Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,269
questions
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Sequential separability on $C_p(X)$
Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
1
vote
1
answer
75
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Initial and final Theorem for upper and lower limits?
Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$.
Can we use upper limits or lower limits to ...
6
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1
answer
170
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Some special sequence in $C(\mathbb{R})$
Let us consider $C(\mathbb{R})$, the space of continuous functions on the reals.
Q. Does there exist a sequence $\{f_n\}$ in $C(\mathbb{R})$ such that for every $f\in C(\mathbb{R})$ one may find a ...
11
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2
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L'Hopital rule for upper and lower limit?
I am reading the following paper 1998(H.Hudzik) P.574
It reads using L'Hopital rule$$\liminf_{u\to\infty} \frac{1/\varphi(1/u)}{\psi(u)}=\liminf_{u\to\infty}\frac{\varphi'(u)}{\psi'(u)u^2[\varphi(1/u)]...
2
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1
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On which subspace $W\subset C^{\infty}[0,1]$ is $(Df)(x)=xf'(x)$ a bounded operator provided all functions in $W$ are flat functions?
I have already asked this question on MSE; now I repeat it on MO.
https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator
First we ...
2
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0
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130
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Is there a characterization of tempered functions whose Fourier transforms are also tempered functions?
A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions.
This MSE question asked ...
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0
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111
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How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
1
vote
1
answer
500
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First derivative of cut off function
I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in ...
1
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0
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88
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Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
1
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1
answer
102
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Given an increasing function, need to construct a continuous increasing function equivalent to given function
Given an increasing function $f:[0,\infty)\to[0,\infty)$, we can define $$F(x)=\int_x^{x+1} f(t)dt,$$
which is continuous, increasing function satisfying $$f(x)\leq F(x)\leq f(x+1).$$
Question) For ...
6
votes
1
answer
454
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A limit problem
Let $f$ be a bounded and continuous function, $0<a < 1$. $U(x,r)$ is the neighborhood of $x$ with diameter $r$. Can we prove the following equation of two limits
$$ \lim_{r\rightarrow 0} \sup_{...
3
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0
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120
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If $u_n\rightharpoonup u$ in $L^2(0,T;L^2)$ then there is a subsequence such that $u_n(t)\rightharpoonup u(t)$ almost everywhere?
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?
I'm not sure if this question is trivial or not,...
3
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1
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298
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Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
9
votes
1
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335
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Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
10
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3
answers
840
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Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses
The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...
1
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0
answers
69
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Elliptic systems with two dimensions
Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\...
1
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0
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74
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Existence of solutions to $\alpha(s)=\mathbb P[Y_s>0] + \int_0^s \dot{\alpha}(t)\mathbb P[Y^{t,0}_s>0] dt$
Let $\alpha:\mathbb R_+\to\mathbb R_+$ be a "nice" function with $\alpha(0)=1$. Define the process
$$Y_t=Y_0+t+\int_0^t\frac{dW_u}{1+\alpha(u)},\quad \forall t\ge 0,$$
where $Y_0>0$ has a ...
1
vote
1
answer
189
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Proof of extended version of non-random "almost supermartingale"
In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
3
votes
1
answer
202
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Deriving inequalities from other inequalities
My questions come from the proof of Theorem 5.14 in section 5.7 of Boucheron, Lugosi, and Massart - Concentration inequalities. My first question can be stated as follows:
Suppose for positive ...
1
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0
answers
74
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Boundary estimates for elliptic systems
Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\...
9
votes
2
answers
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Is $\mathbb{Q}$ the orbit of a rational function under iteration?
In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the ...
2
votes
0
answers
73
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Is there a finite set of polynomials generating all rational numbers by iteration?
In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices.
The ...
2
votes
0
answers
80
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A complicated equation of integro-differential type
Consider the following equation of $\beta$ : $\beta(0)=2$ and
$$\dot\beta(t)\beta(t)^2 = \frac{1}{2}\int_0^t\frac{\dot\beta(s)(t-s)}{A_{\beta}(s,t)^{3/2}}f\left(\frac{t-s}{\sqrt{A_{\beta}(s,t)}}\right)...
33
votes
2
answers
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What is the smallest set of real continuous functions generating all rational numbers by iteration?
I recently came across this problem from USAMO 2005:
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
5
votes
0
answers
645
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Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
1
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0
answers
99
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Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
2
votes
0
answers
67
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Core for Neumann Laplacians
Let $d$ be a positive integer. We write $\mathbb{H}^d$ for the closed $d$-dimensional upper-half space: $\mathbb{H}^d=\{(x_1,\ldots,x_d) \in \mathbb{R}^d,\,x_d \ge 0\}$. We consider the Neumann ...
1
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1
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269
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Expectation of the maximum of a lognormal distributed variable and zero
I need to find an algebraic expression for E(max{X-a,0}), where X has a lognormal distribution with mean mu and standard deviation sigma. So far, I have derived the following expression, but I could ...
1
vote
1
answer
250
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Can we invoke "almost supermartingale" Theorem for deterministic sequences?
Perhaps stupid question.
Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems?
Attempt ...
0
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0
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110
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Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
2
votes
1
answer
244
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A question about pushforward measures and continuous Borel isomorphisms
It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
2
votes
0
answers
52
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An integral average condition and its relationship with BMO, VMO, and Sobolev spaces
Let $V: \mathbb R^n \to \mathbb R^n$ be a vector field which satisfies
$$
\lim_{l \to \infty} \sup_{x \in \mathbb R^n} \left|\frac{1}{l^n} \int_{[0,l]^n}V(x+y) dy \right| = 0
$$
What is the ...
5
votes
2
answers
608
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Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?
I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
3
votes
0
answers
119
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Question on the model completeness of the real field expanded by restricted Pfaffian functions
Currently I'm reading "Model completeness results for expansions of the ordered field
of real numbers by restricted Pfaffian functions and the exponential function" by Wilkie, and I do not ...
0
votes
1
answer
223
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Is this a contraction mapping for small $T$?
Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$
$$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$
For $T>0$, let $\mathcal H_T:=\{h:[0,T]\...
1
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0
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73
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Representing a function in terms of higher order differences
I want to write a function in terms of its mollification and higher order
forward differences. Given a function $u:\mathbb{R}\rightarrow\mathbb{R}$ and
$h>0$, we set $u_{h}(x):=\frac{1}{h}u\left( \...
7
votes
2
answers
467
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Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$
Let $f:\mathbb{R}\to\mathbb{R}$ be the function $$f(x) = \sum_{k=1}^{\infty} \sqrt{\max\left\{1 -\frac{k^2}{x^2},0\right\}}.$$
Numerical experiments suggests that there exists $n\in\mathbb{N}$ and a $...
3
votes
0
answers
138
views
Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
2
votes
1
answer
142
views
Given an increasing function $f$, to find a continuous function satisfying properties of $f$
Let $f:[0,\infty)\to [0,\infty)$ be an increasing function satisfying
$$\int_0^\infty f(x)\frac{dx}{1+x^2}=\infty.$$
Can we find a continuous increasing function $F$ on $[0,\infty)$ satisfying
$$\...
4
votes
1
answer
244
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Is $C_n$ infinitely log-convex?
A sequence $a_n$ is called log-convex if $\mathcal{L}(a_n):=a_{n+1}a_{n-1}-a_n^2\geq0$ for all $n$; it is infinitely log-convex provided that all the iterates $\mathcal{L}^k(a_n)$ are still log-convex,...
2
votes
2
answers
165
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Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces
Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that:
For $n \geq 0$, let $E_{n}$ ...
0
votes
1
answer
160
views
Lower bound related to derivative of $j$-invariant
Recall the $j$-invariant function, namely,
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{2\pi i \tau}$ and the coefficients $(c_k)_k$ are in the OEIS sequence A000521.
By using some ...
0
votes
0
answers
139
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
4
votes
1
answer
182
views
Maximal increasing behaviour of $\sqrt{-\log x}\operatorname{Li}_{1/2}(x)$
This is an extension of a problem in mathematical biology. It appears that
For any $\varepsilon>0$, there exists an interval $I\subset(0,1)$ on which $\sqrt{-\log x}^{1+\varepsilon}\operatorname{...
2
votes
1
answer
68
views
Subordination principle
Assume that $f,g:[0,1]\to [0,1]$ so that $f(0)=g(0)=0$ and $f(1)=1=g(1)$, $f$ is increasing and $f'$ is decreasing in $[0,1]$ and that $g(x)\le f(x)$ for $x\in[0,1]$. Can we conclude that $$\liminf_{x\...
2
votes
0
answers
123
views
Taylor series with less than differentiability
I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite
$$
F^1 := \lim_{x\to \infty} x \cdot f^0(x)
$$
Then I consider $f^1(x) := x \cdot ...
2
votes
2
answers
240
views
Measure of non-commutativity of two invertible functions
I need to estimate $|x - f^{-1}(g^{-1}(f(g(x))))|$ for various values of $x$ for two smooth invertible functions $f$ and $g$ on $\mathbb{R}$ (actually some other spaces, but $\mathbb{R}$ will do.) Are ...
3
votes
1
answer
121
views
Estimate the homogeneous components of a polynomial against its maximum
Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over ...
1
vote
0
answers
71
views
Error estimates for orthogonal polynomial approximation
tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials?
There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
14
votes
2
answers
1k
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Generalisation of Cauchy's mean value theorem
I apologise in advance if this is an elementary question more fitted for Math Stack Exchange. The reason why I have decided to post here is that the question I am used to seeing on that site are not ...