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).

learn more… | top users | synonyms

1
vote
0answers
38 views

A question on convergence of derivative of power series

This is a question from Fourier Analysis with Applications by Folland. First we write Fourier series for $$e^{\theta}=\sum c_ne^{in\theta}$$ We differentiate this series term by term to obtain ...
1
vote
3answers
35 views

Fourier Series Transformation

I have a question regarding the following: Compute the Fourier transform of $f(x)=xe^{-2x^2}$, $x\in\mathbb{R}$. The Fourier Transform of $f(x)$ is given by ...
0
votes
1answer
15 views

Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
3
votes
2answers
106 views

Approximation of a $L^1$ function by a dominated sequence of continuous functions

Consider $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$ and the Lebesgue measure on it. Denote by $L^1(\mathbb{T})$ the set of integrable functions on $\mathbb{T}$ and by $C(\mathbb{T})$ the set of ...
1
vote
1answer
38 views

Fourier Series of Real-valued Functions

Context: For a $2\pi$-periodic bounded function $f:\mathbb{R}\to\mathbb{C}$, we define the complex Fourier coefficients of $f$ by $$ \hat{f_k}:=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}\,dx. $$ We call ...
0
votes
1answer
18 views

Is there any way to use a Fourier Transform or a variant to find periodic increases?

Suppose I have a staircase function, which has a periodic increase but no periodic decrease. I've been playing with Fourier transforms recently, and I know one main use is to pick out frequencies ...
1
vote
1answer
24 views

Band-limited function vanishing on a set of positive measure

I'm working my way through a Fourier analysis textbook and came across the classic result that a function cannot be both band- and time-limited, that is, if both $f$ and $\hat{f}$ are compactly ...
0
votes
0answers
13 views

Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
1
vote
0answers
17 views

Fourier Cosine series expansion for two dimensional function

I have a two dimensional function with its values and range. I need to expand the function in Fourier cosine series. The function as follows: $$f(x,y) = \begin{cases} A &, -\frac{L}{2} + 2nL < ...
1
vote
1answer
28 views

Find the Fourier coefficients of $cos^2(x)sin^2(x)+2cos^3(x)$

So I know that the Fourier coefficients are expressed as: $$a_0 = \frac{1}{\sqrt{2}\pi} \int_{-\pi}^{\pi}f(t)dt$$ $$c_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(kt)dt$$ $$b_k = ...
0
votes
1answer
19 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
1
vote
1answer
21 views

Formulation of Fourier transform

I would like to know about Fourier transform more. I attended a standard lecture of mathematics, but we did not talk about Fourier transform on $L^2$ much, nor the theory of $L^2$. We only defined it ...
0
votes
0answers
12 views

A question about time series analysis with fast fourier transform

I have a question about time series analysis with FFT. from here I have understood, to calculate a periodogram of a time series we should do these to steps: Fast Fourier trasnfomation of time ...
0
votes
1answer
14 views

Fourier transform for specific signal

Could someone show me how to calculate fourier transform for the following signal? s(t) = e^(-3*t^2) Thanks!
2
votes
1answer
49 views

Find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$.

Suppose a $X$ is a random variable, I am asked to find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$. I am trying to use the inversion formula for the ...
3
votes
0answers
31 views

Fourier transform of $\frac{1}{r}$ (Coulomb potential)

When calculating the Fourier transform of a function of the form $f(\vec{r}) = \frac{1}{4 \pi \left|\vec{r}\right|}$, one encounters the problem that the resulting integral does not converge, i.e. ...
0
votes
1answer
48 views

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 ...
0
votes
1answer
38 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. ...
1
vote
0answers
20 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 ...
1
vote
1answer
18 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 ...
2
votes
1answer
33 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. ...
3
votes
0answers
67 views

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 ...
0
votes
1answer
18 views

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 ...
5
votes
1answer
79 views

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
votes
1answer
28 views

CTFT and DTFT in MATLAB

Is it possible to plot CTFT and DTFT in MATLAB? I know of DFTs(FFTs) in MATLAB since I am using them but what if I want CTFT and DTFT? If yes, then what function shoulf I use?
0
votes
1answer
31 views

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 ...
2
votes
1answer
31 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 ...
2
votes
2answers
25 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.
1
vote
0answers
42 views

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. ...
1
vote
1answer
38 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 ...
3
votes
1answer
64 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 ...
0
votes
0answers
23 views

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 ...
0
votes
1answer
30 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 ...
4
votes
1answer
69 views

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 ...
1
vote
2answers
34 views

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 ...
4
votes
1answer
82 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 ...
3
votes
1answer
66 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 ...
3
votes
1answer
35 views

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 ...
0
votes
2answers
69 views

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 $u_{t}(x,0) = g(x)$. ...
0
votes
1answer
14 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 ...
2
votes
1answer
40 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 ...
0
votes
0answers
39 views

Apply delta function on Fourier transforms

I would like to filter out a particular frequency $\omega_o$ from the Fourier transform of a function in the form of $\left<\delta(\omega-\omega_o), \mathcal{F}(f)\right>$. If $f(t) ...
0
votes
1answer
29 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?
3
votes
1answer
62 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 ...
1
vote
0answers
12 views

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 ...
0
votes
0answers
23 views

Two definition of Fourier's transformation agrees? [duplicate]

Definition 1: If $f\in L^1(R^n)$, $\hat{f} (s)=\int _{R^n} e^{-isx}f(x)dx$ Definition 2: If $f\in L^2(R^n)$, let $f_i \in$ {Schwartz functions} such that $f_i$ converges to $f$ in $L^2$, then ...
0
votes
1answer
26 views
3
votes
0answers
19 views

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 ...
1
vote
1answer
21 views

Smoothness of Fourier transform of a measure

Is the Fourier transform of a finite Borel measure on $\mathbb{R}$ necessarily a smooth function?( $\widehat{\mu}(x)=\int_\mathbb{R}e^{-i\pi xy} d\mu(y)$)
0
votes
1answer
36 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 ...