Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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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 $x=\lim x_{n_k}$. ...
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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 ...
Fractional analysics's user avatar
6 votes
1 answer
170 views

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 ...
ABB's user avatar
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11 votes
2 answers
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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)]...
Fractional analysics's user avatar
2 votes
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204 views

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 ...
Ali Taghavi's user avatar
2 votes
0 answers
130 views

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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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(...
xiuhua's user avatar
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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 ...
Mr. Proof's user avatar
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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(\...
asrxiiviii's user avatar
1 vote
1 answer
102 views

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 ...
Wilderness's user avatar
6 votes
1 answer
454 views

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_{...
Watheophy's user avatar
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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,...
demlevi33's user avatar
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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{...
Mr. Proof's user avatar
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9 votes
1 answer
335 views

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 $(f(x_i+x_j))_{i,j=1}^n$...
Hans's user avatar
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10 votes
3 answers
840 views

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 ...
Ivan Meir's user avatar
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1 vote
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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\...
Luis Yanka Annalisc's user avatar
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0 answers
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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 ...
GJC20's user avatar
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1 answer
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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 ...
user550103's user avatar
3 votes
1 answer
202 views

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 ...
Fei Cao's user avatar
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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\...
Luis Yanka Annalisc's user avatar
9 votes
2 answers
623 views

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 ...
Ivan Meir's user avatar
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2 votes
0 answers
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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 ...
Ivan Meir's user avatar
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2 votes
0 answers
80 views

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)...
GJC20's user avatar
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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 ...
Ivan Meir's user avatar
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5 votes
0 answers
645 views

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 (...
TPC's user avatar
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0 answers
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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 \, ...
Riku's user avatar
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0 answers
67 views

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 ...
sharpe's user avatar
  • 701
1 vote
1 answer
269 views

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 ...
Jelle van Lin's user avatar
1 vote
1 answer
250 views

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 ...
user550103's user avatar
0 votes
0 answers
110 views

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 $...
GJC20's user avatar
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2 votes
1 answer
244 views

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 ...
O-Schmo's user avatar
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2 votes
0 answers
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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 ...
Riku's user avatar
  • 819
5 votes
2 answers
608 views

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 ...
D.R.'s user avatar
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3 votes
0 answers
119 views

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 ...
Bytegear's user avatar
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0 votes
1 answer
223 views

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]\...
GJC20's user avatar
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1 vote
0 answers
73 views

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( \...
Gio67's user avatar
  • 401
7 votes
2 answers
467 views

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 $...
Onur Oktay's user avatar
  • 2,263
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 $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
Noah Schweber's user avatar
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 $$\...
Wilderness's user avatar
4 votes
1 answer
244 views

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,...
T. Amdeberhan's user avatar
2 votes
2 answers
165 views

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 $E_{n}$ ...
ScienceAge's user avatar
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 ...
Jean's user avatar
  • 515
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)...
Dattier's user avatar
  • 3,737
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{...
TheSimpliFire's user avatar
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\...
MathArt's user avatar
  • 333
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 ...
Kolodez's user avatar
  • 335
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 ...
Jps's user avatar
  • 21
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 ...
fsp-b's user avatar
  • 421
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 ...
user13322's user avatar
14 votes
2 answers
1k views

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 ...
carfog's user avatar
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