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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339 views

Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
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92 views

Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs. Let $u(x,t): \mathbb{R}^2 \to \mathbb{C}$, which solves the differential equation $\hat{L}u(x,t)=0$. $u(x,t)$ is ...
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4answers
356 views

Find the Fourier transform of $\dfrac{\sin(x)}{x}$

"Let $\chi : \mathbb{R}\to \{0, 1\}$ be the characteristic function of the interval $[−1, 1]$ and let $f(x)=\sin(x)/x$. find the Fourier transform of $\chi$, find the Fourier transform of $f$, ...
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3answers
397 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
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1answer
53 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
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1answer
186 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
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1answer
544 views

Fourier transform of a continuous non-periodic function in matlab

I would like to use matlab to find an Fourier transfom of a function which is known only on a grid. As an example I take the function f(x) = exp (-x^2), which Fourier transform is known and is equal ...
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1k views

Can you provide help with interpreting this periodogram?

I'm trying to track down the source of some wonky data. The data are response times (RTs) collected from humans using a computer keyboard. Here's a histogram of the RTs, binned to 1ms: Obvious is a ...
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1answer
20 views

Zeros of a finite Fourier integral implies the same for all derivatives

Let $\phi(x)\in C_0^{\infty}(\mathbb{R})$ be an infinitely differentiable finite function with support $\operatorname{supp} \phi \subset [-c,c]$ and let $\mu_0$ be a zero of the function ...
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2answers
2k views

Fourier transform of a function of compact support

My professor occasionally assigns optional difficult problems which we do not turn in from Stein and Shakarchi's Complex Analysis. I am currently studying for a test in that class and try to get all ...
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1answer
71 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= ...
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39 views

Formal inverse of a matrix ressembling Fourier's matrix

What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$? When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and ...
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1answer
41 views

How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
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42 views

Avoiding FFTs by reusing prior FFT results

Background From a mathematical point of view, the formulas similar to the following were produced: $F_1(f) = \mathcal{F}(T(t))$ $F_2(f) = \mathcal{F}(T(t)\times sin\Theta t)$ $F_3(f) = ...
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99 views

Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
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1answer
87 views

Where is the symmetry of Fourier transform in its implementation in Maxima and Wolfram Alpha?

From Wikipedia I saw that there is a symmetry of the Fourier transformation $F(F(f))(x) = f(-x)$ This matches the graphical explanation of the (German) Youtube video (9:15 to 9:45). I tried to see ...
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82 views

Understanding the Quantum Fourier Transform

I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. ...
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1answer
279 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
4
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1answer
160 views

Using Fourier Transform to solve heat equation

the heat equation of $U(x,t)$ on $-\infty<x<+\infty$ and $t>0$ is $$U_t=U_{xx}+\exp\left({\frac{-x^2}{2}}\right)$$ where $$U(x,t)\rightarrow 0 \quad as\quad x\rightarrow\pm\infty$$ and the ...
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1answer
363 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
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1answer
282 views

$f\in L^{\infty}(\mathbb{R})$ as Fourier transform

i need to know if one can view a function $f\in L^{\infty}(\mathbb{R})$ as a Fourier transform of a certain function, say g? If the answer is positive please state the proof, or help me find one. ...
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1answer
46 views

Positive-definite + continuous at 0 $\Rightarrow$ continuous?

Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$. Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$? ...
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1answer
177 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
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1answer
142 views

Fourier transform of $F(x)=\exp(-x^2/(2 \sigma^2))$

I am looking for the fourier transform of $$F(x)=\exp\left(\frac{-x^2}{2a^2}\right)$$ where over $$-\infty<x<+\infty$$ I tried by definition $$f(u)={\int_{-\infty}^{+\infty} ...
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1answer
94 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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1answer
55 views

Fourier Transform Properties Proof

If: $y(t) = x(t)*h(t)$ and $g(t) = x(9t)*h(9t)$ (Where * is convolution) How can I use properties of the Fourier transform to show: $g(t) = Ay(Bt)$ and find constants? I think A should be $1/81$ ...
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0answers
42 views

How to find the global minimum of a function from its Fourier transformed function.

i.e Can $\min{f(t)}$ be expressed by $F(\omega)$? I have a series of data in frequency space. I can do discrete Fourier transform to time space to find its minimum. But I am wondering if there is a ...
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1answer
75 views

Fast Fourier Transformation: inverse transform of the product of polynomials

I have managed to implement and understand most of the Fast Fourier Transformation. However, I have one last question. If one has two polynomials, say $A(x)$ and $B(x)$, and one computes DFT of ...
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48 views

Fast Fourier Transformation, explanation of $A(x)=A_0(x^2)+x A_1(x^2)$

I was looking at the Fast-Fourier Transformation today, on this site [if you cannot read Russian, simply use Google Translate, which is what I am doing right now]. http://e-maxx.ru/algo/fft_multiply ...
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1answer
70 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
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2answers
70 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
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1answer
81 views

Functions for which $\mathcal{F}g = f \ast f$

Suppose one is given $f \in L^{2}(\mathbb{R})$, my question is whether or not there exists a $g \in L^{1}(\mathbb{R})$ such that $f \ast f = \mathcal{F}g$ where $\mathcal{F}$ is the Fourier transform. ...
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63 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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1answer
289 views

Amount of sound near a specific frequency at a specific time

I have a sound signal, sampled at 48000 hz. Now I want to know 'how much sound' there is of a specific frequency (or near that frequency) at a specific time. For example, I want to know at 10s how ...
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1answer
170 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
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1answer
28 views

Is it true that, $x\rho(x/t)\in H^{s}$ for $\rho\in \mathcal{D}(\mathbb R)$ and $s>3/2$?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support; and the Sobolev space $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} ...
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1answer
66 views

Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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4answers
2k views

How to determine the periods of a periodic function?

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period ...
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1answer
37 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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1answer
83 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
3
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1answer
58 views

Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
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1answer
46 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
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2answers
76 views

Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
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1answer
44 views

Paley-Wiener theorem for a sector $\{\zeta:-\epsilon<\arg(\zeta)<\pi+\epsilon\}$

One of the variations of the Paley-Wiener theorem yields: If $f\in L^2(\mathbb R_+)$, then the Fourier transform $F$, defined by $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ is a holomorphic function ...
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110 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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1answer
1k views

The meaning of the connection between power spectral density and auto correlation

I know that if we have a signal $x(t)$, then its Fourier transform would be the signal in the frequency space, which I understand to be how much of each frequency exists in the x(t) signal. $ ...
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3answers
65 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
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1answer
40 views

Help for understanding Danielson-Lanczos lemma

The Danielson-Lanczos lemma is the basis for fast Fourier transform algorithms. Now, I do understand this step $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N} = \sum_{n=0}^{(N/2)-1} ...
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1answer
538 views

Square Summable functions

Can somebody please help me understand the notion of square summable functions intuitively?? I have been self studying Hilbert Spaces and Fourier Transform for DSP. Any help is appreciated. Thanks.
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1answer
66 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...