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Boundedness of Riemann-like sums on unbounded interval

Hi I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that: \begin{equation} \sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+...
Francesco Mina's user avatar
0 votes
0 answers
92 views

Lower bound for double sums with power law decay terms.

This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion. The motivation to ask here if the inequality below ...
Leandro's user avatar
  • 2,044
0 votes
0 answers
345 views

Jacobian of the inversion map

Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\...
Diego Sulca's user avatar
0 votes
0 answers
700 views

Sigma algebra generated

Let $\mathcal{L} \subset \mathbb{R}$ the Lebesgue sigma algebra and $\mathcal{B} \subset \mathbb{R}^{n}$ the Borel sigma algebra. I'll denotes by $\mathcal{L} \times \mathcal{B}$ the smallest sigma ...
Santos's user avatar
  • 11
0 votes
1 answer
659 views

Under what condition will this set contain a limit point of [0,1)?

Let $T_1,T_2,....T_n$ be numbers such that $T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. ...
nb1's user avatar
  • 230
0 votes
0 answers
165 views

minimizing the integral of a function over square sets.

Hi! I'm interested in some problems, but to be honest i'm not sure of the field they belong to. Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
kaleidoscop's user avatar
  • 1,352
0 votes
1 answer
122 views

Existence of an eigenpair for d-bar operator in the unit disck

Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem: $$ \overline{\...
Ali's user avatar
  • 4,135
-1 votes
1 answer
354 views

The grail of functional analysis?

Does $g\in C([0,1],[0,1])=A$ exist such that $\{g^n ,n\in\mathbb N\}$ is dense in $A$ provided with the uniform norm? with $g^2=g \circ g $ If we can find $g$ then $F$ a closed of $A$, $id \in F$ ...
Dattier's user avatar
  • 4,074
-1 votes
1 answer
369 views

Would this go to 0 [closed]

Let $t_{m}$ be the sup of the sum of the pairwise distances between any $2m$ points in the unit disk. Does $t_{m}/m^{2}$ go to $0$ as $m\rightarrow\infty$?
u51245's user avatar
  • 1
-1 votes
3 answers
589 views

on compact support distributions [closed]

If $f$ a distribution with compact support then they exist $m$ and measures $f_\beta$,$|\beta|\leq m$ such that $$f=\sum_{|\beta|\leq m}\frac{\partial^\beta f_\beta}{\partial x^\beta}$$ how to ...
deval sidi's user avatar
-1 votes
1 answer
113 views

Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds? $$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
Lao's user avatar
  • 217
-1 votes
1 answer
227 views

Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]

We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows. Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
Nilotpal Kanti Sinha'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
-1 votes
1 answer
152 views

Topological characterization of invertible real matrices [closed]

Let $n\geq 2$ be an integer. Consider the topological space $M_n$ of $n$-by-$n$ matrices with real entries. Can you give a short non-constructive proof of the existence of a continuous function $M_n\...
orname's user avatar
  • 23
-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
-1 votes
1 answer
223 views

Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball

Hi I just happened to have a small question. If we have $$\frac{\int_\Omega x}{|\Omega|}=\frac{\int_{\partial\Omega} x}{|\partial\Omega|}$$ for a simply connected set $\Omega$ with analytic boundary. ...
Holden Lyu's user avatar
-1 votes
1 answer
110 views

Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$

Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying $$ 0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2 $$ where $\Delta$ denotes the Laplace ...
J. Swail's user avatar
  • 437
-1 votes
1 answer
96 views

Limiting points of elementary set

I consider the following set $$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$ Is it possible to identify the closure of $A$ in the reals?
Pritam Bemis's user avatar
-1 votes
1 answer
1k views

A question about pointwise convergence of Fourier transform in $N$-dimensions

I am retreating back on this statement, after some explorations and calculation Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
Rajesh D's user avatar
  • 698
-1 votes
1 answer
4k views

Lipschitz condition on the first derivative of a function? [closed]

If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?
Hafiz ul Asad's user avatar
-1 votes
1 answer
167 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 349
-1 votes
1 answer
80 views

Regions when a concave function is smaller than another concave function

Let $f_1,f_2:[0,1]\mapsto\mathbb{R}$ be two bounded and concave functions. Assume $f_1(0)<f_2(0)$ and $f_1(1)<f_2(1)$. I want to investigate the set $\mathcal{X}\triangleq\{x\in[0,1]: f_1(x)>...
Eggplant's user avatar
-1 votes
1 answer
102 views

Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]

Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
Hiro's user avatar
  • 131
-1 votes
2 answers
129 views

Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]

Under which condition can it form a Hilbert space? Or what space can it form? You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
Nan Zhang's user avatar
-1 votes
1 answer
120 views

Existence of sequence of measurable sets with prescribed densities

Consider Lebesgue measure $m$ on $[0, 1]$. Fix a countable sequence $a_i, 0 < a_i < 1$ such that $\sum_i a_i = 1$. Is there a sequence of disjoint measurable subsets of $[0, 1]$, $E_i$ whose ...
James Baxter's user avatar
  • 2,069
-1 votes
1 answer
230 views

Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
Peng's user avatar
  • 31
-1 votes
1 answer
237 views

Theorem with an example [closed]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
Vrouvrou's user avatar
  • 277
-1 votes
1 answer
61 views

Asking for some references on correlations of joint optimization problems

Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
Aaradhya Pandey's user avatar
-1 votes
1 answer
204 views

Cauchy reduction formula with measure (a variation)

The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
Math2024's user avatar
  • 141
-1 votes
1 answer
189 views

f a continuous function satisfying $\sqrt{xy}(f(x) + f(y)) \leq 1 \; \forall x,y \in [0\; 1]$ Show that $\int_0^1 f(t) dt \leq \frac{\pi}{2} $ [closed]

Let $f :[0 \; 1] \rightarrow R $ be a continuous function satisfying $ \sqrt{xy}(f(x) + f(y)) \leq 1 \; \forall x,y \in [0\; 1]$ ....(1) Show that $\int_0^1 f(t) dt \leq \frac{\pi}{2} $ .... (...
Math XO's user avatar
-1 votes
1 answer
179 views

Real number which is different from all rationals [closed]

By diagonalization, it is possible to construct a real number $r \in [0,1]$ such that for every rational $q \in [0,1]$, there exists an index $i \in \mathbb{N}$ such that $r_i \neq q_i$ (where $x_i$ ...
Larry's user avatar
  • 1
-1 votes
1 answer
338 views

Capacity and harmonic measure

Suppose $D$ is a bounded domain of $\mathbb{R}^{n}$ with $n>1$ and $E$ a subset of its boundary. We know that if $E$ has capacity zero I.e. it is a polar set , then the harmonic measure of $E$ ...
M. Rahmat's user avatar
  • 411
-1 votes
1 answer
70 views

Is this kind of interpolation correct?

Let $f=\sum f_j$ be a finite sum. Assume that $$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$ $$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$ Then can we conclude that for $2<p<\infty$ $$\|f\|_p\le C^{1-\...
xsbb2001's user avatar
-1 votes
1 answer
149 views

How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]

It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$ Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
Math Learner 's user avatar
-1 votes
1 answer
208 views

Does this function belong to $L^2(\mathbb{D})$?

Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question. The unit ...
Ali Taghavi's user avatar
-1 votes
1 answer
196 views

Closed form for sum involving digamma? [closed]

Let $\Gamma(n)$ be Euler's Gamma function and $\psi_0$ = $\frac{\Gamma'(n)}{\Gamma(n)}$ be the Digamma function. Is there a closed form for $$\sum_{n=1}^{\infty} \frac{\psi_0(n)}{n^2}=?$$ I've done ...
user99466's user avatar
-1 votes
1 answer
505 views

Derivative of smooth function change sign infinitely on [0,1]? [closed]

Can the derivative $f^\prime$ of a smooth function $f\in C^\infty[0,1]$ change sign infinitely many times (or $f$ have infinitely many isolated critical points)? If yes, how about an analytic function ...
Asdf's user avatar
  • 113
-1 votes
1 answer
346 views

An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
robert caro's user avatar
-1 votes
1 answer
2k views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
user avatar
-1 votes
1 answer
222 views

Does the divergence of the sum of reciprocals of a set of integers imply this density statement about the set?

Suppose $A \subseteq \mathbb{N}$ is such that $\displaystyle{\sum_{n \in A} n^{-1}} = \infty$. Suppose $B \subseteq \mathbb{N}$ is infinite. Is there a set $X \subseteq [1,\infty)$ and a increasing ...
Jason Sawyer's user avatar
-1 votes
1 answer
1k views

derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has $$f(x)=\sum_{n\...
joaopa's user avatar
  • 3,996
-1 votes
2 answers
418 views

An inequality involving multi-index [closed]

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For $x \in \mathbb{R}^{n}$ and $\alpha = (\alpha_{...
Vishal Gupta's user avatar
-1 votes
1 answer
187 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} f(x)=...
Sue's user avatar
  • 25
-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
-1 votes
1 answer
213 views

Building a smooth function from a rapidly decreasing sequence

Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function? More precisely: Let $\lbrace c_k\rbrace _{k \...
Peg Leg Jonathan's user avatar
-1 votes
1 answer
142 views

A pathological (?) function involving powers

This is inspired by a recent math.SE question. Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide ...
Wolfgang's user avatar
  • 13.4k
-1 votes
1 answer
168 views

Searching the roots of a self-consistent transcendental equation

I have the equation $$M = c_1 + c_2M - c_3T\ln\left(\left|\frac{e^{(c_4M + c_5)/T}-1}{e^{(c_6M + c_5)/T}-1}\right|\right)$$ where $c_1, \dots, c_6$ are constants. I am interested in the roots of $$M\...
Essa Ibrahim's user avatar
-1 votes
1 answer
155 views

Is this recurrent sequence decreasing?

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
Shivin Srivastava's user avatar
-1 votes
1 answer
80 views

Minimal covering sets of continuous endomorphisms

For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ covers $\text{End}(X)$ if for every $f\in \...
Dominic van der Zypen's user avatar
-1 votes
1 answer
103 views

Sum of $\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}$ [closed]

I want to calculute or estimate of order $O(n^{2-\varepsilon})$, where $\varepsilon>0$, of the following sum for $0<\alpha<1$ $$\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}.$$
yassine yassine's user avatar