Questions tagged [fourier-analysis]
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
1,159
questions
2
votes
0answers
69 views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
2
votes
1answer
75 views
Estimate of Hölder Norms (Littlewood–Paley theory)
I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem:
Recall that ...
3
votes
2answers
154 views
Vanishing convolution between density and compactly supported function
Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:
$f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
$g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
11
votes
1answer
285 views
Does there exist an upper bound on the Fourier coefficients of the reciprocal theta function $\frac {1}{\theta}$?
Define the theta function as
$$
\theta(x) = \sum_{n=-\infty}^\infty e^{-\gamma(x+n)^2}
$$
where $\gamma>0$. Clearly, $\theta$ is 1-periodic, non-zero and smooth. Therefore, the reciprocal map $x \...
0
votes
0answers
48 views
Inverse of a Function Using its Fourier Series
Let ${f}:{\mathbb{{{R}}}}\to{\mathbb{{{R}}}}$ be a real, isomorphic function defined on the interval ${\left[{0},{l}\right]}$ yielding a meromorphic analytic continuation to the complex plane ${\...
4
votes
1answer
123 views
Scaling of double convolution
I am interested in the scaling of
$$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$
In particular, I suspect that
$$F(...
3
votes
1answer
117 views
A convolution type singular integral operator with log
Define a convolution type operator $T_m$ by
$$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer.
Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
3
votes
2answers
175 views
What is the distribution of the following limit?
Assume $x \in \mathbb{R}$. We already know that
$$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$
Here $\delta_x$ denotes the Dirac distribution. If we ...
2
votes
2answers
175 views
Decay estimate of Fourier transform of a compactly supported function
Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate
$$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$
for some $\...
2
votes
0answers
59 views
Half-integer Fourier transform
Suppose we want to carry out the (generalized) Fourier transform of a function defined in the domain $\mathbb{T}\times\tfrac12\mathbb{Z}$, i.e. dependent on the arguments $\phi\in\left(-\pi,+\pi\right]...
0
votes
0answers
33 views
Hamiltonian of Benjamin-Ono equation
Could someone please tell me why the Hamiltonian of the Benjamin-Ono equation
$$\partial_t u=H \partial_x ^2 u -\partial_x(u^2),$$
is given, on the Torus $\mathbb R / 2\pi \mathbb Z$ , by
$$\mathscr{H}...
5
votes
1answer
312 views
Schwartz regularity for the density of a stochastic process
Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables
$$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$
It ...
5
votes
1answer
213 views
Generalization of Bernstein’s inequality
I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim:
Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...
2
votes
1answer
68 views
Uniform convergence of Eigenfunction decomposition on Riemannian sphere?
Let $\{u_k\}_{k=1}^\infty$ be a sequence of ($L^2$ normalized) mutually orthogonal eigenfunctions of the operator $-\Delta$ on the sphere $\mathbb{S}^n$ (here $\Delta$ is the Laplace Beltrami operator)...
2
votes
0answers
94 views
$L^p$ estimate of a multiplier operator
I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies:
$$\sum_{n \in \...
6
votes
1answer
84 views
Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?
Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm
$$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$
...
0
votes
0answers
38 views
Can I write this series in a recursive way?
I would like to know given the following definition of the function X(n) if it is possible to express two consecutive values of X(n) (example: X(1) and X(2) ) to obtain a recursive expression.
What I'...
3
votes
2answers
263 views
Fourier transform of eigenvalue distribution of GUE matrices
I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
6
votes
0answers
86 views
Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
2
votes
1answer
88 views
Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces
I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise:
For any function $f \in L^{1}(\mathbb{R}^{d})$ ...
2
votes
0answers
68 views
Definition of a continuous Gabor frame
I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\...
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.
...
3
votes
1answer
159 views
“Reversed” Bernstein Inequality
I'm studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below:
...
1
vote
0answers
70 views
Decay conditions on the Fourier coefficients ensuring that a smooth function doesn't vanish on some interval
Let $f$ be a non-zero smooth $1$-periodic function. Is there a decay condition on its Fourier coefficients ensuring that it won't vanish identically on any interval?
I mean a condition weaker than &...
-1
votes
1answer
646 views
What does $O(N)$ mean in this article and how does it imply this lemma?
In this article the author proves the following lemma:
LEMMA: $\forall N \in \Bbb N$, there exists $v=v_N$ with compact support so that
$$[M_S(M_S v)^\delta(x)]^{1/\delta} \geq c_\delta NM_S v(x), \...
1
vote
1answer
76 views
Integrability of fractional heat kernel
In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
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
$$
f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},
$$
...
1
vote
0answers
63 views
When can one bound the Hilbert transform on the torus in $L^1$?
If $f$ is a function on $\mathbb T$ then the conjugate $\tilde f$ of $f$ satisfies Zygmund's bound
$$\int |\tilde f|\leq A \int |f|\log(e+|f|)+B.$$
I am curious if there are any conditions where one ...
4
votes
1answer
178 views
Smallest regular $m$-gon covering a regular $n$-gon
I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.
Let us fix a regular $n$-gon with area $1$. What is the smallest ...
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
131 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 ...
6
votes
1answer
251 views
Wiener Corollary in “An introduction to harmonic analysis” by Yitzhak Katznelson
I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows:
Corollary. Let $\mu\in M(\mathbb T)$. Then
$$\sum\limits_{\tau\in\...
0
votes
0answers
51 views
Context for this discrete Cauchy integral formula
Notation: I will use the following conventions for discrete Fourier transforms (DFT) and discrete time Fourier transforms (DTFT):
$$\mathcal{D}_N[x_j](k) := \sum_{j=0}^{N-1} e^{-2\pi i j k} x_j$$
$$\...
1
vote
0answers
33 views
Function of several variables whose hessian is a Hankel matrix
First of all, let me apologize because I asked this question a few days ago on https://math.stackexchange.com, but I did not get any reply.
I am studying a function $f:\mathbb{R}^n\rightarrow\mathbb{R}...
1
vote
1answer
92 views
How many Fourier coefficients of a sparse signal $f=\sum_{n=1}^Nc_n\delta_{t_n}$ are needed to determine $f$ uniquely?
Let $N \in \mathbb N$ and $c_n \in \mathbb C$, $t_n \in \mathbb R$ for $n=1, \dots, N$. Suppose that $f$ is a linear combination of dirac-deltas with locations $t_n$ and coefficients $c_n$, i.e.
$$
f=\...
0
votes
0answers
51 views
Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
7
votes
1answer
291 views
What makes Gaussian distributions special? Local field version?
This question is inspired by the recent one about Gaussian measures over the reals:
What makes Gaussian distributions special?
I would be interested in a similar list of characterizations for the ...
5
votes
0answers
48 views
Does every compact abelian group contain a Kronecker set generating a dense subgroup?
Let $G$ be a compact metrizable abelian group with infinite exponent.
Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
7
votes
0answers
66 views
Smoothing property of a certain singular integral operator of non-convolution type
For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by
$$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
1
vote
0answers
57 views
Factoring a complex function such that it is analytic in upper and lower plane
Consider this function $$\frac{k^{2}-\xi^{2}}{k^{2}+1}$$
which has singularities at $k=\pm i$, the strips where it is analytic are
$$
-1<k^{\prime \prime}<0 \quad \text { or } \quad 0<k^{\...
0
votes
1answer
60 views
Can a Fourier transform be performed on irregularly sampled data with timestamps?
Normally, when I think of performing a Fourier transform, I imagine that my samples are spaced regularly in time (or space).
If I have a set of samples that are spaced irregularly, but have accurate ...
1
vote
0answers
58 views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
1
vote
0answers
23 views
Deriving periodical processes from a finite time series
Suppose we have a finite time series of real-world events measured at $(t_k), k \in \mathbb{N}$ with $(t_{k-1} < t_k)$. The content of the actual events is irrelevant.
I would like an automated ...
1
vote
0answers
48 views
Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
1
vote
0answers
101 views
Integration by parts formula for the spectral fractional Laplacian
Let $f,g:C^\infty_c(0,1)$. Is there a formula similar to
$$ \int_0^1 (f_{xx})^2g \ dx = \int_0^1 \frac{1}{2} (f_x)^2 g_{xx} \ dx - \int_0^1 f_{xxx}f_x g \ dx $$
for the spectral fractional Laplacian ...
3
votes
1answer
145 views
Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$
Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
2
votes
0answers
147 views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
1
vote
1answer
266 views
Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers
When counting the number of integers $n(x)$ below a certain non-integer number $x$, the following series could be used:
$$n(x) = x-\frac12 + \sum_{n=1}^{\infty} \left(\frac{e^{x \mu_n}} {\mu_n}+\frac{...
