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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8
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1answer
63 views

Fourier transform with $\sin(t^2)$

This exercise gave me nightmares this night. I have $$ x(t)=\sin(t^2)e^{-2|t-2|} $$ to Fourier transform. First I though about solving the integral. (should I divide the signal in $2$, first for ...
3
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2answers
66 views

Replicating Kolmogorov's Counterexample for Fourier Series in Context of Fourier Transforms

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More ...
2
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2answers
62 views

On the Fourier transform of $f(x)=e^{-x^2+2x}$

So, I have the $f(x)=e^{-x^2+2x}$ and to take the FT of it, I complete the square: \begin{equation} f(x)=e^{-x^2+2x \pm1}=e^{-(x-1)^2}e \end{equation} Then, by knowing that the FT of $g(x)=e^{-x^2}$ ...
1
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2answers
25 views

Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
3
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0answers
61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
7
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2answers
104 views
+50

$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
0
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1answer
6 views

Increasing order of fourier coefficients on the boolean cube

Given a function $f:\{0,1\}^n\rightarrow \{0,1\}$, is it true that for any $S,T\subseteq[n]$, such that $S\cap T =\phi$, then $\hat{f}(S\cup T)\leq \hat{f}(S)$? It seems so to me cause, if if you just ...
0
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0answers
14 views

How to introduce a new equivalent using two separate boxes in variables?

I am reading a paper in mathematics about Fourier Transform. It consider two boxes A and B with centers of $ x_0(A) $ and $p_0(B)$ respectively. It introduce a new function $R^{AB}(x,p)$ as follows: ...
2
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1answer
45 views

Representation of Heaviside function's Fourier transform

I've seen here that the Fourier transform of Heaviside function $\Theta(t)$ is $$ \Theta(\omega) = \frac{1}{i\omega} + \pi \delta(\omega) \tag{1}$$ But in some physics texts and here I've seen the ...
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0answers
32 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
2
votes
1answer
47 views

Solving wave equation by fourier method

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
7
votes
1answer
95 views

Sets of Divergence for Fourier Partial Integals

It is a consequence of Carleson's theorem together with a transference argument that (see Section 4.3.5 in L Grafakos, Classical Fourier Analysis for proof) that the Fourier partial integrals of a ...
1
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1answer
47 views

Fourier transform of cosine with square root

In relativistic mechanics, i came across the Fourier transform of the following function : $\cos \left(t \sqrt{x^2+m^2} \right)$ or $e^{it \sqrt{x^2+m^2}}$ ($t$ and $m$ are constants). Is there a way ...
0
votes
1answer
27 views

When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
2
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4answers
82 views

Dirac Delta function inverse Fourier transform

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
2
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0answers
27 views

Dirac Delta function inverse Fourier transform [duplicate]

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
0
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1answer
20 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
1
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1answer
38 views

An equality about Fourier transform

I have read an equality about Fourier transforms which I can not proof. It is as following: Let $u\in C_0(\mathbb{R}^n)$ and \begin{equation} g(x_1,x_2,...,x_{n-1}):=u(x_1,x_2,...,x_{n-1},0). ...
1
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1answer
48 views

What if the Fourier series of a periodic function also has periodic coefficients $a_k$

If given that $x(t)$ is a periodic continuous time signal, with periodic $T$. It can be expressed by the Fourier series, i.e. $x(t)=\sum\limits_{k=-\infty}^{+\infty}\,a_k\cdot e^{j k \frac{2 ...
0
votes
1answer
27 views

In what sense is the Schwarz class of functions a “class”?

As an engineer who has not learned very much modern algebra, I recently learned about "class" in the algebra sense. Then I remembered our professor calling the set of Fourier transformable functions ...
1
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1answer
40 views

Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
4
votes
1answer
68 views

Fourier transform is real if $f$

I want to prove that the Fourier transform $F(\xi)$ of a function $f$ will be a real function when, and only when, $f(x)$ is an even function. I'm using the following definition of Fourier ...
0
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1answer
36 views

Fourier analysis notation - Sh and Ch

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!
1
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1answer
20 views

Bessel equation of half-order (asymptotic)

Not really optimistic about getting a reply for a question tagged under "Bessel function" but here goes, I have $$J_{\frac{1}{2}} = (a_1 \cos(z) + a_2 \sin(z))Z^{-\frac{1}{2}} $$ and ...
1
vote
1answer
67 views

What is $\lim_{n \to \infty} n^3 a_n$? [duplicate]

$a_n$ is the Fourier coefficient of $$f(x) = \left(1 - \frac{|x|}{\pi}\right)^4$$ The answer is infinity, but can someone give an answer that doesn't require explicit computation of the $a_n$? I'm ...
0
votes
1answer
18 views

1D FFT on rotated image column by column

I am facing a problem: performing 1D FFT on a rotated column by column on a rotated image, described as following: Original Image: Rotated Image: What I have: original image convolution ...
2
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1answer
45 views

A Hölder continuous function whose Fourier coefficients do not decay very fast

At Stein's book of Fourier analysis (Chapter 3, page 91, exercise 15) I was trying to solve the following problem I have to prove that the result ...
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0answers
13 views

When would a Fourier Product (made up term) exist for a finite sequence of the form $C_{\text{Max}}\prod _{i=1}^k A_i \cos \left( B_i n\right)$

Let us say that we are given a finite list of points of the form C = {i,$x_i$} where i goes from 0 to the card(C) that when plotted in the Euclidean plane has some vertical axis that splits the graph ...
0
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0answers
12 views

The bond between Fourier Transform and Epicycle theory

Can someone help me understanding the bond between the Fourier Transform and the epicycle theory? I have searched in many places such as: http://math.stackexchange.com/a/72479/185138 ...
1
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1answer
36 views

Fourier transform division theorem in $\mathbb R^n$

It is known that if $f \in L^1(\mathbb R)$, $\widehat f(\xi) \neq 0$ for any $\xi \in \mathbb R$, then for any $h \in L^1(\mathbb R)$ such that $\widehat h$ is compactly supported there exists $g \in ...
2
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0answers
15 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
0
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0answers
28 views

Central Limit theorem: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected [duplicate]

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
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0answers
6 views

A theorem regarding epicycles

Can somebody help me understanding the theorem on the last page of that article about Fourier Series and Epicyles? ...
0
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1answer
17 views

Functions of polynomial growth and the Schwartz space

A smooth function $m \in \mathcal C^\infty(\mathbb R^n)$ is said to be slowly increasing if for all $\alpha \in \mathbb N^n_0$ there exists $C_\alpha, k_\alpha$ such that $|\partial_\alpha f(x)| \leq ...
1
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1answer
64 views

Central Limit Theorem proof: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
3
votes
2answers
57 views

Fourier transform of the 1-d Coulomb potential

Though it may sound like a physical problem, but the thing I will introduce is rather mathematical. For the Fourier transform of Coulomb potential $$ V(\vec{x})=\frac{1}{\vert x\vert} $$ I can ...
1
vote
1answer
51 views

Finding the limit that involves Fourier coefficients,

Given the function $f(x) = 1 - \dfrac{|x|}{\pi}$, I had computed its Fourier coefficients, using integration by parts and got: $$ a_n = \begin{cases} 0, & \text{for $n$ even}, \\[6pt] ...
1
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1answer
34 views

Fourier series and transform (epicycles)

Let $\gamma:[a,b]\to\mathbb{C}$ be a continuous curve. 1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times ...
0
votes
0answers
86 views

Hausdorff-Young / Restriction Inequality

Let $\lambda$ denote Lebesgue measure on $\mathbb{R}^d$. The Hausdoff-Young inequality is that $$ \| \widehat{f} \|_{L^{q}(\lambda)} \leq \| f \|_{L^{p}(\lambda)}. $$ when $1 \leq p \leq 2$ and ...
0
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0answers
16 views

FFT differential equations

Given a generical differential equation what is the procedure to solve it using fft command. Can anyone explain me how to do it? For example: $$\frac{d^2y}{dt^2}+10\cdot \frac{d\:y}{dt}=-5\cdot ...
0
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1answer
22 views

Fourier Transform existence

Let $\varphi:\mathbb{R}^2\to\mathbb{R}$ be a continuous function. Moreover, consider that $f:\mathbb{R}\to\mathbb{R}$ is a schwarzian function, i.e. $f\in C^{\infty}$ and $\lim\limits_{x\to\pm\infty} ...
1
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1answer
29 views

Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
0
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0answers
37 views

Inverse of a toeplitz matrix with fft based methods

I have a covariance matrix, Q and I need to find out Q^-1. Here, Q is a Toeplitz matrix. Now, I want to calculate the inverse of the matrix with fft based methods rather than the conventional ones ...
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0answers
29 views

Bounding a sum on lattice points in an annulus

How do we bound $\Sigma_{\beta<|\vec{k}|<\alpha}|\vec{k}|^{-2}$ by using an integral comparison type method ? How about $\Sigma_{\beta<|\vec{k}|<\alpha}|k|^{-4}$ Here ...
2
votes
1answer
40 views

Proof of elementary Wiener's tauberian theorem

I want to proof the following (simple case/version of) Wiener's tauberian theorem: The (span of the) set of translates $\{f_a | a \in \mathbb R^n\}$, $f_a(x) = f(a+x)$, is dense in $L^2(\mathbb R^n)$ ...
0
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1answer
32 views

Applying Fourier transform to heat equation with source

I haven't had any experience with applying of FT to heat equation with source. But this popped up in an exercise. Any help in the right direction would be great. Consider: $$\frac{\partial ...
1
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1answer
28 views

What is Fourier Transform of $\phi(x,y) = 2x $

How to calculate Fourier transform of this 2D function? $\phi(x,y)=2x$ for $-1<=x <= 1 ; and -1<=y<=1$ and $\phi(x,y)=0$ ; otherwise I tried like this: $\psi(u,v) = ...
1
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1answer
53 views

Condition for existence of Fourier transform?

We can convert signal into frequency domain using Fourier transform. But I think we can't compute Fourier transform of any signal . Fourier transform also should have some limits. So I want to ask ...
0
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1answer
40 views

Fourier transform of a piecewise

How should I go about seeking the Fourier transform for the piecewise function: $$f(x) = \left\{\begin{matrix} 0 ,&|x|>a \\ 1 ,&|x|<a \end{matrix}\right.$$ Is this the correct ...
0
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0answers
16 views

Discrete Fourier Transform of a function

Is it possible to find the discrete Fourier Transform of a non-discrete function? I've been asked to find out the discrete Fourier Transform of the function $g(x)=\mathcal{X}_{[-2,2]}f(x)$, where ...