Questions tagged [fourier-transform]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
0answers
51 views

Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...
2
votes
2answers
172 views

Decay estimate of Fourier transform of a compactly supported function

Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate $$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$ for some $\...
1
vote
1answer
47 views

Positivity of exponentially bounded characteristic functions

I've noticed that for the classical examples of exponentially bounded, symmetrical distributions (Gaussian, Laplace, Double Exponential, Uniform), their characteristic functions are positive for all ...
0
votes
2answers
88 views

Which is the difference between Fourier Transformation and discrete wavelet transformation? [closed]

I'm sorry, I'm pretty new about these argoments. Is there a difference between Fourier Transformation and discrete wavelet transformation? I looked in Internet but I didn't understand if there is a ...
0
votes
0answers
16 views

Magnitude spectrum of a cascade of filter

Given is a input vector $x=[x_1 x_2 x_3] \in \mathbb{R}^{3N}$ with 3 consecutive sub-blocks $x_1,x_2,x_3 \in \mathbb{R}^{N}$, which goes through a cascade of filtering operations defined as Step 1 (1-...
3
votes
1answer
230 views

What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?

I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks. ...
2
votes
1answer
177 views

Find a collection of values of polynomial

Given a polynomial $f(x)\in \mathbb C[x]$ where $\deg f(x)=n-1.$ Assume that we need to find a collection of values of this polynomial corresponding to the following set of $x$-values: $\{ e^{ik} \}$ ...
2
votes
0answers
69 views

What are the necessary/sufficient conditions for a Fourier transform to have at least $k$ roots?

Let $f(x)$ be a symmetric function from $\mathbb{R}\to \mathbb{R}$, and $\hat f(k)$ be it's Fourier transform. What are the necessary and sufficient conditions for $\hat f(k)$ to have at least $n$ ...
4
votes
1answer
168 views

Vanishing of the product of a function and its own Fourier transform

I have found the following question to be surprisingly hard: Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that $$ f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere}, $$ ...
-3
votes
1answer
225 views

Is delta function symmetric against real axis? [closed]

Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$? I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis. We can write Delta function as $$\delta(z) = \...
3
votes
0answers
172 views

Finding (and saturating) a sharp Babenko-Beckner inequality for finite fields

My question is a follow-up to Abdelmalek Abdesselam's recent post What makes Gaussian distributions special? Local field version? asking about various characterizations of (real-valued) Gaussian ...
0
votes
1answer
55 views

Variance of spectral density is related to the gradient of signal?

Define the frequency variance as: $$ \sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$ Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore, $...
1
vote
1answer
129 views

Does Bochner's Theorem apply to Fourier coefficients?

Let $f $ be a periodic function and denote by $c_n$, for $n \in \mathbb{N}$, its Fourier coefficients, i.e. $$ c_n := \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{inx}\ dx. $$ It is well known that Bochner's ...
0
votes
0answers
44 views

Where does the order of conormality comes from?

Let $X^n$ be a closed manifold, and let $Y^k \subseteq X$ be a closed embedded submanifold. A distribution $ u \in C^{-\infty} \left(X\right)$ is said to be conormal of order $m$ along $Y$ if its ...
0
votes
1answer
60 views

Can a Fourier transform be performed on irregularly sampled data with timestamps?

Normally, when I think of performing a Fourier transform, I imagine that my samples are spaced regularly in time (or space). If I have a set of samples that are spaced irregularly, but have accurate ...
1
vote
0answers
48 views

Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?

Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form: $$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$ where the ...
1
vote
0answers
76 views

Fourier Transform; half space baby problem (new)

This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem. Essentially I am asking the same question now but taking a lot more care. So lets ...
4
votes
0answers
155 views

Arbitrarily long compositions of a functions that stay within $[0,1]$

This question is inspired by this MSE post. (in short, it is a continuous strengthening of the case of their claim for $k=1$, which is the only case which has been fully resolved discretely) Given $c\...
2
votes
1answer
315 views

Deconvolution using the discrete Fourier transform

Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
1
vote
1answer
176 views

When are Fourier cosine coefficients convex?

In the question When are Fourier coefficients monotonic it was determined that, if a function $f$ is (the restriction to $[0,2\pi]$) of a completely monotone function, then its Fourier coefficients, ...
12
votes
2answers
1k views

Intuitive explanation why “shadow operator” $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
18
votes
2answers
1k views

When are Fourier coefficients monotonic?

Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by $$ \hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
0
votes
0answers
47 views

Parseval's equivalent of Norm that includes a Projection matrix

I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...
0
votes
1answer
148 views

Joint distribution of random Fourier coefficients

Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_{...
2
votes
1answer
188 views

Fourier cosine transform of $\frac{\sin(b\sqrt{a^2+x^2})}{a^2+x^2}$

I've noticed it is not in Erdelyi or Gradshteyn, although the version with the sine replaced by a cosine is in Erdelyi (page 26 eq. 33). I've tried using the substitution $x=a \sinh z$ to avoid a ...
2
votes
0answers
67 views

Fourier Transform ; half space elliptic baby problem

I am attempting to look at some Liouville type theorems via a Fourier analysis approach and after looking at a baby problem I seem to be very confused. I assume this doesn't count as a research ...
0
votes
1answer
123 views

Fourier transform of $\frac{1}{t^2} \sin( at)^{2}\cos (at) \sin (2 a t)$ [closed]

What is the Fourier transform of this function: $\frac{1}{t^2} \sin( at)^{2}\cos (at) \sin (2 a t)$ I tried to solve it with trigonometric identities but i got a strage result..
1
vote
1answer
163 views

Well-known conditions for the Fourier inversion formula

Let $f\in L^1(\mathbb{R})$. One may easily check that $$(*)~~~f', f''\in L^1(\mathbb{R})\Rightarrow \int_\mathbb{R}|\hat{f}| ~\text{is finite} \Rightarrow \int_\mathbb{R}\hat{f}(s)e^{2\pi is x}ds ~\...
1
vote
2answers
96 views

Estimate for a simple oscillatory integral

If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that $$ \Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\...
0
votes
1answer
136 views

Convolution, Fourier transforms, and area preservation [closed]

Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
1
vote
0answers
105 views

Integrable functions that may not satisfy the inversion Fourier formula

Let $f\in L^1(\mathbb{R})$. We define $\phi_f(x)=\int_{\mathbb{R}} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$ if the improper Riemann integral is finite otherwise, $\phi_f(x)=\infty$. Does there exist ...
3
votes
2answers
417 views

On the Fourier inversion formula

For a given function $f\in L^1(\mathbb{R})$, suppose that the $$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$ almost every where converges in $\mathbb{R}$. Then, can we say that ...
1
vote
2answers
149 views

Two classic problems concerning Fourier transform of an integrable function

I am looking for the following questions: (1) True or false? for every $p<q$, one may find a function $f\in L^1(\mathbb{R})$ such that $\hat{f}\in L^q (\mathbb{R})$ but $\hat{f}\notin L^p (\...
0
votes
0answers
60 views

Making sense of a Fourier transform and proving that a function is differentiable and is in $W^{1,1}_\text{loc}(0,\infty)$

I have a function $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that $V(x,\cdot)\in C^0([0,\infty))$ and $V(x,0)=u(x)$, $\forall x\in\mathbb{R}^n$, where $u\in\mathcal{S}(\mathbb{R}^...
1
vote
0answers
53 views

The meaning of the frequency in continuous signals

Suppose that for a given signal $x:\mathbb{R}\to \mathbb{C}$ both of the following Fourier identities hold. $$ \hat{x}(\omega)=\int_\mathbb{R} x(t)e^{-it\omega} dt~~~,~~~x(t)=\frac{1}{2\pi} \int_\...
1
vote
0answers
63 views

Fourier transform of a Sobolev function dependent on a “parameter”

Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that $$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$ and ...
1
vote
0answers
31 views

Extension problem of fractional laplacian and Fourier transform of $L^1_\text{loc}$ function?

I have understand the proof oh the lemma 4.1.9 "SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY" by C.Bocur, there is a link. In this paper, we define, for $u\in L^1_\text{loc}(\...
2
votes
0answers
52 views

Fourier Transform diagonalizes time-invariant convolution operators [closed]

I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2. The Fourier transform is everywhere in physics and mathematics because it diagonalizes ...
0
votes
1answer
51 views

Integration against a certain Fourier transform

I asked the following question on mathstack but didn't receive any answers. I suspect that this question has a simple answer but I haven't thought about Fourier transforms in a while so am being ...
0
votes
0answers
78 views

A question about Fourier transform of a function defined by an integral

I have the function: $$ G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta}, $$ for all $x\in\mathbb{R}^n$ and $k>0$....
1
vote
0answers
104 views

Fourier transforms exhibiting symmetries about their critical points

Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
6
votes
1answer
334 views

General Fourier inversion formula (Gil-Pelaez)

Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{...
1
vote
1answer
153 views

Posterior expected value for squared Fourier coefficients of random Boolean function

Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by $$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
0
votes
0answers
75 views

Characterization of convolution operators via the Fourier transform

Let $\mathcal{L}$ be a linear and continuous operator from the space of tempered distributions $\mathcal{S}'(\mathbb{R})$ to itself. The Fourier transform of a tempered distribution $f$ is denoted by $...
6
votes
0answers
105 views

Fourier transformation of a distribution

We have no idea how to tackle the following Fourier transformation of a distribution: $$ \lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
2
votes
0answers
78 views

Failure of Strichartz estimates for the wave equation: elaboration of a counter-example

One can read in Oh - Probabilistic perspectives in nonlinear dispersive PDEs (Proposition 64, p. 60) that there exists a function $F \in L^2_tL^{1}_x (\mathbb{R}_t\times \mathbb{R}^3_x)$ which is ...
2
votes
0answers
91 views

About solutions of Klein-Gordon equation

I wonder how to solve the Klein-Gordon equation $$\left\{\begin{split}&\partial_t^2u-\Delta u+u=f\\&u(0,x)=u_1(x),\quad \partial_t u(0,x)=u_2(x)\end{split}\right.$$ where $u(t,x)$ defined on $\...
0
votes
0answers
38 views

Computing the matrix of first order marginals from low frequency terms of the Fourier Transform

Let $S_n$ be the symmetric group and $f:S_n \to \mathbb{R}$ be a function on $S_n$. We define the Fourier Transform of $f$ as the collection of matrices $$ \hat{f}_{\lambda} = \sum_{\sigma \in \...
2
votes
0answers
64 views

Particular Ehrenpreis factorization for covariance function

Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite). Is there a compactly supported function $g:\mathbb{R}^d\...
0
votes
1answer
79 views

How to use inverse Fourier transform to get the original signal

Suppose I have a signal $u(x)$ where $x\in[0,2\pi]$ and the signal has following form in Fourier space, $\hat{u}(\kappa)=\begin{cases} \frac{1+i\tan{(\varphi_u(\kappa))}}{\sqrt{1+\tan^2{(\varphi_u(\...

1
2 3 4 5
7