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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Solving the wave equation bounded by one free end and one fixed end

Given that $\{\sin\left[\frac{(2n-1)\pi}{2L}x\right] : n\in\mathbb N\}$ is the complete set of eigenfunctions of a regular Sturm-Liouville with boundary points $0$ and $L$ and weight function $1$, and ...
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Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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Fourier transform and Z transform question?

Lets suppose we have an exercise where I have to find the Z transform and its region of convergence.I find the Z transform and the region.How do I determine if the Fourier transform exists from this ? ...
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70 views

How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
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Mellin transform with compact support

Mellin transform for $f(x)$ is usually defined as: $$F(s)=\int_0^\infty f(x)x^{s-1}dx$$ Is there a Mellin transform with compact support? For example like $$F(s,a,b)=\int_a^{b} ...
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35 views

Convergence of a sum of sines

If $ s_N(x) := \sum_{n = 1}^N c_n \sin(n x) $ converges uniformly on $[0, \pi]$ as $N \to \infty$ then $c_n = o(n^{-1})$. a) Is $c_n = o(n^{-1})$ sufficient for uniform convergence? b) Is $\sum_n n ...
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53 views

How to get fourier series of 8-bit character to be transmitted?

I have been reading this in a book, but can't understand how he used the 8-bit in fourier series equation to get the result below. The transmission of the ASCII character ‘‘b’’ encoded in an 8-bit ...
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327 views

Solving a tough integral

I am studying telecommunications theory and I was doing an exercise where it's required to find the (infinite) taps of a zero forcing equalizer. Here's the point where I am stuck at: $$ ...
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28 views

Fourier transform of $e^{if(x)}$

I'm trying to find an explicit result for the following Fourier transform: $$\mathcal{F}\left[e^{if(x)}\right](k)=\int_{\mathbb{R}^n} e^{if(x)}e^{-ik\cdot x} dx$$ So far I could come up only with a ...
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60 views

An entropy inequality

Let $f:[0,2\pi]\to \mathbb{R}$ be a smooth, positive function such that $f(0)=f(2\pi)$, and $\int_0^{2\pi}fd\theta=2\pi.$ Is it true that $$2\int_0^{2\pi}f\ln fd\theta- 2\int_0^{2\pi}\ln ...
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Need a closed form for fourier coefficients (if it exists)

i have a set of 53 fourier coefficients. the dc term is 0. the 26 positive frequency amplitudes (coefficients) are given below. the 26 negative frequency amplitudes are the same. {0.014451, ...
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3answers
36 views

Fourier Series Trig Functions

I need assistance finding the fourier series for the following function: $$ f(x)=3\cos^2(5x) $$ I know that $$ a_0={1\over 2\pi}\int_{-\pi}^\pi 3\cos^2(5x)\,dx={3 \over 2} $$ and $$ ...
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a question about integration by parts

Suppose that $t f(t) \to 0$ when $t \to \infty$ and $t f(t)\to 0$ when $t \to 0$. For the following integral, $$I(z)=\int_0^{\infty} f(t) \cos (z t) \mathrm{d}t,\qquad z>0 \tag{1}$$ We can apply ...
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1answer
50 views

Fourier series of rescaled cosine function [closed]

How would I find the Fourier series of $\cos\left(\, 5x/2\,\right) $ on $\left[-\pi,\pi\right]$? Progress $$A_0={1\over 2\pi}\int_{-\pi}^\pi \cos(5x/2)dx={2\over 5\pi}$$ $$A_n = {1\over \pi} ...
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Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
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Need some help computing Fourier Transforms of a couple of functions

I have the following functions and would like to find the Fourier Transform: a) $x(t) = (t + 2)^2 e ^ {-2t} $ b) $x(t) = e^{-5it} [σ (t+7) - σ (t+1)]$ I don't really know what $σ$ stands for so I ...
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30 views

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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1answer
23 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
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19 views

Frequency response of unit impulse function

Could someone throw some light on how to get the frequency response of unit impulse function. I am not from EE, but I need it for my wavelet study.
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17 views

Heat Equation Two Conditions

I'm currently working on solving the Heat Equation in a one dimensional rod of length $L$. However, instead of the 'usual' singular condition $u(x,0)=f(x)$ for all $0\leq x\leq L$, I am given ...
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29 views

Finding inverse of a general linear transform

I'm not a mathematician, so I may abuse some notation here. Please comment for any clarification. Let's define a general linear transform as $$\int_XK(\mathbf{\omega},x)f(x)dx$$ where $X$ is some ...
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Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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58 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
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1answer
34 views

Power series and Fourier identity approximated in two or three iterations

I understand that Fourier has proven that the sum of sines and cosines can be used to describe (almost) any curve. The power series describe that the sum of polynoms can be used to describe (almost) ...
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70 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
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33 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
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1answer
34 views

Fourier transform of a continuous non-periodic function in matlab

I would like to use matlab to find an Fourier transfom of a function which is known only on a grid. As an example I take the function f(x) = exp (-x^2), which Fourier transform is known and is equal ...
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57 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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Computing Fast Fourier Transform in R

This is a math question with programming application. So I am trying to find 3 things given a certain function in the x domain when transformed into the spectral domain. 1) the Amplitude 2) The ...
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17 views

Zeros of a finite Fourier integral implies the same for all derivatives

Let $\phi(x)\in C_0^{\infty}(\mathbb{R})$ be an infinitely differentiable finite function with support $\operatorname{supp} \phi \subset [-c,c]$ and let $\mu_0$ be a zero of the function ...
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Formal inverse of a matrix ressembling Fourier's matrix

What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$? When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and ...
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43 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= ...
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Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
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questions regarding Huseynov’s 2009 preprint titled ”On a class of entire functions, all the zeros of which are real”.

I bumped into this 2009 preprint by Huseynov titled ”On a class of entire functions, all the zeros of which are real” when I was reading Prof. Terry Tao's blog on "Tate’s proof of the functional ...
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Avoiding FFTs by reusing prior FFT results

Background From a mathematical point of view, the formulas similar to the following were produced: $F_1(f) = \mathcal{F}(T(t))$ $F_2(f) = \mathcal{F}(T(t)\times sin\Theta t)$ $F_3(f) = ...
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Where is the symmetry of Fourier transform in its implementation in Maxima and Wolfram Alpha?

From Wikipedia I saw that there is a symmetry of the Fourier transformation $F(F(f))(x) = f(-x)$ This matches the graphical explanation of the (German) Youtube video (9:15 to 9:45). I tried to see ...
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How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
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Understanding the Quantum Fourier Transform

I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. ...
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1answer
83 views

Using Fourier Transform to solve heat equation

the heat equation of $U(x,t)$ on $-\infty<x<+\infty$ and $t>0$ is $$U_t=U_{xx}+\exp\left({\frac{-x^2}{2}}\right)$$ where $$U(x,t)\rightarrow 0 \quad as\quad x\rightarrow\pm\infty$$ and the ...
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Positive-definite + continuous at 0 $\Rightarrow$ continuous?

Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$. Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$? ...
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Fourier transform of $F(x)=\exp(-x^2/(2 \sigma^2))$

I am looking for the fourier transform of $$F(x)=\exp\left(\frac{-x^2}{2a^2}\right)$$ where over $$-\infty<x<+\infty$$ I tried by definition $$f(u)={\int_{-\infty}^{+\infty} ...
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Is there a way to characterize the range of a chebyshev series through its coefficients?

Let $f$ be a Chebyshev series of order $n$ $$ f(x) = \sum_{i=0}^n a_i \cos\left( i \arccos\left(x\right)\right), x \in \left[-1, 1\right]. $$ Is it possible to characterize all the $\lbrace a_i ...
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Is 2D FFT separable?

Suppose I have a 2D matrix (or image). Can I loop on the columns - compute the FFT of each column and then loop on the rows (of the result matrix) and compute the FFT of that? Would that be equivalent ...
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Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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Fourier Transform Properties Proof

If: $y(t) = x(t)*h(t)$ and $g(t) = x(9t)*h(9t)$ (Where * is convolution) How can I use properties of the Fourier transform to show: $g(t) = Ay(Bt)$ and find constants? I think A should be $1/81$ ...
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Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
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Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs. Let $u(x,t): \mathbb{R}^2 \to \mathbb{C}$, which solves the differential equation $\hat{L}u(x,t)=0$. $u(x,t)$ is ...
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How to find the global minimum of a function from its Fourier transformed function.

i.e Can $\min{f(t)}$ be expressed by $F(\omega)$? I have a series of data in frequency space. I can do discrete Fourier transform to time space to find its minimum. But I am wondering if there is a ...
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Fast Fourier Transformation: inverse transform of the product of polynomials

I have managed to implement and understand most of the Fast Fourier Transformation. However, I have one last question. If one has two polynomials, say $A(x)$ and $B(x)$, and one computes DFT of ...
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About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...