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0 votes
1 answer
197 views

Analyze a complicated double summation

Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$F(t,f(x)):=\dfrac{1}{t}\Big(\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\...
4 votes
1 answer
1k views

Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
5 votes
2 answers
565 views

Geometry of Level sets of elliptic polynomials in two real variables

Updated: A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
-1 votes
1 answer
81 views

Closed on generic set implies closed set whole set [closed]

Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
1 vote
1 answer
608 views

Radius of convergence of multivariate Taylor series

Consider the function $f$ on $\mathbb{R}^{l}$ given by \begin{eqnarray}\left(x_{1},...,x_{l}\right)\mapsto\left(\sum_{i=1}^{l}\frac{1}{\left(1+x_{i}\right)^{k_{i}}}-\left(l-1\right)\right)^{-1} \end{...
0 votes
0 answers
91 views

Does $L^1$ convergence preserve the regularity of this sequence of functions?

Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges $...
3 votes
0 answers
238 views

Dominated convergence Theorem

I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with Generalized Spatiotemporal Gaussian Process Models. Theorem 2.1 in the page 33 uses ...
6 votes
0 answers
267 views

Convergence of $\sum_{n=1}^\infty x_n^k$

I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
1 vote
1 answer
151 views

Original examples of functions of slow increase in the spirit of Jakimczuk

I believe that it is possible to prove that $$f(x)=e^{\operatorname{Ai}(x)}\log x$$ is a function of slow increase in the spirit of the definition given by the author of [1], where $\operatorname{Ai}(...
4 votes
2 answers
145 views

Understanding equiprobable trinomial identity

With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
1 vote
0 answers
59 views

Uniform lower bound on a sequence of functions on $[0,1]$

Consider the sequence of functions $\{F_n(\cdot)\}_{n \in \mathbb{N}}$ on $[0,1]$, where for each $n$, $F_n(\cdot)$ is defined as \begin{equation} F_n(x) = \sum_{i=2}^n \Big( x^{b_n} \frac{i}{(i+1)^{...
1 vote
0 answers
51 views

Mean value of a function with binomial coefficients as weights

Is the following true? Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$ $$ \beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}...
0 votes
1 answer
126 views

Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$

For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that $$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$ converges?
2 votes
1 answer
230 views

Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?

Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
1 vote
1 answer
64 views

Effect of repeated subtraction of the average of average function values in coordinate directions

Questions: assuming $$a\lt b,\ c\lt d;\ \ (x,y)\in [a,b]\times[c,d];\ \ f_0: (x,y)\mapsto z\in\mathbb{R};\ \ |a|,\ |b|,\ |c|,\ |d|,\ |z|\lt\infty$$ $$0\quad\lt\quad\left|\int_a^b{f_0(x,y)dx}\right|,\ \...
1 vote
0 answers
51 views

Convergence acceleration of a series by using optimal parameters

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
2 votes
1 answer
184 views

Why is this series summable?

Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$. Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that $\lim \sup_{k \rightarrow \infty} \frac{...
8 votes
2 answers
559 views

How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
8 votes
1 answer
620 views

Is "weakly good" series in a finite-dimensional Banach space "good"?

Let us call a series $\sum_n x_n$ in a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges. Find a ...
-2 votes
1 answer
1k views

Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
2 votes
2 answers
152 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
0 votes
0 answers
299 views

When convolution with exponential kernel is bounded

Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying $...
1 vote
0 answers
314 views

Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?

In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given: The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
6 votes
1 answer
396 views

Well definition of a function

I've edited, just skip the first attempt and go to the second one. THE FRAMEWORK: let us consider a real topological vector space $V$. We denote with $\mathscr C_k(V)$ the set of all continous ...
2 votes
0 answers
1k views

Is there an infinite product like this for $\cos x$?

There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example $$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
5 votes
0 answers
313 views

Uniqueness of a SDE with non-negativity constraint

I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed): \begin{equation}\label{sde}%sde x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...
4 votes
0 answers
459 views

Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie? $$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$ Or at least a proof that it is an irrational number. The ...
2 votes
0 answers
147 views

Interchanging limit and infinite product in Euler product for Dedekind function s=1

For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation $$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
1 vote
1 answer
262 views

Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)

Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
3 votes
1 answer
174 views

How does the function g(x) behave as x tends to 1?

Suppose $g:[0,1)→R$ is a continuous function satisfying $g(x^2)=x−g(x)$ for every x on interval $[0,1)$. How does the function g(x) behave as x tends to 1?
1 vote
0 answers
71 views

Proving an Algorithm that generates minimal $\|x\|_0$ for the underdetermined system $Ax=b$

Let $A \in \mathbb {F}^{m \times n}$ with $m< n,$ $b \in \mathbb{F}^m$ and let $x$ be unknown in $\mathbb{F}^n.$ Assume $0<p<1.$ Then $$\arg \min\limits_{x: Ax=b} \|x\|_0 = \lim\limits_{p \to ...
3 votes
1 answer
167 views

Recovering residue using local real information

Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$. Compute the residue of $f(z)$ at z = 0 using just the ...
1 vote
1 answer
186 views

Almost binomial sum limit

I got the following sum with which I want to prove one limit fact: $$ f_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} (a^t)^{n-t} $$ I want to prove that $f_n(a) \to 1$ while $n \to \infty$ for $a\...
1 vote
1 answer
1k views

properties of orderd upper and lower semi continuous functions [closed]

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$. If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it ...
5 votes
1 answer
2k views

Baire's simple limit theorem "almost everywhere"

The Baire's simple limit theorem states that if the functions $f_n : \mathbb{R} \to \mathbb{R}$ are continuous and converge everywhere to a function $f$ then $f$ has a dense set of continuity points. ...
3 votes
1 answer
317 views

Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$...
3 votes
1 answer
105 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} \sqrt{...
3 votes
1 answer
367 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum_{...
4 votes
2 answers
4k views

Pointwise convergence for continuous functions

Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set $f(x)=\...
5 votes
1 answer
680 views

When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$ I don't know of any references or methods for this -- not even for $x=1$, for which the ...
1 vote
0 answers
150 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
4 votes
1 answer
262 views

Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field. Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials $P_n:...
5 votes
1 answer
229 views

Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible. Let $a_k >0$ be an increasing sequence ...
1 vote
1 answer
657 views

Local Uniform Convergence

Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as $x\...
2 votes
2 answers
253 views

finding the limit $\lim_{a\rightarrow \infty} \frac{a^N}{\log a} \int_{0}^\infty \frac{e^{-x}}{(1+ag(x))^N}dx = c$

I am realy stuck in solving the following limit problem. Can you find any function $g(x)$ by which $$\lim_{a\rightarrow \infty} \frac{a^N}{\log a} \int_{0}^\infty \frac{e^{-x}}{(1+ag(x))^N}dx = c$$ ...
-2 votes
1 answer
395 views

non-trivial convergent sequence [duplicate]

I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$) can you give me a example of ...
1 vote
2 answers
654 views

Limit with theorem of dominated convergence

Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$) I have to calculate this limit $$\lim_{|x-y|\to 0}\int_{\...
2 votes
1 answer
412 views

Convergence in norm of Sobolev spaces

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\lbrace{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\rbrace}$ and I have to show that the function $$f(x)=\...
1 vote
1 answer
393 views

On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$. By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...
1 vote
1 answer
496 views

Convergence of Difference Quotients

Let $\gamma_{\varepsilon} \rightharpoonup \gamma$ in $W^{1,\infty}(0,1)$. Then for any fixed $s \in \mathbb (0,1)$ does the limit $\lim_{\varepsilon \rightarrow 0} \frac{\gamma_{\varepsilon}(s\...