Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
280 questions from the last 365 days
0
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Explosion of the moment of double stochastic exponential
We consider the stochastic system
$$\frac{dS_t}{S_t}=-R_t\,dW_t,$$
with
$$dR_t=-R_t\,dt-R_t\,dW_t, \quad R_0>0.$$
We conjecture, and would like to show that
$$\mathbb{E}[S_t^2]
=
S_0^2\,\mathbb{E}\...
- 696
1
vote
0
answers
76
views
Reducing net convergence to sequence convergence
As mentioned in a previous question of mine, I came across Royden's proof of the density of $C^\infty_c(\mathbb{R})$ in $L^p(\mathbb{R})$, in which he shows that $$ \lim_{\epsilon \to 0^+} \|j_\...
- 59
-2
votes
0
answers
218
views
On construction of certain class of analytic functions with given properties
(I don't know how naïve this question is / could be.)
Is there a class of function with following properties:
$f: \mathbb{R} \to \mathbb{R}$ is analytic.
$f$ is such that each positive integer input ...
- 802
4
votes
0
answers
119
views
Is the one-sided concentration function of a non-negative integrable function absolutely continuous?
For $g\in L^1_+\!(0,1)$ define the one-sided concentration function
$$
Q_g(\delta)\;:=\;\sup_{0\le x\le 1-\delta}\int_x^{x+\delta} g(t)\,dt,
\qquad 0\le \delta\le 1.
$$
It is easy to check that $Q_g$ ...
- 12.4k
4
votes
0
answers
225
views
Density of convergent subsequences of any $s:\mathbb{N}\to [0,1]$ [migrated]
Starting point. Early on in the first analysis class, it is taught that any sequence $s:\mathbb{N}\to [0,1]$ has a convergent subsequence. This question is about whether every sequence has a ...
6
votes
2
answers
424
views
Does the functional equation $(f(x+y) - f(x))/y = (f(x+y) - f(y))/x$ imply that $f$ is affine?
Let $f:(0,\infty) \to \mathbb{R}$ be a function satisfying $$ \frac{f(x+y) - f(y)}{x} = \frac{f(x+y) - f(x)}{y} $$ for all $x,y>0$.
Does this already imply that $f$ is an affine function, i.e., $f(...
- 61
0
votes
0
answers
51
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Monotonicity of a unique root curve arising from coupled cumulative entropy functionals
I would like to ask about the following monotonicity problem.
For $g>1$ and $\rho>0$, define
$
\bar b(g,\rho):=\frac{\rho g}{1+\rho g}
$
and
$$
\Phi_\ell(b;g,\rho):=
\begin{cases}
\left(\dfrac{1}...
9
votes
1
answer
233
views
Approximation of multivariant Lipschitz functions by piecewise linear functions
Let $f\colon \mathbb R^n \to \mathbb R$ be a Lipschitz functions.
Is it possible to approximate $f$ by piece-wise linear functions $f_k$ with almost the same Lipschitz constant as $f$?
More precisely, ...
- 221
16
votes
3
answers
2k
views
Category-theoretic foundations of analysis
I want to know if there is a solid, category-theoretic foundation underlining the study of analysis and what has been done in this direction over the past years.
Category theory was pioneered by Mac ...
- 415
0
votes
0
answers
107
views
Boundedness of a function orthogonal to a subspace of polynomials
Let $\mu$ be a probability measure on $\mathbb{R}$ having moments of all orders and assume that $\mu$ is moment determinate. This holds in particular when $\mu$ has exponential moment of some order. ...
- 537
20
votes
1
answer
497
views
Which non-constructive principle is equivalent to the isomorphism between Dedekind reals and Cauchy reals?
When I say that $x$ is a Dedekind real, I mean that $x$ is a left Dedekind cut of the rational numbers; that is, $x \subseteq \mathbb{Q}$ satisfying the following properties:
$\exists r,s \in \mathbb{...
- 1,361
2
votes
1
answer
114
views
Proving a compact embedding involving $L^2(0,T;H_0^1(\Omega))$ and $L^2(0,T;L^{p^+}(\Omega))$
I am reading the paper with DOI:10.1080/00036811.2025.2450692, and I would like to understand the proof of the following compact embedding result.
The paper assumes that the exponent $p(x)$ is ...
- 63
1
vote
0
answers
147
views
Removable singularities and continuous extension of a parametrization
Informal description of the problem
This problem has been bugging me for a long time. In summary, I have a real-analytic parametrization of a surface defined on $\mathbb{R}^2$ minus the coordinate ...
- 257
5
votes
2
answers
454
views
Pointwise limit of cadlag functions
Suppose I have a sequence of cadlag functions $f_{n}$ defined on $\mathbb{R}$ and a function $f$ such that $f_{n} \to f$ pointwise. Can the limit $f$ have uncountably many points of discontinuity? ...
- 145
3
votes
1
answer
200
views
Does the mean value $\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tf(t)\,\mathrm dt$ exist if $f$ is given by a function in $L^1(\mathrm b\mathbb R)$?
Let $\mathrm b\mathbb R$ denote the Bohr compactification of $\mathbb R$, let $\mu$ be its normalized Haar measure, and let $\iota:\mathbb R\to\mathrm b\mathbb R$ be the usual embedding of $\mathbb R$ ...
- 133