All Questions
Tagged with harmonic-analysis integration
29 questions
23
votes
1
answer
528
views
A characterization of constant functions
In How to recognize constant functions. Connections with Sobolev spaces (Russian Math Surveys 57 (2002); MSN), H. Brezis recalls the following fact:
Let $\Omega\subset{\mathbb R}^N$ be connected ...
- 52.3k
15
votes
1
answer
602
views
Integrability property of polynomials in several variables
This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
- 151
6
votes
1
answer
741
views
Is the following integral nonzero?
Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...
- 1,997
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
6
votes
1
answer
339
views
Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$
Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to ...
- 1,284
6
votes
1
answer
193
views
Oscillatory integrals with a decaying factor in the integrand
Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased):
Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
- 63
6
votes
1
answer
360
views
Stationary phase in spherical integral
I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that:
If $\lambda\gg 1$...
- 377
5
votes
3
answers
718
views
Speed of convergence for Weyl's Equidistribution theorem
If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...
- 65
5
votes
1
answer
163
views
A question about integration of spherical harmonics on $(S ^ 2, can)$
Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that
$$ \int_{\mathbb{...
- 245
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
5
votes
0
answers
243
views
Is there a way to solve this integral on the sphere explicitly?
Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that
$k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral
$$f(y):=\int_{\...
- 852
4
votes
2
answers
310
views
interpretation of a singular integral
There is a post on MSE about a principal value integral in this paper. It has not received much attention even with a bounty, and since it concerns a published paper, I believe this is a better forum ...
- 323
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
3
votes
2
answers
353
views
Calculation of Riesz energy for balls
I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
- 87
3
votes
1
answer
480
views
To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$
I wants to understand the integrals of the form
$$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$
where $\mu(\alpha)$ is a non decreasing function such ...
- 1,051
3
votes
1
answer
93
views
Does $p$ integrability in n-1 dimensions give higher integrability in $n$ dimensions?
Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that
$$
\int\limits_t\left(\int\limits_x|...
- 391
3
votes
0
answers
168
views
Efficient integration over part of a compact group
I am trying to find the matrix coefficients $\{\hat{c}^{\alpha}_{i j}\}$ that minimize the mean squared error against a function $f(g)$ over a compact group to some bandwidth cutoff $\ell$.
$$\...
- 121
2
votes
2
answers
399
views
Asymptotic decay rate of an oscillatory integral
Consider the following oscillatory integral
$$
I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $...
- 2,712
1
vote
1
answer
247
views
Convergence of a sum to the integral
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
$$\frac{1}{n}\sum_{\ell=0}^{n-1}f(a+b\...
- 65
1
vote
1
answer
135
views
Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$
Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$
and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
- 123
1
vote
1
answer
120
views
estimate involving Gaussian data
Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...
- 131
1
vote
2
answers
152
views
Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 3,625
1
vote
2
answers
687
views
High dimensional beta integral (a typo in Stein's book "singular integrals")
Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} |y|^{-n+\beta}=\frac{\gamma(\alpha)\gamma(\beta)}{\gamma(\alpha+\beta)},...
- 1,649
1
vote
1
answer
222
views
Multidimensional improper Riemann integrals with oscillatory kernels: Existence
I have asked this question three weeks ago here
https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930
but received no relevant answers.
Let $n\geq 2$ ...
- 143
1
vote
2
answers
245
views
Numerical calculation of Fourier transform with a nice error bound
I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and $...
- 195
1
vote
0
answers
338
views
Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 21
0
votes
1
answer
193
views
$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$
Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
0
votes
0
answers
47
views
Oscillatory integral independent of a parameter
Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral
$$Q(t) :=...
- 222
-1
votes
1
answer
128
views
Riesz energy for open sets in dimension $1$
This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by
$$I_s(B(...
- 87