Questions tagged [limits-and-convergence]
Convergence of series, sequences and functions and different modes of convergence.
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Generalized limits
Cross-posted from Math SE.
The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:
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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
$...
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Convergence in the narrow topology of measures and strongly converge for signed measures
We say that a sequence $(\mu_{n})$ of measures in $M_{b}(Q)$ converges tightly (or, equivalently, in the narrow topology of measures) to a measure $\mu$ in $M_{b}(Q)$ if
$$\lim_{n\to\infty}\int_{Q}\...
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Counterexample to uniform convergence of Laplace series (expansion in spherical harmonics)
Expansion of a real function on 2-sphere in spherical harmonics, so-called Laplace series, converges uniformly for continuously differentiable functions (see e.g. https://projecteuclid.org/euclid.bbms/...
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Convergence of the series of Legendre polynomials
Consider the generating function of Legendre polynomials:
$$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}_{n=0} P_n(x)t^n$$
Is it true that for $0<x<1, t=1$ series of Legendre ...
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Dominated convergence Theorem
I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...
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A problem inspired in the definition of tau numbers and a divisibility relationship related to powers of two
It is (I assume that this easy fact is well-known) obvious that an integer $n>1$ is a power of two $n=2^{\alpha}$, where $\alpha\geq 1$ is integer, if an only if $n$ satisfies the divisibility ...
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Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
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Original examples of functions of slow increase in the spirit of Jakimczuk
I believe that it is possible to prove that $$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}(...
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The growth of a sequence related to Liouville numbers [closed]
I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends ...
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Question on operator topologies convergence
Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of bounded operators on $H$. It is known that the strong operator topology and the norm topology on $\mathcal{B}(H)$ ...
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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/...
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interchanging limits and summation
So I am stuck at this situation . Suppose $m:B_2(H_1)\times B_2(H_2)\to \mathbb C$ be bilinear form given by $m(S,T)=\left<T,\phi(S)\right>$, where $\phi: B_2(H_1)\to B_2(H_2)$ be a bounded ...
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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)^{...
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Ideal characterization of almost convergence
$\bullet$ A real sequence $x=(x_n)_n$ is called convergent to $\alpha$ in usual sense if for any $\epsilon>0$ the set $\{n\in\mathbb N:|x_n-\alpha|\geq\epsilon\}$ is finite.
$\bullet$ A real ...
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The mysterious numbers $ \frac{13}{20} $ and $20$?
Let $g(x) = x^6 - 30 x $
Let $h(x) = x^6 $
Let $f(x) = x^2 - 2 $
Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$
Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
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Convergence in Product Formula for Tamagawa Number
Let $G = \operatorname{SL}_n$, and recall that its Tamagawa number is $\tau(G) = 1$ and is given by the product expansion
$$\tau(G) = \operatorname{Vol}(G(\mathbb{Z})\backslash G(\mathbb{R})) \cdot \...
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A question involving a summation of eigenvalues of the Laplacian operator on $\mathbb{S}^2$
Infinite series involving eigenvalues of the Beltrami-Laplace operator on Riemannian manifolds as well as $L^p$-estimates of eigenfunctions arise in the study of the nonlinear Schrödinger equation (...
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From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes
In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...
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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 $t_n$
$$
\beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}...
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Logarithmic vs sequential density of a sequence
Given a sequence of complex numbers $\{a_n\}_n$, one says that this sequence admits $a$ as a sequential density if
$$\underset{N_s\to\infty}{\lim}\frac{1}{N_s}\sum_{n=1}^{N_s} a_n = a$$
where $N_s = ...
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Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$
For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that
$$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?
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Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?
Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
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Convergence of gPC expansions for random variables in the total variation distance
Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
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Effect of repeated subtraction of the average of average function values in coordinate directions
Questions:
assuming
$$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|,\ \...
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Prove an existing formula for a limit of a specific sum
Prove that$$\lim_{n\to\infty}\frac1n\sum_{i_1,i_2,...i_k=1}^n\lambda_1^{|i_1-i_2-s_1|}\lambda_2^{|i_2-i_3-s_2|}...\lambda_k^{|i_k-i_1-s_k|}$$is equal to$$\sum_{j=1}^k\lambda_j^{S+k-1}\prod_{l=1,l\ne j}...
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Taking limits in stochastic partial differential initial value problems
Background: A (stochastic) Cauchy problem I am interested in looks like this:
$$
(1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
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What is the current fastest method to calculate Lerch's Phi transcendent?
Lerch's Phi transcendent is
$$
\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s}
$$
I am trying to compute this for the following parameters:
$z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (...
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Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$
Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
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Limits of the wave equation with piecewise constant propagation speed
This question is cross-posted from math.stackexchange.com, where it did not (yet?) get any answers despite a +100 bounty.
Consider a wave equation
$$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\...
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Largest eigenvalue scaling in a certain Kac-Murdoch-Szegö matrix
I'm looking at $N\times N$ matrices $M_N$ with elements
$$M_N=\left( \rho^{|i-j|} \right)_{i,j=1}^N,$$
where $\rho$ is a complex number of unit modulus.
These matrices with $\rho\in\mathbb R$ and $|\...
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Limiting property of polylogarithm ratio
I try (without success) to figure out what could be the following limit if any...
For real $s$ strictly $> 1$ and $x \rightarrow +\infty$ (x real) the limit of the polylogarithm ratio $\frac{Li_{s-...
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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 ...
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Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
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Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
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Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$
We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim_{n\to\infty}y_{\alpha n}=0$) of rational numbers with the property: $\forall\...
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Sum of reciprocal quadratic
Is there a general method or formula for calculating the infinite sum $\sum_{n=1}^{\infty} 1/(an^{2}+bn+c) $?
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Does the intercept converge if we fit a best fit line to points with prime coordinates?
A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here.
Let $p_k$ denote the $k$th prime such that $...
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Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes
Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
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Asymptotic rate for the expected value of the square root of sample average
I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$.
Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.
I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.
My first ...
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Integrals of power towers
Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
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If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^*$ = $\mathfrak{M}(G)$
Let $\mathfrak{M}(G)$ be the set of all means on $L_\infty (G)$
If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^* = \mathfrak{M}(G)$
My attempt:
We know ...
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Limit Behavior of a Graph Iteration
,Let $G(V,E)$ be a weighted complete graph.
Let further $\min_k(v_i)$ denote, depending on whether the context is arithmetic or set theoretic, either the set of the $k$ smallest edges adjacent to $...
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Is the density of 1's in the Fibonacci word uniform?
The Fibonacci word is the limit of the sequence of words starting with $0$ and satisfying rules $0 \to 01, 1 \to 0$. Equivalently, it is obtained from the recursion $S_n= S_{n-1}S_{n-2}$ under ...
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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{...
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Does this definition of the Fourier intensity measure make sense?
Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$.
For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...
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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?$$ $f$ ...
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Showing $o(1)$ convergence for ratio of successive binomial tail probabilities
For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
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Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$
Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$
Let $V= \...
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A limiting sequence of positive definite matrices
Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...
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