\lim_{n \rightarrow \infty} \displaystyle\int_{0}^{1} f(x) \sin (nx) \ dx = 0 \ (n \in N)$$ is obvious to ...
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Inverse Fourier Transform of the output, Y(f)

A linear system is defined by the differential equation: $$ y''(t) + 4y'(t) + 25y(t)= x(t) $$ The transfer function of this system is: $$ H(f) = \frac{Y(f)}{X(f)}= \frac{1}{(2\pi fj)^{2}+ 4(2\pi ...
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Fourier- Lebesgue space and Fourier transform

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
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Fourier transform of complex Gamma function

I am wondering if it is known how to evaluate the Fourier transform of the complex Gamma function, i.e. $$ ...
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123 views

Laplace Transform and Fourier Transform of a function

I have this transfer function: $$ h(t)= -\frac{1}{16}te^{-2t} $$ and the Laplace Transform is: $$ H(s) = \frac{-\frac{1}{16}}{(s+2)^{2}} $$ I know that to find the Fourier Transform, I would ...
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Sobolev spaces and integrability of Fourier transforms

I have a Lemma from a text that states that if $g\in W^{1,2}(\mathbb{R})$ ($W^{k,p}$ a Sobolev space) and the weak derivative $Dg\in L^2(\mathbb{R})$ then the Fourier transform $\mathcal{F}g\in ...
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Is Fourier transform of a $L^{1}$ integrable function is $L^{1}$ integrable?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. Let ...
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62 views

Complex Fourier Transform

Quick question, say I have a complex series of data $f(t)$, so that at each data point $t_i$ have a real and imaginary number, is it correct to calculate the power spectrum of that series (so I want ...
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252 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
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Limits of a Fourier transform

Consider a function $f(t)$ satisfying the following properties $$ \lim_{t\to\pm\infty} f(t) = f_0,~~~~~\lim_{t\to\pm\infty} f'(t) = 0 \tag1 $$ Consider now the Fourier transform of this function $$ ...
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solve laplace equation by fourier tranform

If $$ ∇^2 u=0$$ ,for $$ x≥0$$ and if $$u=f(y)$$on $$x=0$$ show that $$u(x,y)=x/π ∫_-∞^∞]〖f(ξ)/(x^2+(y-ξ)^2 ) dξ〗$$ solve by fourier tranform
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Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
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62 views

Fourier Series Approximations of Functions

From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series ...
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spectral structure of sinusoidal model

let us consider following code ...
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How to estimate (compute) Fourier transform?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. ...
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Subspace of Tempered Distributions

Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all ...
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Fourier - Are sinusoidals strictly required?

We can define all signals as a sum of sinusoidals by taking fourier transform of the signal. Thats OK. My question is, why sinusoidals.? Can there be an another transform like Fourier somewhere in the ...
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What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
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25 views

Fourier on discrete but not sequential data

I have time series data, which is discrete as it is timestamped with microsecond resolution. It is not sequential, as in not every microsecond has a value. How would I go about Fourier in such a case? ...
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Necessary conditions on $f$ if $\int|\hat{f}(\xi)|^2+\frac1{|\xi|}|\hat{f}(\xi)|^2d\xi<\infty.$

Let $f:\mathbb{R} \to \mathbb{R}$ such that $\int|f|<\infty$. Consider the Fourier transform $$\hat{f}(\xi)=\frac1{\sqrt{2\pi}}\int e^{-i\xi > x}f(x)dx.$$ Give necessary conditions on $f$ ...
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How do the Fourier Transform of sampling and the Frequency-domain convolution match?

The Fourier Transform(FT) is $X(\upsilon) = \int_{-\infty}^{\infty}x(t)e^{-2{\pi}i{\upsilon}t}dt$. The impulse train is $\delta_1(x)=\sum\limits_{k=-\infty}^{\infty}\delta(x-k)$, and its FT is ...
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DFT Vs DCT - spectrum differences

When you are transforming a signal in to a frequency domain using both DFT and DCT, say for a function sin(x), how the spectrum will be? will both frequency values (of DCT and DFT) same? Can anyone ...
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51 views

Cosine transform of a Sine function

We all know that cos and sin are two orthogonal functions hence their dot product is zero over a domain. Then in the cosine transform how can a sine function be represented by a cosine basis ...
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Evaluating sinusoid at Chebyshev points

Suppose I have a sinusoid $f(t) = A \cos(\omega t + \theta)$ and I want to evaluate it at Chebyshev points of the second kind ($\cos(\frac{2 \pi i}{N}), 0 \le i \le N, i \in \mathbb{Z}$), and then ...
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On the root of $\cos (a_1x) + \cdots + \cos (a_nx) = 0$

This is a problem I was trying to solve for a while with no succeed. Show that the equation $\cos (a_1x) + \cdots + \cos (a_nx) = 0$ has at least one solution in $[0,\frac {\pi}{a_1}]$, where $0 < ...
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Finding the complex fourier series of the function $x^2\sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
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389 views

Contour Integral of Exponential

I want to show the following for $a > 0$: $$e^{-a} = \int_{0}^{\infty}{\frac{e^{-x}}{\sqrt{x}}e^{-a^{2}/(4x)}dx}.$$
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Solve Basel problem with Fourier series of $[-\pi,\pi]\to \mathbb{R}:\theta \mapsto |\theta|$

A problem from Stein/Shakarchi's Fourier Analysis: Consider the function $f:[-\pi, \pi] \to \mathbb{R}:\theta \mapsto |\theta|$. Show $$\hat{f}(n)=\begin{cases} \pi/2& n=0 ...
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Inhomogenous Heat equation using fourier transform

Is it possible to transform the inhomogenous heat equation: $ u_t = u_{xx} + h(x,t)$ for $ - \infty < x< \infty , t > 0$ and $u(x,0) = 0$ to the integral equation: $$\int_0^t ...
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Fourier series and Fourier transform of a periodic function

The Fourier coefficient $C_n$ of a periodic function $s(t)$ with period $T$ is given by $$C_n= \frac{1}{T} \int_0^T s(t) e^{-2\pi int/T} \,dt$$ Now consider the Fourier term ...
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Upper bound for the norm of inverse Fourier tansform

Recall Hausdorff-Young inequality: For any $f\in L^p(\mathbb{R}^n)$, we have $||\hat{f}||_q\le ||f||_p$, where $p$ and $q$ are conjugate exponents and $p\in[1,2]$. It seems to me that it follows ...
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Bound of Fourier transfrom of a cutoff function

I am reading an article, I found that \begin{align} \mathcal{F}\chi_{[0,t_0]}(s) \end{align} is bounded by $\sqrt{1+s^2}^{-1}.$ I do not know how to get this ?
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Determining the amplitude of $x(t) = 3\cos^2(\omega t - \frac{\pi}{3})$ at $2\omega$ with fourier series

The given function is $x(t) = 3\cos^2(\omega t - \frac{\pi}{3})$. I have to determine the amplitude of the component with frequency $2\omega$ in the fourier series of the function. I can only do it ...
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132 views

Denoise using wavelet transform

My mathematical class task is to de-noise a function using wavelet transform. I am to select a function $f(x)$ and noise function with zero-mean $n(x)$. I am to add noise like this: $$f_{noise}(x) = ...
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Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...
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Is there a combinatoric identity for the multiplicities of the following set?

Are you ready for some psychedelic pictures? Define the multiset$$S_n=\left\{\sum_{j=1}^n(-1)^{\left\lfloor(k-1)/2^{j-1}\right\rfloor}u_n^j\mbox{ for }1\leq k\leq2^n\right\}$$ where ...
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Period of a multivariable function

consider a function $$f(x_1, x_2, \ldots, x_n) $$ is it possible to compute the period of the function as a vector $$\langle l_1, l_2, \ldots, l_n\rangle$$ where each $l$ denotes the period of the ...
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331 views

Shifting using fouriertransform

I'm doing a small mathematical exercise where I take a function, perform a Fourier transform on it and then multiply the result by $e^{i\alpha w}$. And then take the inverse Fourier transform of the ...
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Why is the transition band of a least-square linear-phase FIR filter seems always monotonic

Given desired magnitude response and linear-phase constraint in predefined pass band and stop band, we can get the desired frequency response in both bands. By sampling the frequency in both bands, we ...
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What does this sentence of Greek means in the book of Modern Fourier Analysis?

I am reading Loukas Grafakos' book Modern Fourier Analysis and found this apparently Greek text in the first pages What does it mean?
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Is there a relation between $l_p$-norms of functions with same Fourier spectra but w.r.t different measures on the Hamming cube?

Informally, I want to ask if two functions $f$ and $g$ on the Hamming cube have the same Fourier spectra but w.r.t different measure and basis, then is $||f||_p$ related to $||g||_p$? (Where each ...
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280 views

Why is the DTFT (Discrete Time Fourier Transform) unique to each input?

As the title implies. I know the DFT of a signal is unique due to the matrix, but can anyone give a solid explanation as to why the DTFT is unique for each signal input? Thanks for your time!
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a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
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Understanding Dirac delta integrals?

I'm confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$ and am told this evaluates to 8. Can anyone please ...
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112 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...