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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every continuous signal being modelled as a function

Can every coninuous signal be modelled as a function, which then can be converted into a series of sine and consine functions with unique frequencies? And let us say that we have some arbitrary ...
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303 views

Properties of the Toeplitz matrices formed by square-summable sequence (as opposed to absolutely summable)

I've been reading a wonderful monograph by Robert Gray on the Toeplitz and circulant matrices and am curious about the assumption (4.3) of absolute summability of the sequences $\{t_k\}$ that form the ...
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192 views

fourier transform of $\operatorname{sinc}$ function

I have to do the fourier transform of this signal $\left(\frac{1}{10}\right)\operatorname{sinc}\left(\frac{t}{10}\right)$ where sinc function is defined as $\frac{\sin(\pi x)}{\pi x}$. the transform ...
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735 views

Fourier series /spectrum of different cosine functions

I was given the following task. b) In this task you will concatenate the seven cosines from task a) into one 7 sec long vector. To concatenate vectors in MATLAB use: x=[x1 x2 x3 x4 x5 x6 x7]; ...
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229 views

A function in BMO space

Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ...
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90 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
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227 views

Do Fourier transforms of $\min$ and $\max$ exist (in closed form)?

I am wondering if there are Fourier transforms of $\min(x,a)$ and $\max(x,a)$ functions. Please forgive me if this is a dumb question, I don't normally use Fourier transforms. I attempted to simply ...
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308 views

How to solve a linear PDEs with trigonometric function as coefficients

Is a general method for solving a system of linear partial differential equation with trigonometric function as coefficients exist ? For example something like that: $q$ is the unknown function, $2 ...
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65 views

Power Spectral Density excercise

Compute the power spectral density of $x(t) = \operatorname{sgn}(t)$ Hint: $$\lim_{t\to\infty} t(\operatorname{sinc}(ft))^2 = \delta(t).$$ Please help me I have to solve this exercise urgently. ...
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455 views

Fourier transform independent of kernel?

I've tried computing a windowed Fourier transform using various kernels that were all made from periodic signals of the form a + bi with b being a shifted version of a. I used square waves, sin waves, ...
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757 views

Square wave transform

One can define an analogous transform to the Fourier transform that uses square waves as the basis instead of sinusoids. Everything seems to work out in parallel and I imagine one can even come up ...
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104 views

Dealing with integrals and Fourier transforms.

I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int ...
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128 views

Finding Transfer Function with an intermediate variable

How do I find the transfer function (using the bilateral z-transform) of the problem below. A stable LTI system with input x[n] and output y[n] is modeled by the difference equations c[n] + ...
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64 views

Slowly varying vectors and coefficients of a sine transform

Let $u_k$ be the vector in $\mathbb{R}^n$ whose $i$'th entry is $\sin(\pi ki/n)$. The vectors $u_1,\ldots, u_n$ are orthogonal and correspondingly every vector in $\mathbb{R}^n$ can be decomposed as a ...
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83 views

Fourier analysis confusion

I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory: Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty ...
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128 views

Inequality for Trigonometric Polynomials

Problem statement: Define $p(t) = \sum\limits_{j=-N}^{N}c_{j}e^{ijt}$ be a real-valued trigonometric polynomial. Suppose there exists an $x_{0}\in\mathbb{R}$ such that $p(x_{0}) = \|p\|_{\infty}$. ...
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254 views

Laplace Eigenfunction: Show Eigenvalue is Positive Using Fourier Transform

Problem: Let $ \lambda\in\mathbb{R}, u $ a smooth function, not identically zero, defined on a neighborhood of the unit disc satisfying $ \Delta u+\lambda u = 0 $ in the interior of the unit disc and ...
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143 views

Estimate the Hilbert transform

Let $1\leq p<∞$: Suppose that there exists a constant $C>0$ such that for all $f\in S(\mathbb{R})$ with $L^p$ norm one we have $$\biggl|\{x:|H(f)(x)|>1\}\biggr|\leq C.$$ Here $H(f)$ is ...
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116 views

How to know if the Fourier Transform is surjective

I am studing if I could modell a function using LUTs (ROMs) for a electronic digital design. I have two functions, a FFT and other function, S. S function system recive as input the output of the FFT ...
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234 views

Link between time and complex amplitude argument, DFT

I have some discrete points, with time $t_n$ and value $u(t_n)$. I perform discrete fourier transormation using values $u(t_n)$. Now I have some complex values, as I understood absolute value is ...
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10 views

Fast Fourier transfrom

What are the prerequisites for understanding the fast fourier transform for fast multiplication? What topics should I be familiar with first?
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18 views

DFT of vector $(0, 1, 2, 3)$

The problem is that my answer is different from answer i get in MATLAB. My answer is $(6, -2-2i, -2, -2+2i)$ while MATLAB answer is $(6, -2+2i, -2, -2-2i).$ In MATLAB i use command ...
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8 views

Where can I find material on polynomial filters?

Most students and mathematicians probably know a fair amount on roots-of-unity filters, or on Fourier analysis. The basic notion of this "filtering" is, given a polynomial, we can find the $n$th ...
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14 views

Fourier transform of $be^{i k y^b}/y^{1-b}$

I'm trying to compute the Fourier transform of $$ \frac{ be^{i k y^b}}{y^{1-b}}$$, i.e. $$ F(z) = \int_{-\infty}^\infty \frac{ be^{i k y^b}}{y^{1-b}} e^{i z y}dy$$ I tried using Mathematica for ...
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30 views

How to prove $\hat f$ is uniformly continuous in $R^n$?

Let the Fourier transform be defined by $\hat f(\xi)=\int_{R^n}f(x)e^{-ix\xi}dx$. Suppose $f\in L^1(R^n)$. How to prove $\hat f$ is uniformly continuous in $R^n$?
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17 views

Plancherel's theorem variants

How would you prove a variant form of Plancherel theorem: If $(c_n)_{n\in\mathbb{Z}}$ are coefficients and $\sum_{n\in\mathbb{Z}}|c_n|^2<\infty$, then there exists a unique function $g\in L^2(0,1)$ ...
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11 views

different way to compute power spectral density

I am writting a piece of code to compute power spectral density (psd) of a signal and wanted to compare two approaches : compute the FFT of the signal and square its amplitude compute the biased ...
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14 views

Solution of a differential equation with problem of Cauchy

The question is the next: What can I say from the existence, uniqueness and continuos dependence of the solution? Is this a strongly continuos one-parameter group or a semigroup. $ \left\{ ...
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23 views

Function of bounded Fourier degree; bounding on subinterval

Suppose $f(x)=\sum_{|k|\le d} a_ke^{2\pi i kx}$, and it is given that $|f(x)|\le 1$ on $[0,L]$. Over all such functions, what is the maximum possible value of $$\max_{x\in [0,1]} |f(x)|?$$ (For ...
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13 views

Fourier series qn determine the fourier series coefficients

Can someone please help me with this Fourier series $q_n$: determine the fourier series coefficients of $x(t)$ given as $x(t) = \cos4t + \sin8t+3$?
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10 views

Extensions to higher dimensions by tensorization. Unitary DFT in 2D?

I have problem understanding the underlying concept of tensoration (if there is such term). Fist of all the unitary DFT is NxN. Is it 1D ? How does it look when we increase the dimension let say to ...
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25 views

Fourier Transform of a function with sinusoidal sampling

What is the relation between the Fourier Transform (FT) of $f(x)$ with regular sampling and the FT of $f(x)$ with sinusoidal sampling? In other words, it's a FT of a function composition $f\circ ...
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23 views

Hankel transform of shifted Gaussian function

I'm trying to find Hankel transform of the function $$e^{-(r-r_0)^2}$$ Could please anyone confirm that it doesn't exist ?
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31 views

A highly oscillatory integral

I am considering the following integral $$ \int_{-\infty}^{\infty} \text{d} z' e^{-i\alpha(z-z')}e^{iV(z')(z-z')}\text{sign}(z-z'), $$ where $\alpha\in\mathbb{R}$ is a (large constant) and ...
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how to show that Kirchhoff's formular solves wave equation.

There is one exercise in text book, Fourier Analysis: An Introduction, Stein p.211 Ex# 11. I have no idea how to handle that formula. please help me!! this is Theorem 3.6 The definition of ...
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12 views

estimate on a convolution

Let $\psi$ be a non-negative Schwartz function on $\mathbb{R}$ such that supp$\hat{\psi}$ is contained in $[-0.1, 0.1]$ and $\hat{\psi}(0)=1$. Define $\psi_k(x)=2^k\psi(2^kx)$ for any integer $k$. Let ...
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34 views

Integration by parts with Bessel function $j_0$

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
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6 views

Power law in power spectrum and memory.

If we generate white noise and do the FFT of it, we get the same amplitude for each of the frequencies. Therefore, the output of the FFT of the noise follows approximately the power law ...
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18 views

Hoe can I find the Inverse FourierTransform for 1/(1+w^4)?

I have the expression $S(w)=\frac{1}{1+w^4}$. I am trying to find its inverse FourierTransform. I know that I have to get a sin-cos expression, but I haven´t found the way to do it. On the tables that ...
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18 views

Solve this PDE, using Fourier transforms

PDE: $v_t(x, t) = kv_{xx}(x, t) + bv_x(x, t)$, $v(x, 0) = f(x)$, $\quad-\infty < x < \infty$ I can't apply the inverse Fourier in this case. If someone could help me, because i can't find a ...
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13 views

Fourier coefficients of three times differentiable functions.

I was wondering, can we determine somehow the decay of the Fourier coefficients of a function $g \in C^3\mathbb{(T)}$ /three times continuously differentiable/ as $|n|\rightarrow \infty$? Any help ...
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11 views

fourier transforms of $e^{(-kw^2+b)t}$

I was solving and PED using fourier transforms and reached this point: $$v(x,t)=f(t)*F^{-1}\big[e^{(-kw^2+b)t}\big]$$ $F^{-1}$ denotes inverse fourier transforms, and $*$ is used for convolution. I ...
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22 views

Continuity of Fourier-Stieltjes transform

I read in Kolmogorov-Fomin's (p. 419 here) that, if $F$ is a function having bounded variation on $\mathbb{R}$ then the Fourier-Stieltjes transform$$g(\lambda):=\int_{-\infty}^\infty e^{-i\lambda ...
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Inverse Fourier Transform with Duality

I need to find the inverse Fourier transform of the following equation using the duality property: $X(w)=\begin{cases} 2w+2 &\mbox{if} -1<w<0 \\ -2w+2 &\mbox{if }0<w<1 \\ 0 ...
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25 views

Inversion formula for $\int_{\mathbb{R}}f(x)e^{-izx}dx$

Let $f:\mathbb{R}\to\mathbb{C}$ be a measurable function such that$$\forall x\ge 0\quad|f(x)|<Ce^{\gamma_0 x}$$$$\forall x<0\quad f(x)=0$$I must specify that all the integrals I am going to ...
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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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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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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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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 ...