Questions tagged [fourier-analysis]

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).

Filter by
Sorted by
Tagged with
1
vote
1answer
40 views

What is the value of $\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{k=1}^\infty\Big(1-p+p\exp \frac{it}{2^k}\Big)dt$?

Here $0<p<1$. If $p=\frac{1}{2}$ the value is $1$. In all cases the value is a positive real number. This integral is associated with the inverse Fourier transform of some characteristic ...
-1
votes
0answers
24 views

Show $L^2([0,1])$ with $|f|^2 = \int_0^1 (f(x)^2 + 0.5 \, f'(x)^2 ) \, dx$ is isomorphic to $L^2([0,1])$ with the standard norm

Let's consider two Hilbert spaces, copies of $L^2([0,1])$ with two different norms. $H_1 = L^2([0,1])$ with norm $|f|^2 = \int_0^1 f(x)^2 \, dx $ $H_2 = L^2([0,1])^2$ with norm $|f|^2 = \int_0^1 \...
1
vote
2answers
39 views

Book recommendation on Fourier analysis technique for PDE

I want a graduate-level textbook which discusses Fourier analysis techniques for solving PDE. To elaborate what I want to study, consider the Laplace operator $-\Delta$. Since the differential ...
0
votes
0answers
26 views

Fourier transform of product of exponential decay and cumulative normal

I am trying to find the Fourier transform $ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(t) \cdot e^{i\cdot\omega\cdot t} dt $ of the following function: $$ f(t) = e^{-a\cdot t}\cdot \mathcal{N}\...
1
vote
3answers
41 views

If $\{x\}:=x-\lfloor x\rfloor$, then are these the same? $\{l+m\}$, $\left\{l+\{m\}\right\}$, $\left\{\left\{l\right\}+\{m\}\right\}$

I am currently working on some signal processing project and come across this particular problem: Define $\{x\} := x - \lfloor x\rfloor$ and consider $$y_1 = \{l+m\}$$ $$y_2=\left\{l+\{m\}\...
1
vote
0answers
24 views

Estimate of a certain Hardy-Littlewood maximal function at infinity

$f\in L^1_{loc}(\mathbb{R}^n)$,prove that if $f(x)=O(|x|^{-n}),|x| \rightarrow \infty$,then $Mf(x)=O(|x|^{-n}\log|x|),(|x|\rightarrow\infty)$. where $Mf$ is the Hardy-Littlewood maximal function of f. ...
0
votes
0answers
21 views

implication of $\int _\mathbb{R} \left(|\hat{f}(x)|^2+|\hat{f}(x)|^2/|x|\right)dx < \infty$

Let $f$ be in $L^1(\mathbb{R})$ and $\hat{f}$ be its fourier transform. Then, what can be the implication of the condition $$\int _\mathbb{R} \left(|\hat{f}(x)|^2+|\hat{f}(x)|^2/|x|\right)dx < \...
0
votes
0answers
28 views

“Steady-state” of Error function using Fourier Transform

I am interested in studying the steady state dynamics of a function, by means of looking at the Fourier transform. As a way to illustrate, suppose that the signal I want to study is the error function....
5
votes
2answers
83 views

Why are variance and expected value all we care about?

My statistics background is almost $\varnothing$, so I apologise if my question has already been asked, and I just didn't know the right terminology to find it here. Suppose $p_X$ is the probability ...
2
votes
0answers
19 views

error estimation for the Fourier series of the fractional part of $x$

I'd like to prove the error estimate for the factional part of $x$ as given above. To begin with, we apply the Abel's formula and get $$(1):\left|\sum_{n=N+1}^M \frac{\sin(2\pi n x) }{n}\right|=\...
0
votes
1answer
22 views

For every $f \in C(\mathbb{T})$, there exists a corresponding solution $u(x,t)$ to the heat equation on a ring under specific conditions

The following is an old exam question I'm stuck on: Show that for every $f \in C(\mathbb{T})$ (where $\mathbb{T} = [-\pi, \pi]$) there is an initial condition $g \in C(\mathbb{T}$ for which there ...
0
votes
0answers
10 views

Coefficient of variation of a periodic signal

Is there a way to estimate the coefficient of variation $\frac{\sigma}{\mu}$ of a non-zero mean periodic signal $x(t)$ from its Fourier Transform? I'd like to estimate the ``flatness'' of the signal
0
votes
0answers
34 views

Steady-state of a system and Fourier Transform

I successfully found the Fourier Transform of a function $f(t)$. Suppose the Fourier transform looks like the one below, so its not easy to calculate the Inverse transform: $$ F(w) = a ^{c - \sqrt{b -...
0
votes
0answers
5 views

Derivative of Fourier Expanded Variable

Suppose I have a vector field that is periodic in one direction such that it can be simplified as: $$\mathbf{u'}(x,y,z,t)=\int_{-\infty}^{\infty}\mathbf{\hat{u}}(x,y,t)e^{ikz} dk,$$ where $\mathbf{u'...
4
votes
2answers
876 views

Can FFT be adapted for deconvolution of non-periodic functions?

Can a non-periodic function be padded at the boundaries and deconvolved with inverse FFT? Since a Toeplitz matrix can be embedded in a circulant matrix to perform the deconvolution, is there an ...
0
votes
0answers
20 views

How to visualize complex Dirichlet Kernel?

We know that the Dirichlet Kernel is defined as $$ D_{N}(t) = \frac{1}{2\pi} \sum_{n=-N}^{N} e^{int}$$ for $N \in \mathbb{Z}$ and $t \in \mathbb{R}$ (for Fourier series, we usually just consider the ...
3
votes
1answer
57 views

How to calculate $c_a$ where $\left(f\mapsto\int_{\mathbb{R}}\frac{f(t)-f(0)}{|t|^{a}}dt\right)=c_a\mathcal{F}_x(|x|^{-1+a})$

From this question I know that for every $a\in\mathbb{R}$ there exists a unique radial positive homogeneous tempered distribution of degree $a$, up to a multiplicative constant. Also, it is easily ...
0
votes
1answer
44 views

Fourier transform of $\frac{1}{\sinh(x+a)}$ for complex $a$

I am trying to understand how to generally compute the Fourier transform of the function $\frac{1}{\sinh(x+a)}$, where $a$ is a general complex number. Plugging the equation into Wolfram Alpha gives a ...
1
vote
2answers
66 views

Distributional Fourier Transform of $\frac{1}{|x|^a}$

Let $h_a=\frac{\Gamma(\frac{a}{2})}{\pi^{\frac{a}{2}}}|x|^{-a}, x \in R^d$ Then $\hat{h_a}=h_{d-a}$ in the sence of $L^1+L^2-$Fourier transforms if $\frac{d}{2}<Re(a)<d,$ and in the sence ...
0
votes
0answers
35 views

Fourier Expansion of this Wave.

I've been trying to Fourier expand $\psi(t)=\sin\left(2\pi at\ +\ \sin\left(2\pi bt\right)\right)$ skip the below paragraph if you like to I think the methods I've tried all along evolved with time. ...
3
votes
1answer
49 views

If $f \in L^1(\mathbb{R})$ and its Fourier transform has compact support, then $f(x) = \sum_{n \in \mathbb{Z}} f(n)\operatorname{sinc}(x-n)$

An old exam question I'm practicing with: Let $f \in L^1 (\mathbb{R})$ and assume that $\hat{f}$ (the Fourier transform of $f$) is supported on the interval $[-1/2, 1/2]$. Let $\operatorname{sinc} (x)...
1
vote
1answer
27 views

$2\pi$-periodic $L^2$ functions on $R^1$ approximated by its Fourier series

I'm reading section 4.26 in Big Rudin, but I have two questions. Suppose $f$ is in $L^1(T)$. This means $f$ is the class of all complex, $2\pi$-periodic, and Lebesgue measurable functions on $R^1$ for ...
0
votes
0answers
44 views

Fourier Series for a Dirac Train

I'm trying to find out by myself the Fourier Series of a Dirac Train, but I'm getting after Integration by Parts that Dn equals to 0 and not to 1 as needed to be. Could you please help me find my ...
1
vote
1answer
737 views

Can you integrate sine or cosine of a cubic polynomial?

I would like to be able to integrate the sine (and cosine of a cubic polynomial) e.g. $\int_X^Y \sin{(ax^3 + bx^2 + cx + d)} {\rm d}x$ Does anyone know if an analytical solution to this exists? ...
8
votes
4answers
9k views

“Every function can be represented as a Fourier series”?

It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series. So this got me thinking about the ...
1
vote
0answers
33 views

Fourier transform of a Weierstraß product

Consider $E_1:\mathbb{C}\rightarrow \mathbb{C}, s\mapsto(1-s)e^s$ and the infinite (and on $\mathbb{C}$ holomorphic) product: $$\Pi_k(s)=\prod_{q \in \mathbb{Z}^\times\setminus{k}} E_1\left(\frac sq\...
1
vote
1answer
221 views

Zeros of Fox-Wright Function

This question is stimulate by the previous two question here and here. We are interested in studying the following special case of Fox-Wright function \begin{align} \Psi_{1,1} \left[ \begin{array}{...
0
votes
0answers
27 views

Fourier coefficients of $\frac {a-b \cos x}{a-be^{-ix}}H_1^{1} \left(\sqrt{a^2+b^2-2ab \cos x} \right)$ [closed]

The function is this $$\frac {a-b \cos x}{a-be^{-ix}}H_1^{1} \left(\sqrt{a^2+b^2-2ab \cos x} \right)$$ where H is a Hankel function. Is there any way to get the complex Fourier series coefficient of ...
0
votes
1answer
88 views

Paley–Wiener theorem for generalized functions

If I understand correctly, Paley–Wiener theorem says that if a function $F:\mathbb{R}\to \mathbb{C}$ is compactly supported, its holomorphic Fourier transform is entire. Just wonder, whether this ...
0
votes
1answer
38 views

The fourier series of periodic and real analytic function

Let $f$ be a real analytic and periodic function defined on the interval $[0, 2\pi]$. Then $f$ is infinitely differentiable for sure. Therefore, the fourier coefficients of $f$ decay faster than any ...
2
votes
1answer
3k 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 ...
2
votes
1answer
49 views

Proof of Theorem 9.10 in Rudin's Real & Complex Analysis

This theorem essentially states that an $L^p$ function can be approximated in the $L^p$ norm by convolutions with a suitable kernel, and I'm having a bit of trouble seeing how Rudin obtains one of the ...
0
votes
0answers
18 views

Wiener-Khinchin theorem problem

It is stated by the Wiener-Khinchin theorem that you can obtain the spectral density of a stochastic process $X_t$ merely by taking the Fourier-transform of its auto-correlation function \begin{...
6
votes
1answer
654 views

Connection between SVD and Discrete Fourier Transform for Denoising

Denoising signals (in particular, 2D arrays, such as images) can be done by removing the high frequency components of the discrete Fourier transform (which is related to convolution with a Gaussian ...
0
votes
1answer
18 views

Multidimensional Fourier series

On $\mathbb{T}$, we have approximation series $\{f_n\}$ of $f\in L^p,p\in[1,\infty)$ in the form of $f_n(x)=\sum_{|k|\le n}a_{n,k}e^{ikx}$, converging to $f$ in $L^p$. (for example, we can consider ...
3
votes
2answers
42 views

Help to find complex Fourier series coefficient of this periodic function

I'm having big trouble finding the complex Fourier series coefficient of the following periodic function $$\frac{a-b\cos\varphi}{\sqrt{a^2+b^2-2ab\cos\varphi}}$$ Mathematica is unable to compute it!!...
0
votes
0answers
35 views

Impossible to find the complex Fourier series coefficient for this periodic function [duplicate]

I'm having big trouble finding the complex Fourier series coefficient of the following periodic function $$ f(\varphi) = \frac{a-b\cos{\varphi}}{\sqrt{a^2+b^2 - 2ab\cos{\varphi}}} $$ Mathematica is ...
2
votes
1answer
137 views

A finite field problem

$\mathbb F$ is finite field, $\mathbb A$ is a subset of $\mathbb F$, and $|\mathbb A|>|\mathbb F|^{\frac 34} $. Proof $\forall x \in \mathbb F$, there exist $a,b,c,d,e,f\in\mathbb A$ that makes $x ...
1
vote
1answer
2k views

How do I find the magnitude and phase of the frequency response function for a third order system using transfer functions?

A third order system is described by: $$\frac{{\rm d}^3 y}{{\rm d}t^3} + 2 \frac{{\rm d}^2 y}{{\rm d}t^2} + 6 \frac{{\rm d} y}{{\rm d}t} + 5y = u + 2 \frac{{\rm d} u}{{\rm d}t}.$$ Determine the ...
0
votes
0answers
9 views

Find a certain function of moderate decrease

I am trying to solve the following problem: Find a function $f$ of moderate decrease such that $$\int_{-1}^{1} f(x-t) dt = e^{-2\pi |x-1|} - e^{-2\pi |x+1|} .$$ The first idea I had was to use the ...
0
votes
0answers
14 views

Elliptic Fourier fit coefficients

I am trying implement a function that could fit an elliptic fourier curve on a set of border points of a detected object. I am using cv2.findContours to acquire border points from a binary image. Next ...
0
votes
1answer
21 views

Uniform convergence of fourier series of step function [closed]

Consider the series $f(x) = \sum_{n=1}^{\infty} a_n \sin nx$, where \begin{align} a_n = \begin{cases} \frac{4}{n \pi}, &n \text{ odd} \\ 0, &n \text{ even} \end{cases}. \end{align} It is ...
4
votes
2answers
3k views

Fourier representation for $\tan(x)$

Q: Which Fourier representation is suitable for $f(x) = \tan(x)$: Fourier trigonometric series, Fourier half-range expansion, or Fourier integral and why? Well I searched and found that: $\tan(x)$ ...
3
votes
1answer
57 views

Is there a Mellin transform or an analogue on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$?

From Wikipedia, the Mellin transform is an isometry $M : L^2(\mathbb{R}^+) \mapsto L^2(\mathbb{R})$, $$\{M f\} (s) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}^+} x^{-1/2 + \mathrm{i} s} f(x) dx.$$ ...
0
votes
1answer
2k views

Fourier analysis notation - Sh and Ch

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!
0
votes
1answer
718 views

Fourier Transform of sinc function (confused about $\operatorname{sinc}(\pi x)$ and $\operatorname{sinc}(x)$)

I am confused about the fourier transform of the $\operatorname{sinc}$ function. First I don't know if $$\operatorname{sinc} (x) = \frac{\sin(\pi x)}{(\pi x)}$$ or $$\operatorname{sinc} (\pi x) = ...
7
votes
1answer
80 views

Carleson-Hunt Theorem on $\Bbb R$.

Wikipedia states the following: If $p\in(1,\infty)$ and $f\in L^p(\Bbb R)$ then $f(x)=\lim_{A\to\infty}\int_{|\xi|<A}\hat f(\xi)e^{i\xi x}\,d\xi$ almost everywhere. This seems clearly ...
1
vote
1answer
29 views

Twice continuously differentiable wavelets with compact support

Do there exists wavelets such that both mother wavelet and the scaling function have compact support and they are both twice continuously differentiable?
0
votes
1answer
39 views

Fourier Series of Brownian Motion $(B_t)_{t \in [0,1]}$ wrong?

In our lecture about Brownian motion, we calculated the fourier series of Brownian motion $(B_t)_{t \in [0,1]}$. We did the following: We first defined the Brownian bridge $(\beta_t)_{t \in [0,1]}$ ...
5
votes
2answers
184 views

About Rudin's outline to proving that Lipschitz functions have converging Fourier Series

I'm trying to do the following exercise from Rudin's Real and Complex Analysis: Suppose $f\in C(T)$ and $f\in \text{Lip }\alpha$ for some $\alpha>0$. Prove that the Fourier series of $f$ ...