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
0 votes
0 answers
11 views

Fractional Fourier Transform of $\sqrt{c} x( c(t - \tau))$

I am trying to figure out what the Fractional Fourier Transform of the signal $\sqrt{c} x(c(t-\tau))$ would be with respect to that of $x(t)$. According to the paper "The Fractional Fourier ...
user avatar
  • 1,011
1 vote
0 answers
17 views

Solving the discrete Fourier Transform of $\sin(x)+\sin(2x)$

I'm new to the discrete Fourier Transform and have managed to do a few examples by hand, (for example the Fourier Transform of $\sin(x)$). I now want to try the transform on signals that are made up ...
user avatar
  • 11
0 votes
1 answer
42 views

The Fourier Transform of a Radial Function on $\mathbb R^n$

I am trying to understand the following computation for the Fourier Transform of a Radial Function on $\mathbb R^n$. I shall ask questions in-line. Suppose $n\ge 2$, $f\in L^1(\Bbb R^n)$, and $f(x) = ...
user avatar
-1 votes
0 answers
18 views

Fourier series coefficient question from a past paper [closed]

f'(x)= a0 + ∑(ancos(nx)+bnsin(nx)) f and f' are continuous on [0,2pi] domain[0,2pi] determine a formula for b8 in terms of f can anyone help please.
user avatar
0 votes
0 answers
15 views

Discrete Fourier Transform, even and odd components

Let $v = [v_0, v_1, \dots, v_{n−1}]^T$ is the Discrete Fourier Transformation, $V = [V_0, \dots, V_{n-1}]^T $. Define $w = [w_0, w_1, \dots, w_{2n−1}]^T$ where $v_k =w_k$ for $0 \leq k \leq n−1$ and $...
user avatar
  • 1
2 votes
0 answers
15 views

Counstruct a sequence of Schwartz functions

Here is an exercise from Wolff's lectures on harmonic analysis, p26: Using translation and multiplication by characters, construct a sequence of Schwartz functions $\lbrace\phi_n\rbrace$ so that Each ...
user avatar
  • 51
0 votes
1 answer
30 views

Expression $f$ on $\mathbb{T^n}$ in terms of $\hat{f}$.

I'm studying Real analysis, Folland section 8.4 page 257. I can't undersfand the below highlighted statement. If $f\in L^1(\mathbb{T^n})$ and $\hat{f}$ $\in l^1(\mathbb{Z^n})$, then the Fourier series ...
user avatar
2 votes
1 answer
25 views

How can you derive the spacetime Fourier transform of the free Schrodinger evolution rigorously?

I'm trying to compute the spacetime Fourier transform of the free Schrodinger evolution. Consider $f\in L^2(\mathbb{R}^d)$ and $e^{it\Delta}f=:\mathcal{F}^{-1}(e^{-it|\xi|^2}\hat{f}(\xi))$ its free ...
user avatar
  • 43
4 votes
1 answer
104 views
+50

Does a bounded function like this exist?

is it possible to find a function $f$ such that $f \in L^{\infty}(\mathbb{T}^1)$ and $$\sum_{j=0}^{+\infty} \left(\sum_{n \in \mathbb{Z}: 2^j \le |n|<2^{j+1}}|\widehat{f}_{n}|^{2} \right)^{1/2}=+\...
user avatar
  • 222
0 votes
1 answer
35 views

Fourier transform of a class of interesting functions to optimize a numerical algorithm

I try to speed up a numerical algorithm and I came to a class of real functions where I need the Fourier transform or the coefficients of the Fourier series with a large interval of them. The class of ...
user avatar
  • 111
4 votes
1 answer
51 views

The functionals $\|\cdot\|_{L^{p,q}}$ do not satisfy the triangle inequality.

Given a measurable function $f$ on a measure space $(X,\mu)$ and $0<p,q\leq \infty$, define $$\|f\|_{L^{p,q}}=\left\{ \begin{array}{ll} \displaystyle{\left(\int_{0}^\infty\left(t^{1/p}f^*(t)\right)...
user avatar
1 vote
0 answers
43 views

Formula connecting Fourier transforms of function and its derivative

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, differentiable and suppose $f$ and $f'$ are both summable. Then $$\mathcal{F}(f')(x) = -ix \mathcal{F}(f)(x)$$ (Here $\mathcal{F}$ represents ...
user avatar
1 vote
0 answers
24 views

Cesaro summation of the inverse Fourier transform.

Let $f\in \mathcal{L}(\mathbb{R}^1).$ Prove that $$f(x)=\frac{1}{\sqrt{2\pi}}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat fdy\,dt$$ for almost all x including points of continuity ...
user avatar
3 votes
2 answers
106 views

Let $f \in C^1(\mathbb{R})$ be such that both $f$ and $f'$ belong to $L^2(\mathbb{R})$. Show that $\hat{f} \in L^1(\mathbb{R})$

The exercise is the following: Let $f \in C^1(\mathbb{R})$ be such that both $f$ and $f'$ belong to $L^2(\mathbb{R})$. Show that $\hat{f} \in L^1(\mathbb{R})$. What I want to prove is that $\int_\...
user avatar
  • 61
0 votes
0 answers
12 views

How differentiate signals

I am doing a course called Signals & Transform and I am having difficult time understanding the concept of unit step functions and how to use it to differentiate signals. Here is the exercise: ...
user avatar
  • 153
0 votes
0 answers
14 views

Fourier transform of $sinc(cos(t)) $

Given Linear system $Q $ with input $x(t)$ and output $ y(t) $ and impulse response to $ \delta(t-\tau) $: $$ h\left(t,\tau\right)=sign\left(\cos\left(\omega_{0}t\right)\cos\left(\omega_{0}\tau\right)+...
user avatar
-1 votes
0 answers
19 views

This is Stein fourier analysis. In section 3.3,I really don't understand how he gets to the final result(equation (4)),can someone explain in detail?

enter image description here the equation 3 mentioned i ll also post here (but actually it is of little use) enter image description here
user avatar
1 vote
0 answers
20 views

Interfering higher frequency waves to create a lower frequency one

Once I saw a demo where multiple speakers were outputting inaudible sounds (I assume very high frequency) that would interfere at certain locations to be audible. It has fascinated me for some time ...
user avatar
  • 611
0 votes
0 answers
18 views

MSE of the Phase Estimation of a Noisy Complex Quantity

Presentation of the Problem I am making a measurement which yields me the complex quantity: \begin{equation} S_k = e^{i \cdot \phi} + X_k + i \cdot Y_k \quad \text{with} \quad X_k, Y_k \stackrel{iid}{...
user avatar
  • 101
0 votes
0 answers
18 views

Is the function $(1+|\ln |x||)^{-1/2}$ on $\mathbb{R}$ a Fourier multiplier?

How to judge whether the function $(1+|\ln |x||)^{-1/2}$ is a Fourier multiplier on $\mathbb{R}$? Should I use some multiplier theorems?
user avatar
0 votes
0 answers
28 views

Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$?

Question: Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$? My attempt: the period of $\sin(t)$ is $2\pi$, so the period of $\sin(2t)$ is $\pi$ and the period of $\sin(qt)$ is ...
user avatar
  • 153
0 votes
0 answers
6 views

Energy of a signal simplification

I'm learning about signals from the textbook Signals and Systems Second Edition, by Oppenheim and Willsky in order to learn more about Fourier Analysis and I'm having difficulty with the Energy ...
user avatar
4 votes
1 answer
49 views

problem on $L^2$ (pointwise) convergence and Carleson's Theorem

I am having some problem on my arguments and I want to see where it fails. It deals with pointwise convergence in $L^2[-\pi,\pi]$: pointwise convergence is given by Carleson's Theorem (so it is a hard ...
user avatar
0 votes
0 answers
15 views

Help understanding an application of the inversion theorem (limit version) for the Fourier transform

On page 5 of this paper, it is stated in the proof of Prop. 4 that for a random variable$X$ with distribution function $F$ and characteristic function $\phi$ (i.e. scaled two-sided Fourier transform $\...
user avatar
0 votes
1 answer
74 views

How to compute the Fourier transform of a zero-centered Gaussian pulse over a specified range, say $[0,1]$?

Given an even Gaussian kernel $f : \Bbb R \to \Bbb R^+$ $$f(x) = \exp \left(- \frac{1}{2\sigma^2} x^2 \right)$$ and a probability density function (or, if you discretize $[0,1]$, then a probability ...
user avatar
0 votes
0 answers
41 views

Estimation for the integral of square function from its integral

Let $f$ be a real valued smooth function which has compactly supported Fourier transform and $$2\leq \int_{\mathbb{R}} f(x) dx \leq 2+ c, $$ for some sufficiently small $c>0$ with $\chi_{[-1,1]}(s) ...
user avatar
  • 1,036
0 votes
0 answers
23 views

Error in solution of 3-45 in Solution Manual of Signals and Systems Oppenheim 2nd edition

Hello I was solving the question 3-45 of the book Signals and Systems 2nd edition and I assume there is an error in the solution manual, kindly if someone confirm that am I right? or the solution ...
user avatar
0 votes
0 answers
19 views

Prove that if $TM_{e^{2\pi ix}}=M_{e^{2\pi ix}}T$ then $f=0$ where $T=C_f$ (convolution), $f \in C[0,1]$ and $M_g(h) := gh$

I am trying to solve the following problem: Let $H=L^2[0,1]$, $T=C_f$ (convolution) where $f \in C[0,1]$ (continuous function). Define $M_g(h) = gh$. Prove that if $TM_{e^{2\pi ix}}=M_{e^{2\pi ix}}T$ ...
user avatar
  • 723
1 vote
0 answers
172 views

Support of Fourier transform and Inverse Fourier Transform

Suppose $f \in L^2(\mathbb{R})$ and let $X \subseteq \mathbb{R}$ be a set of finite measure. Here let $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier transform. Q: If the support of $\mathcal{...
user avatar
0 votes
0 answers
11 views

Discrete fourier series from impulse train

Im struggling to derive the DFS from a periodic impulse train: let x(t) be a periodic signal over $NT_s$. As such, its sampled, impulse train version is: $x_p(t) = \sum_{n\in Z}x(nT_s)\delta(t-nT_s) = ...
user avatar
  • 65
0 votes
0 answers
34 views

Why do we use {+1, -1} in place of {0, 1} for the Fourier analysis of boolean functions?

I want to know what will change if we keep on using {0,1} for our Fourier analysis of boolean functions? What are the things, which can not be performed with {0,1} and can be done with {+1, -1}?
user avatar
  • 9
0 votes
0 answers
20 views

About the Dirichlet conditions and the Fourier series of the sine function

One of the Dirichlet conditions, at least in the texts I've used, is: "$f$ must have a finite number of maxima and minima." This is not true for the sine function as its derivative is the ...
user avatar
  • 379
2 votes
0 answers
39 views

Need help understanding proof of the functional equation for the theta function.

I am following the proof of theorem 1.3 above until at the point the author finds the recurrence relationship $a_n = a_{n-k}e^{b - 2\pi ni\tau}$. When $k = 0$ we have $a_n = a_{n}e^{b - 2\pi ni\tau} \...
user avatar
  • 184
2 votes
1 answer
212 views

prove that the sequence $A_n$ satisfies a particular formula

Let $D = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2\leq 1\}.$ Let for each $n\in\mathbb{Z}, f_n(r,\theta) = r^{|n|} e^{in\theta}$. Consider $f(r,\theta) = \sum_{n\in\mathbb{Z}} A_n f_n(r,\theta)$, where $...
user avatar
  • 883
0 votes
0 answers
27 views

What is the Fourier transform of $xf(2x)$?

Given $f\in L^1(\mathbb{R})$ is a generic function, how do you find the Fourier transform using tables? Thus far I've gotten: $F(xf(2x))=F(xg(x))=(\frac{i}{2n})^1*\frac{d}{d\omega}\hat{g}(\omega)$ $\...
user avatar
1 vote
0 answers
119 views

Prove that the Fourier series of a function converges uniformly

This question is asking for a proof of a claim I don't understand (namely the pointwise convergence of the partial sums to a Fourier series). Alternatively, a completely different approach that's not ...
user avatar
  • 883
0 votes
2 answers
19 views

How to prove $|-\Delta_{x} ((-x)^{\alpha} \phi(x))|\leq A_{j,\alpha}(1+|x|)^{-n-1}$

How to prove $|-\Delta_{x} ((-x)^{\alpha} \phi(x))|\leq A_{j,\alpha}(1+|x|)^{-n-1}$ Hi all, i am reading Pseudo-differential Operators, singularities, applications. Y Egorov, on page 2. The inequality ...
user avatar
0 votes
0 answers
10 views

Inverse transform of the exponential covariance function

Let $C(h)$ be the exponential covariance function, Bochner's theorem says that $C$ is positive definite if and only if its the Fourier transform with respect to a spectral measure $\chi$. Now, under ...
user avatar
1 vote
0 answers
16 views

proofs about a norm involving an integral and fourier coefficients

Let $C([-\pi, \pi])'$ be the set of continuous functions $f$ from $[-\pi, \pi]$ to $\mathbb{C}$ such that $f(\pi) = f(-\pi)$. For $f \in C([-\pi, \pi])', n \in \mathbb{Z}$, define $a_n = \frac{1}{2\pi}...
user avatar
  • 883
0 votes
0 answers
20 views

Frequency peak always appearing at half of Nyquist frequency in Fourier transform

When taking the FT of a signal I always get a sharp peak at exactly half the Nyquist frequency. My signal is shown here: and its FT here: The Nyquist frequency is 36.7 KHz. As can been seen in the ...
user avatar
  • 1
0 votes
0 answers
34 views

Show that $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$.

Problem: $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$. The Problem arises from Walter Rudin's PMA: For the texts ...
user avatar
3 votes
1 answer
63 views

Analytic formula for $f(\alpha):= \int_{-\infty}^\infty\frac{|e^{is}-1|}{|s|^{2\alpha+1}}\,ds$

For any $\alpha \in (0,1)$, define $\displaystyle f(\alpha):= \int_{-\infty}^\infty\frac{|e^{is}-1|}{|s|^{2\alpha+1}}\,ds$. Question. Is there an analytic formula for $f(\alpha)$, say, in terms of ...
user avatar
  • 8,309
1 vote
0 answers
53 views

Question about a Corollary of Grafakos's "Modern Fourier Analysis"

The question is about the proof of Corollary 2.2.5 in Grafakos's book "Modern Fourier Analysis". And the statement can be writen as follows: Let $\phi,\psi \in \mathcal{S}(\mathbb{R}^n)$ and ...
user avatar
  • 11
1 vote
0 answers
57 views

Computing inverse Fourier transform of a step function

I am trying to compute the inverse Fourier transform of this function: $F(x)=\chi_{[-1,1]\times[-1,1]}$. Using the definition of inverse Fourier transform, $$ \mathcal{F}^{-1}(F) \\ =\int_{\mathbb{R}^...
user avatar
  • 1,294
2 votes
1 answer
65 views
+100

Prove a relation involving the Laplace–Beltrami operator and spherical harmonics

Let $P_ℓ$ denote the space of complex-valued homogeneous polynomials of degree $ℓ$ in $n$ real variables, here considered as functions ${\displaystyle \mathbb {R} ^{n}\to \mathbb {C} }$. Let $A_ℓ$ ...
user avatar
  • 7,280
5 votes
1 answer
97 views

Question concerning Tao's proof of uncertainty principle in Z/pZ

In his paper "An Uncertainty principle for cyclic groups of prime order", in the proof of Theorem 1.1, while proving the converse statement, Tao makes the following statement- "It will ...
user avatar
3 votes
1 answer
87 views

Prove the orthogonal decomposition of the space of spherical harmonics

Let $\mathbb{S}^{n}$ denote the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ endowed with the standard metric. For $k=0,1, \cdots$, denote by $\mathscr{H}_{k}^{n}$ the space of spherical ...
user avatar
  • 7,280
-5 votes
1 answer
29 views

Project 'b' onto 'a' to find out the projection of 'b' onto 'a' [closed]

[Question: Project 'b' onto 'a' to find out the projection of 'b' onto 'a'. To solve this, which formula should be used to find out the projection? Scalar Projection or Vector Projection? Lil bit ...
user avatar
1 vote
0 answers
25 views

Spherical Mean of Schwartz Function Decreases Rapidly - Stein, Shakarchi, Fourier Analysis.

I'm studying Stein and Shakarchi's Fourier Analysis book, and I'm stumped on page. 189, where it talks about the spherical mean of a function. The book defines the spherical mean of a complex-valued ...
user avatar
3 votes
2 answers
49 views

Applying Fourier transforms in two variables separately

Let $G(x, t) := (4\pi t)^{-d / 2}e^{-|x|^2 / 4t}$ for all $x \in \mathbb{R}^d$ and $t \in (0, \infty)$. The function $G$ is not integrable on $\mathbb{R}^d \times (0, \infty)$ since $$ \int_0^\infty \...
user avatar
  • 958

1
2 3 4 5
190