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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Why use Hilbert transform for non-stationary time-series

Why is the Hilbert transform preferred over the Fourier transform for non-stationary time series (like amplitude modulated radio signal)?
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431 views

Give an example a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge

Give an example of a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge. That is, for any $a,b \in \Bbb Z_+$, $$ \|f_n\|_{a,b} < \infty, $$ but $$ ...
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236 views

Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$

Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $$ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $$ But ...
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93 views

Suggestion for a project on Harmonic measure and Fourier analysis

I have a course project on harmonic measure and Fourier analysis. The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis. Harmonic measure is a vast ...
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357 views

Proove the Riemann-Lebesgue lemma for the Fourier Transform on $\mathbb R$ using the Riemann-Lebesgue lemma for the Fourier coefficient on the circle

Proove the Riemann-Lebesgue lemma for the Fourier Transform on $\mathbb R$ using the Riemann-Lebesgue lemma for the Fourier coefficient on the circle. Periodize $f \in L^1(\mathbb R)$ and ...
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95 views

Show $\left\lvert \frac{1}{\pi+2} \sum_{0 \le k \le 2n,k\ne n} \frac{i}{k-n} e^{ikt} \right\rvert > c \ln n, c>0$

The partial sum of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ can be written as \begin{align} S_n(f,t) &= \sum_{0<|k| \le n} \frac{i}{k} e^{ikt} \; (1) \\ & = ...
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34 views

Proof that every real function has negative frequency component

I want to know why every real function has negative frequency component. If I am not correct, can anyone tell me how it is really? I heard that it is related to Fourier analysis, though not sure. ...
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89 views

Filter/Removal of periodic 'delta-peaks'

Currently I am measuring data (Counts over Time). Due to measurement problems I have some nasty peaks in this data. These peaks are periodical, very sharp (~3 datapoints over a range of 10000) and ...
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219 views

Projection Slice theorem getting rid of artifacts?

I have employed the fourier(projection) slice theorem in matlab. I have a 3D image, P(x,y,z) defines their pixel intensities at a given location int he image volume, it is discrete and uniform. I ...
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136 views

Order of partial sums in the derivatives of the Fourier series

Given periodic function $f\in C^{w}[0,1]$ with its Fourier series $f(x)=\sum\limits_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$. What can one say about the asymptotic order of ...
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77 views

Bounds on Fourier coefficients of Euclidean distance functions

I am interested in the bounding the Fourier coefficients $a_{m,n}$ of the function $f(x,y)=\sqrt{x^2+y^2}$ defined on the interval $[-1,1]^2$. I am specifically interested in understanding the ...
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282 views

square root of s domain transfer function

I have a question about calculating the square root of the magnitude of a transfer function. When you take the square root, what is happening? My initial idea of a magnitude is a single value, but in ...
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358 views

estimation for Dirichlet kernel.

I have the next questions that I am stuck in them. Let $f = 0 $ for $x\in [0,\pi]$ and $f=1$ for $x\in [\pi,2\pi]$. I need to find some constant $c$ such that for every natural N: $$f*D_N ...
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540 views

What is the fourier transform of $\operatorname{sinc}^4(kt)$?

I have to use Parseval's Theorem. I used it and ended with the integral of $(\operatorname{sinc}^2(kt))^2$. I know the Fourier Transform of $\operatorname{sinc}^2(kt)$ is the triangle function but I ...
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247 views

Complex numbers and Fourier transform

Here I am stuck in solving Fourier transform and the funny part is that I am stuck in the basics, in the complex part. I hope someone can help me solve this part. $$ 3 + 3 ( \cos \frac{4\pi}3 + j ...
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38 views

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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309 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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197 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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744 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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230 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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92 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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230 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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312 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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71 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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466 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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775 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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109 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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130 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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65 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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131 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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256 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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145 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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117 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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24 views

Calculation of Fourier Transform

I am trying to calculate the Fourier Transform (with respect to k) of the following function: $\frac{1}{\sqrt{k^2 + 1} - C}$ where $C$ is some complex number. Does anyone knows how to do that? ...
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9 views

Scaling for Matlab fft operation?

I have a $N$ complex signal samples (QPSK) and I am creating an OFDM signal. When I am doing a IFFT operation in matlab, I use following command: $$Y=(dft/sqrt(N))*ifft(X),$$ where $X$ is the input ...
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18 views

An upper bound for the Fourier coefficient of the “infinite cake” function

Consider a function $x_{s_n} (t) = s_n$ for $t\in[-\frac{s_n}{T_0}, \frac{s_n}{T_0}]$ and $x_{s_n} (t) = 0$ for $t$ everywhere else, with period $2T_0$. Now let $s_n=\frac{1}{n^2}$, and define the ...
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20 views

Computing the accumulated power of an image's Fourier transform

I need to compute the accumulated power of an image. The purpose is to verify the 1/$f$ power law in natural images. I'm not sure how to do this. I've done a fast Fourier transform of the image and ...
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12 views

2D Fourier transform of $x.\frac{\partial f}{\partial x} (x,y)$

What is the simple form of the 2D fourier transform of the following functions: $$ g(x,y)=x.\frac{\partial f}{\partial x} (x,y) $$ And $$ h(x,y)=x.\frac{\partial f}{\partial y} (x,y) $$
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25 views

When has the Fourier transform for some values equal values?

Definition We take a function $F : \mathbb T^n \rightarrow \mathbb R$ that is even ( $F(x)=F(-x)$) and continuous (hence bounded), where $\mathbb T^n$ is the $n$-dimensional Torus. Now we define the ...
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18 views

Give a suitable way to study Fourier Transforms:

Give a suitable way to study Fourier Transforms. In the website called the fourier transform, gives somewhat good approach to meet it. But, I need to clarify onething. I am doing my pure papers ...
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22 views

Fourier transform of delta function $\delta(x)$ and my defined function $s_1(x)$ and $s_2(x)$

$\delta(x)$ and $s_1(x)$ are $0$ if $x\not=0$, if $x=0$, then $\delta(x)=+\infty$ and $s_1(x)=1$, respectively. $s_2(x)=1$ if $1\geq x\geq -1$, otherwise $s_2(x)=0$. What is the Fourier transform of ...
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20 views

Parseval's Identity, problem with $|a_n|^2$

I'm trying to obtain the Fourier Series of this function: $$f(x)=\begin{cases} \pi -x, x\in [0, \pi]\\ \pi+x, x \in [-\pi, 0) \end{cases}$$ It is a even function, so I can write: ...
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24 views

Fourier Series: even extension and Parseval Identity

I'm trying to solve this exercise but I have some problems, because I haven't seen an exercise of this type before. $f(x)= \pi -x$ in $[0, \pi]$ Let's consider the even extension of f(x) in ...
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30 views

Is the inverse Fourier transform a “linear transform”?

Consider the inverse Fourier Transform and the Fourier Transform: $$f(x) = \int_{-\infty}^\infty F(k)e^{2\pi i k x}dk \\ F(k) = \int_{-\infty}^\infty f(x)e^{-2\pi i k x}dx$$ The Fourier transform ...
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14 views

Equality with fourier transform

I have problem with the following equality where the Fourier transform appears: Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, ...
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9 views

Phase difference of two signal of different frequency

Currently, I have two signals, the main components of both signals are 60Hz, but both also have weaker response at 180Hz + small amount of noise. As shown in the photo below, I want to find the phase ...
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18 views

How to prove these two equations

How to prove: $$x(t)*\delta^{(n)}(t) = \frac{d^n}{dt^n}x(t)$$ and $$x(t)*u(t) = \int_{-\infty}^tx(s)ds$$ To the first one, I think I could use the following formula: $$ ...
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14 views

an oscillatory integral with two parameters

Consider $$I(a,b)=\int_{\mathbb{R}}e^{i(ax^2+bx)}\psi(x)\,dx$$ where $\psi$ is smooth and supported in $\{x:|x|\in[1/2,2]\}$. How to control $I(a,b)$ in terms of $a$ and $b$? Moreover, is there an ...
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18 views

construction of a partition function

Let $q$ be a large positive integer. How to construct a smooth function $\phi$ with the following properties? i) $\sum_{a\in\mathbb{Z}}\phi(q(x-a/q))=1$ for any $x\in\mathbb{R}$ ii) For any ...