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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Shifted Fourier transform

Please can some one help and give me a direction to evaluate the following shifted Fourier transform: \begin{alignat}{2} s(x_c) =&\frac{1}{\Delta x_0} \int_{x_c-\Delta x_0}^{x_c+\Delta x_0}...
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53 views

Fourier Differentiation Property

I have been given this problem to solve: Define the function f(t) by $$ f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases} $$ where $k > 0$ is a real number. ...
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55 views

Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on $[0,\pi]$...
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41 views

Some issues for solving differential equations using Fourier transform

Fourier transform is a powerful tool for solving differential equations. But I don't really know when the Fourier transform will give us the full general solution if it can be used. A simple example ...
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58 views

Fourier Transform of $ f(t) = e^{-kt}$

I am trying to calculate the fourier transform of the following function: $$ f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases} $$ where $k > 0$ is a real number. ...
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Special functions, Fourier series

Well known are the Fourier expansions (presented, e.g., in Abramovitz and Stegun): $$ \cos ( A \sin x) = J_0(A) + 2 \sum_{k=1}^{\infty} J_{2k}(A)~\cos(2kx)~~, $$ $$ \sin ( A \sin x) = 2 \sum_{k=0}^{\...
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Using Fourier Transforms to evaluate $\int_{-\infty}^{\infty} x^k \space f(x)dx $

We were asked to show that if the following integral converges: $$ \mu_k =\int_{-\infty}^{\infty} x^k \space f(x)dx \space, k \in \mathbb{N}$$ Then we can obtain $ \mu_k $ from the Fourier Transform ...
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Why it is true for rapid decreasing function $g$ that: $\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\leq A_{l,k}(1+|y|)^{l}$

If $g$ is of rapid decrease, that is $\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x)|<\infty$, then we have: $$\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\...
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Calculation of the Fourier transform of $x/(x^2+1)^2$ using the properties of the transform

I am trying to calculate the Fourier transform of $$f(x)=\frac{x}{(x^2+1)^2}$$ using the property of Fourier transform. So I am trying to use$$\widehat{g_1(x)g_2(x)}=\frac{1}{2\pi}\widehat g_1 *\...
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69 views

How to calculate the residue of the fourier transform?

I have been struggling calculating the Fourier transform of $f(x)=\frac{x}{(x^2+1)^2}$. I tried to calculate $f(t)=\int\frac{x}{(x^2+1)^2}e^{-ixt}\,dx$ directly by integration by parts, but it is not ...
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29 views

Calculation of Fourier tranform

How to calculate the Fourier transform of $f(x)=x$. I know using the formula $f(\varepsilon)=\int_xe^{-ix\varepsilon}x \, dx$. But I have problem calculating this complex integral.
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I am struggling in calculationg Fourier transform?

I need to find the Fourier transformation of $f(x)=\frac{x}{(x^2+1)^2}$ by two different methods. one is using the property of Fourier transformation, another is computing the integral by definition. ...
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77 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 $\Lambda^s:=(I-\...
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98 views

How to show for $\alpha∈(0,1)$, any $f∈ C^\alpha([0,1]/{\sim})$ has a Fourier series $S_nf$ uniformly converging to $f$

Technically homework(a midterm) but its over and I'm itching to know the solution. I know how to show it for $\alpha>1/2$ (the Fourier series will converge absolutely), but apparently its true for ...
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What are the prerequisites to understand Affine Invariant Fourier Descriptors?

I need to implement Affine Invariant Fourier Descriptors on matlab, the objective is to compare two objects one reference and other transformed by affine transformation for recognition, my problem is ...
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46 views

How to restore a function from its Fourier transform on the imaginary axis?

Let $f$ be a `very good' function on the real line; say, infinitely differentiable and compactly supported. We are given its Fourier transform on the imaginary axis: $$g(x)=\int_{\mathbb R}f(t)e^{xt}\,...
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Understand this Fourier transform $\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$

I found the equation $$\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$$ in a 'physics' textbook and I just don't understand what this equation tries to tell me. Is there anybody who ...
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Finding multiple functions with same $f_{even}$ but different $f_{odd}$?

A function can be decomposed as $f(x) = f_{even}(x) + f_{odd}(x)$ where $f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$. If we know only $f_{even}$, how can we find ...
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525 views

Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
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155 views

Problem on Big Rudin about Fourier Transform

Exercise 5: If $f\in L^1$ and $\int |t\hat{f}(t)|<\infty$, prove that $f$ coincide a.e. with a differentiate function whose derivative is $i\int_{-\infty}^{\infty}t\hat{f}(t)e^{ixt}dt$ I know a ...
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The Fourier Transform on Schwartz Space: concerns about $-2\pi ixf(x)\longrightarrow \frac{d}{d\xi}\hat f(\xi)$

Suppose a function $f$ is of rapid decrease. Then, we have $\displaystyle -2\pi ixf(x)\longrightarrow \frac{d}{d\xi}\hat f(\xi)$ It suffices to show $\displaystyle\int_{-\infty}^{\infty}\left\vert f(...
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Expressing solution of this wave equation problem in different Fourier expansion

I have managed to solve the wave equation $u_{tt} = c^{2} u_{xx}$ on the interval $[0,L]$ for $t > 0$, and subject to initial conditions $u(x,0) = f(x)$ and boundary conditions $u_{t}(x,0) = g(x)$. ...
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19 views

Regarding a theorem of S.Bochner

This question is probably answered somewhere in SE or math over flow but counldnt find it. Is there a version of Bochner's theorem ( necessary and sufficient condition for positive definiteness) for ...
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62 views

Showing that $-4\pi^2\sum_{m=1}^\infty me^{2\pi i m\tau}=\frac{\pi^2}{\sin^2(\pi \tau)}$

I'm doing Problem 7b in Chapter 4 of Stein and Shakarchi's "Complex Analysis" for homework. I want to show that if $\tau$ is a complex number with $\mathrm{Im}(\tau)>0$, then $$\sum_{n=-\infty}^\...
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38 views

Can Polynomials be positive definite?

It seems to me that polynomial functions are ,trivially, not positive-definite (for definition )because of growth property of p.d functions. Am I right?
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280 views

How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

Without using Fourier transforms, how do I derive the spectral measure for $A=\frac{1}{i}\frac{d}{dt}$ on the domain $\mathcal{D}(A)$ consisting of absolutely continuous functions $f\in L^{2}(\mathbb{...
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33 views

How to write a fourier series using periodic boundary conditions

Would writing $$ f(x) = x^2 $$ as a Fourier series using periodic boundary conditions on $-L < x < L$ with a basis of $$ e^{\frac{i\pi nx}{L}} $$ be just \begin{align}\bigl\langle e^{\frac{i\...
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differential quotient in Sobolev space by Fourier transform

I am studying Michiel's PDE book. Here I have a question about one theorem in chapter about Sobolev space. At the beginning of this chapter, he define $\tau_y(u)(x)=u(x+y)$ and by assuming $u\in H^s(...
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Calculate difficult Fourier Transform

I have to calculate a quite difficult Fourier Transform for my class $$\int_{-\infty}^\infty dw\frac{(\varGamma-iw)w^3}{(\varGamma-iw)^2+1}\frac{J_{1}(|wr|)}{|wr|}e^{-iwt}$$ $J_1$ is the normal Bessel ...
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Smoothness of Fourier transform of a measure

Is the Fourier transform of a finite Borel measure on $\mathbb{R}$ necessarily a smooth function?( $\widehat{\mu}(x)=\int_\mathbb{R}e^{-i\pi xy} d\mu(y)$)
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269 views

How to prove that cosine squared is a positive-definite function?

I need some help with proving that function $f: \mathbb{R} \to \mathbb C$, $f(t)=(\cos(t))^2$ is a positive-definite function. I know that if $\sum_{k,l\le n}(f(t_k-t_l)z_k\overline z_l)\ge0$ then ...
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Space of multipliers on $C_0(\mathbb{R})$?

What is the space of multipliers (that is translation invariant bounded operators) on $C_0(\mathbb{R})$ (space of continuous functions vanishing at $\pm \infty$)? I suppose the answer should be the ...
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Evaluate the DFT of the vector $(f(x_0),\dots,f(x_{18}),f(x_{19}))$

Let $f(x)=1-x^2$, with $x \in [0,1)$. Evaluate $\hat{f}(x)=\int_0^1 f(y)e^{-2 \pi ixy} \,dy$ (the Fourier transform of $f$). Let $x_j=\frac{k}{10}$, with $k=0,\dots,18,19$. a. Evaluate the DFT of ...
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215 views

Quantum fourier transformation Unitary proof.

I've found a bunch of these proofs online but I am having trouble understanding how the norm of the column/row is 1.
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141 views

Questions about Fourier Series

I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ ...
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113 views

Equivalence of Schwartz Space Definition

I've come across two definitions for what it means for a function to be in $\mathfrak S$, the Schwartz space. A function $f \in \mathfrak S$ if $f \in C^\infty$ and for all $j, k \geq 0$ integers, $\...
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126 views

Mean value of discrete periodic signals

It is clear that a continuous periodic signal always takes at some point $x_m$ the absolute mean value. Then we could define the absolute mean value of the signal as the value that it takes at $x_m$. ...
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112 views

Bounds on the line for entire functions of exponential type

Let $f$ be an entire function on the complex plane $\mathbb C$, assume that $$|f(z)|\le e^{|z|}.$$ Does the property $$|f(x)|\le e^{-|x|}, \qquad x\in\mathbb R,$$ imply $f\equiv 0$? More generally, ...
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Evaluate the DFT of the vectors $(1,1,0,0)$ and $(1,1,1,0,0)$

Evaluate the DFT of the vectors $(1,1,0,0)$ and $(1,1,1,0,0)$ I toke Fourier Analysis last semester but I do not remember how to approach the problem. Can someone give me a re-fresher?
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A bounded monotonic function on an closed interval has Fourier coefficient decaying in O(1/|n|)

On interval $[-\pi,\pi]$, $|f|\leq B$ for $B\geq 0$, and $f$ is assumed to be monotonically increasing. We want to prove that $\hat f(n)=O(1/|n|)$ for $|n|$ large enough, that is for all large $|n|$,...
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804 views

Fast fourier transform and nyquist frequency

Trying to figure out how to use Matlab to calculate the nyquist frequency of a signal. Given a function, lets say $y = 5\sin (2t + \pi /3) + \sin (t + \pi /2)$ for $t > 0$. How do we use a fft in ...
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60 views

Proving that a mass distribution has positive Lebesgue measure

I am confused in this proof about how we obtain $\int f(u) \, d\mu(u) = \int f(u)g(u) \, d\mu(u)$ and how Plancherels theorem has been applied in $(6.6)$. Furthermore, I cannot understand how if $\mu$...
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2k views

Phase shift of two sine curves

How do I determine phase shift of two sine curves (discrete time sampled sine curves) in Matlab. Currently, I have the FFT of these two sine curves, the phase shift is just the delay in time, which ...
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Averaging and approximation

I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$; with $V(t_i,\nu_i)=e^{i\sigma(t_i,\nu_i)}$ then we can ...
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66 views

List of ODE's that can be solved by Fourier transform

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier ...
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64 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
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Applying the Fourier transform to solve an ODE.

We are learning about fourier transfrms in class and I was wondering about solving the following ODE using this method. So, I want to solve the equation $u''(x)+u(x)=0$. Now, it is clear that a ...
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240 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ "...
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206 views

Different Versions of Fourier Series? What about Uniqueness?

Let $f(x)$ be a function, then for its Fourier series $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) $$ I found two different definitions (both yielding different series)....