All Questions
Tagged with fourier-analysis fourier-transform
275 questions
4
votes
0
answers
188
views
Branch cuts, inverse Fourier transform and large time asymptotics
Let the Fourier transform of $f(t)$ be defined as $F(\omega) = \int_{-\infty}^\infty dt f(t) e^{i\omega t}$ for values of $\omega$ where the integral exists. What are the precise conditions on $F(\...
- 1,150
8
votes
1
answer
491
views
Functional equation with Fourier transform and $\frac{1}{x} f(\frac{1}{x}) $
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation:
$$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$
$\...
- 1,422
1
vote
1
answer
197
views
Probability of two Boolean functions being equal expressed in terms of the maximum Fourier coefficient
This paper by Maslov et al. uses that the probability of two $n$-bit Boolean functions $l(x)$ and $g(x)$ being equal is bound in terms of $\hat{g}_\text{max}$, the largest Fourier coefficient of $g(x)$...
- 13
8
votes
1
answer
640
views
Rate of decrease of the Fourier transform of standard mollifiers
What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$,
$$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$
and
$$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
- 6,486
3
votes
0
answers
320
views
Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
0
votes
0
answers
326
views
Precise decay of density through Fourier transform
Suppose $f(x)$ is a probability density on $\mathbb{R}$. Let $\varphi(t)=\int e^{itx}f(x)dx$ denote the Fourier transform (characteristic function). It is well-known that if $\int |x|^p f(x)dx<\...
- 87
1
vote
0
answers
245
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On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)
Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
- 111
4
votes
1
answer
325
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Fourier-positivity of a certain function
I am wondering how to prove the below Fourier transform is non-negative? I did much simulation and it seems to be non-negative.
$$\int_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}...
- 128k
1
vote
0
answers
213
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How to prove the Fourier transform of $e^{-x^p}$ is positive [duplicate]
I wonder how to prove that
$$\int_0^\infty\exp(-x^p)\cos(tx)\,dt\geq 0, \quad \frac{1}{2}<p<1.$$
This conclusion is used in the answer to another question here
Looking for sufficient conditions ...
2
votes
1
answer
250
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 ...
CommunityBot
- 1
6
votes
1
answer
397
views
Absolute values of two functions and absolute values of their Fourier transform coincides
Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or ...
- 13k
0
votes
1
answer
273
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Fourier transform of measures on $\mathbb{T}$
I'm currently working with Fourier transforms of measures on the $\mathbb{T}^n$ (more specifically in dimension two), i.e.
$$
\hat{\mu}(k) = \int_{\mathbb{T}^n} e^{i k \cdot x} d\mu(x)
$$
or something ...
- 2,118
1
vote
0
answers
62
views
Stable deconvolution of a band-limited function from its convolution with a Gaussian
Suppose that $f : \mathbb R \to \mathbb C$ is a band-limited function, i.e. its Fourier transform $\hat f$ has support in a compact interval $[-a,a]$. Let $\phi(t) = e^{-\frac{t^2}{2\sigma^2}}$ be a ...
- 437
12
votes
3
answers
2k
views
Looking for sufficient conditions for positive Fourier transforms
I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words
$$
\int_0^\infty f(x)\cos(x\omega) \, ...
- 178
5
votes
3
answers
2k
views
Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
- 23k
1
vote
1
answer
439
views
Well-known conditions for the Fourier inversion formula
Let $f\in L^1(\mathbb{R})$.
One may easily check that
$$(*)~~~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 ~\...
CommunityBot
- 1
1
vote
2
answers
2k
views
Fourier transform of a holomorphic function
Question: Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?
In order for my question to be well-...
- 169
2
votes
2
answers
1k
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 $\...
- 903
0
votes
1
answer
226
views
Transformation of Fourier Transform
Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...
- 16.2k
3
votes
1
answer
305
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.
...
- 38.6k
2
votes
0
answers
172
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 $n$ ...
- 6,367
4
votes
1
answer
285
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
$$
f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},
$$
...
- 3,403
0
votes
1
answer
344
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,
$...
- 433
2
votes
1
answer
668
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 ...
- 595
1
vote
0
answers
140
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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 ...
- 435
1
vote
1
answer
390
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, ...
- 71
22
votes
2
answers
2k
views
When are Fourier coefficients monotonic?
Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by
$$
\hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
- 6,035
0
votes
1
answer
294
views
Joint distribution of random Fourier coefficients
Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_{...
- 128k
1
vote
0
answers
151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
2
votes
2
answers
333
views
Estimate for a simple oscillatory integral
If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that
$$
\Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\...
- 870
3
votes
2
answers
590
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
...
- 17.2k
1
vote
1
answer
474
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 ...
- 111
1
vote
0
answers
119
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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 ...
- 4,058
2
votes
2
answers
251
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 (\...
- 39.1k
2
votes
0
answers
105
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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 ...
- 4,058
0
votes
1
answer
88
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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 ...
- 17.2k
0
votes
0
answers
129
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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 $...
- 2,306
26
votes
2
answers
3k
views
Image of L^1 under the Fourier Transform
The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
- 846
5
votes
3
answers
2k
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Extension of Poisson Summation formula
Under the condition f continuous, integrable and:
$|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0)
we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet ...
- 113
2
votes
0
answers
127
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 ...
- 489
1
vote
1
answer
487
views
Fourier Transform of an even function
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$, where $f$ is an even continuous function.
Denote
$$
F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\...
- 34.7k
24
votes
3
answers
1k
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Is there a 'certainty' principle?
Heisenberg's uncertainty principle is a restriction on which probability distributions can describe the position and momentum of a quantum particle.
In mathematical terms it says that if $\psi\in L^2$ ...
- 2,369
7
votes
1
answer
1k
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Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 107
3
votes
1
answer
304
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Existence of probability measure on the circle with given Fourier coefficients
We say that a Hermitian symmetric (i.e., $f_{-n} = f_n^*$ for any $n \in \mathbb{Z})$ sequence $(f_n)_{n\in \mathbb{Z}}$ is positive-definite if, for any $N \geq 0$ and any $z_0 , \ldots, z_N \in \...
- 63.9k
12
votes
1
answer
562
views
Fast convolution of sparse functions
Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
CommunityBot
- 1
3
votes
1
answer
763
views
2D Fourier transform of log function
I am studying the paper found here. Halfway in the paper (Equation 6), the inverse 2D Fourier transform of $1/(k_x^2+k_y^2)$ needs to be determined. Is is stated that this is straightforward, and that ...
- 6,035
3
votes
0
answers
309
views
The $2\pi$ factor in the Fourier transform and dimensional analysis
I have been thinking about the $2\pi$ factor in the various conventions of the Fourier transform. For example, I was looking for a way to justify the following:
$(*)$ If we define $\hat f(\xi) = \int ...
- 613
11
votes
0
answers
707
views
What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?
The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions:
$\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
- 111
1
vote
0
answers
353
views
Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
vote
0
answers
107
views
Comparison of two Fourier transforms
I am looking for $\delta>0$, such that
$$
\delta \int_{-\infty}^{\infty} \exp(its)
{ \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\
\int_{-\infty}^{\infty} \exp(its)
{ \Gamma (it+1)\over \...
- 93