# Questions tagged [fourier-transform]

The fourier-transform tag has no usage guidance.

324
questions

**3**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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(\...