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Fourier decay implies what kind of regularity

We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that $$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$ ...
Yizheng Yuan's user avatar
0 votes
1 answer
115 views

Fourier transform of exponential over torus

I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
António Borges Santos's user avatar
1 vote
0 answers
100 views

Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$

$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$. I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
Haidara's user avatar
  • 178
7 votes
1 answer
554 views

Example of continuous function which is not differentiable everywhere in a strong sense

Is there a continuous function $$u\colon (0,1)\to \mathbb{R}$$ such that at every point $x\in (0,1)$ one has $$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$ In particular $u$ is not ...
asv's user avatar
  • 21.8k
0 votes
1 answer
71 views

Upper bound on higher order derivatives of $\frac{1}{v(t)}$

Suppose that $ v(t) >l>0$ and $$ \vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}. $$ Can we give an upper bound for $$ (\frac{1}{v(t)})^{(k)} $$ ? Attempt: We first compute the first fourth order ...
Yidong Luo's user avatar
5 votes
0 answers
204 views

A proof for an $L^p$-$L^p$ inequality

This is a transfer of the question https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
Medo's user avatar
  • 852
3 votes
0 answers
95 views

Deeper reason for why classical orthogonal polynomials have simple generating functions?

Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
Plemath's user avatar
  • 312
0 votes
2 answers
148 views

Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines

I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$: $$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$ where $H_m(x)$ is the $m-$th Hermite polynomial....
Darius's user avatar
  • 21
1 vote
1 answer
188 views

Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?

This question is related to This question. When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
Haidara's user avatar
  • 178
0 votes
2 answers
364 views

Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
Haidara's user avatar
  • 178
2 votes
1 answer
93 views

Reference needed: estimate of the second order derivatives

In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions) $$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
Michael Perelmuter's user avatar
6 votes
1 answer
568 views

Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
Haidara's user avatar
  • 178
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
3 votes
1 answer
309 views

Extremizing sequence consists of two elements

Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
António Borges Santos's user avatar
3 votes
1 answer
240 views

Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $

Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and $$ [f]_{\frac{2}{\...
Luis Yanka Annalisc's user avatar
-1 votes
1 answer
122 views

Divergent summation [closed]

Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series: $$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
Paul Deerock's user avatar
7 votes
1 answer
179 views

More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let $$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
Iosif Pinelis's user avatar
6 votes
2 answers
492 views

Does this polynomial have a real zero less than or equal to $1/2$?

Is the smallest root $x$ of $$ 10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\ +2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
user avatar
9 votes
3 answers
2k views

Smallest root of a degree 3 polynomial

Is it true that the smallest root $t$ of the polynomial $$ 20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
Venus's user avatar
  • 171
2 votes
1 answer
120 views

Difference between finite partial sums from two divergent series

Fix a sequence $(r_i)_{i\in\mathbb{N}} \subseteq (0, 1)$ such that $\lim_i r_i=0$ and $\sum_{i\in \mathbb{N}} r_i=\infty$. According to the answer in this post, for any $c>0$ there exists $N,M\in\...
Sanae Kochiya's user avatar
8 votes
1 answer
449 views

What do smooth signatures give you?

My background is in rough paths theory. In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are ...
user479223's user avatar
  • 1,904
4 votes
1 answer
551 views

Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?

Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, and $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ be the Hausdorff measure in its ...
Arbuja's user avatar
  • 63
2 votes
0 answers
67 views

'Sublinear' and 'superlinear' moduli of continuity

Recall, given a metric space $X$, a function $f:X \rightarrow \mathbb{R}$ has (uniform) modulus of continuity $w:[0,\infty) \rightarrow [0,\infty]$ if $|f(x) - f(y)| < w(|x-y|)$ for all $x,y \in X$....
algebroo's user avatar
  • 135
21 votes
2 answers
2k views

Boundedness of sum of sin(sin(n))

Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$ is bounded. However, I did not succeed in proving this ...
Oleksandr Liubimov's user avatar
0 votes
0 answers
54 views

Inequality between inverses of real functions

Let $s\geq 0$ and $$ f(x)=-\log(x) \quad\text{an}\quad g(x)= \log(\log(1/x)+1)$$ for all $x\in(0,1)$. Is there exists $C_s>0$ such that for all $x,y\in(0,1)$, $$ f^{-1}(s g(x)) \cdot f^{-1}(s g(y))...
B-S's user avatar
  • 39
1 vote
0 answers
100 views

Difference of two completely monotonic functions

We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \...
George Stepaniants's user avatar
2 votes
1 answer
118 views

Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius

This is a follow up from this question. I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
Ryan Hendricks's user avatar
1 vote
1 answer
76 views

Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius

I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
Ryan Hendricks's user avatar
0 votes
1 answer
255 views

Carleson's theorem: proof of a lemma

I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
Alexander's user avatar
4 votes
1 answer
255 views

Asymptotic behavior and of an integral on a d-dimensional torus

I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$: $$ I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...
Ko Hey's user avatar
  • 81
4 votes
1 answer
217 views

$2$ continuous, commuting functions doesn't always have a common fixed point

The question is as such: If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point. -- Zorich ...
Yinuo An's user avatar
  • 183
2 votes
0 answers
97 views

On the second order analog of the upper 1-Lipschitz envelope of a function

Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope $$ \hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
Castoro Moro's user avatar
2 votes
2 answers
192 views

Upper/Lower bounds of real-analytic functions with infinite Taylor series

For example, in 1-D, given some positive increasing polynomial $p(x) = a_1x+\ldots+a_nx^n$, $p(0) = 0$, there exists constants $b_1,b_2$ such that for $x<\delta$, for some $\delta > 0$, we have ...
Doofenshmert's user avatar
0 votes
0 answers
128 views

Lipschitz function approximated by smooth functions with zero a regular value

Consider a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$. Then I want a family of smooth functions $f_\epsilon : \mathbb{R}^n\to\mathbb{R}$, such that $f_\epsilon\to f$ uniformly on compact sets, ...
shadow10's user avatar
  • 1,090
0 votes
0 answers
48 views

First nonzero derivative bounded below (2 dimensions)

Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real-analytic, have only one zero (at $(0,0)$) and be strictly increasing ...
Doofenshmert's user avatar
2 votes
2 answers
173 views

Gronwall-type inequality involving norms of distinct Lebesgue spaces

Let $d \geq 1$, $\Omega \subset \mathbb{R^d}$ be a bounded domain and let $\phi : [0,T]\times \Omega \mapsto \mathbb{R}$ be a measurable and bounded function. Assume that the following differential ...
Theleb's user avatar
  • 213
0 votes
0 answers
29 views

$ \sup_{\theta \in [0,2\pi)}\max_{r\leq \delta}\frac{\log\left(\frac{f(r,\theta)}{f(\delta,\theta)}\right)}{\log(r)}<\infty,$ $f$ real analytic

$\textbf{Conjecture.}$ Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $\delta <1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and suppose that $(0,0)$ is the only ...
Doofenshmert's user avatar
0 votes
0 answers
71 views

Minimum Slice of Real Analytic Function in Two Variables

Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and have only one zero, namely $(0,0)$. Moreover, assume that $...
Doofenshmert's user avatar
3 votes
1 answer
233 views

Analytic solutions to analytic differential equations

Let $U \subseteq \mathbb R^{n+2}$ be an open set for some $n \geq 0$, and let $f: U \to \mathbb R$ be an analytic function. Then we say the equation $f(x,y,y',\ldots,y^{(n)})=0$ is an analytic ...
cubicquartic's user avatar
1 vote
0 answers
67 views

Distribution of zeros for arbitrary Bessel functions

Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
Literally an Orange's user avatar
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
1 vote
1 answer
111 views

How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
207 views

Seeking alternative elementary proof instead of applying Lojaseiwicz's inequality for $f(x,y) \geq c (x^2+y^2)^{\frac{M}{2}}$

Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $0<\delta<1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and contain only one zero in $A$, namely $(0,0)$. In other ...
Doofenshmert's user avatar
7 votes
1 answer
185 views

Question on ODE involving mollifiers from Taylor's book on PDEs

In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form $$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$ with some initial condition $u(...
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
135 views

Estimating an integral of the Green function in the plane

Suppose $\Omega$ is a bounded, simply connected domain, $z_{0}\in{\Omega}$ and for any $z\in{\Omega}$, $d_{z}:=\text{dist}(z,\partial{\Omega})$. I am interested in understanding the behavior of ...
David Pechersky's user avatar
2 votes
0 answers
188 views

A sharp version of a Tauberian theorem

The following Tauberian theorem is true (see Theorem I.11.1 of ''Tauberian theory: A century of developments''). Let $ a_n $ a sequence of real numbers. If $f(x) = \sum_{n=1}^\infty a_n x^n $ ...
an_ordinary_mathematician's user avatar
3 votes
1 answer
346 views

Prove that $\lim\limits_{n\to\infty}\left(\sum\limits_{r=0}^{n-1}\sqrt{1-\frac{r^2}{n^2}}-\frac{\pi}{4}n\right)=\frac{1}{2}$

I came across the above question in a mathematical problem. It is not difficult to see that $$ \lim\limits_{n\to\infty}\left(\frac{1}{n}\sum\limits_{r=0}^{n-1}\sqrt{1-\frac{r^2}{n^2}}\right)=\int\...
Xinyu Wang's user avatar
-1 votes
1 answer
139 views

$L^1$ convergence

Setting For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
Anthony's user avatar
  • 125
0 votes
0 answers
57 views

Projection measure and an integral formula for Lipschitz functions

Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as $$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
Alexander's user avatar
-1 votes
1 answer
113 views

Lipschitz function which is surjective on subset implies that the subset is dense

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a Lipschitz-function. Suppose $A \subseteq \mathbb{R}^n$ is an $(n-1)$-connected subset such that $f(A) = \mathbb{R}^n$. I would like to show that $A\subseteq ...
psl2Z's user avatar
  • 261

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