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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Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
Pietro Majer's user avatar
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Implicit function theorem for non $C^1$ mappings

I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
Lorenzo Vanni's user avatar
8 votes
4 answers
604 views

Distributional derivatives are locally integrable implies the distribution is also locally integrable?

Let $T$ be a distribution on $\mathbb{R}^n$ such that there are functions $f_1,\ldots,f_n \in L^1_\text{loc}(\mathbb{R}^n)$ so that $\dfrac{\partial T}{\partial x_j} = f_j, \forall j=1,\ldots,n. $ My ...
Jinie's user avatar
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0 answers
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How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?

Already asked in SE but no response, I think it also reasonably belongs here. https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions Basically what the title says, plus ...
Clara Torres-Latorre's user avatar
2 votes
1 answer
241 views

Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?

Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
David Walmsley's user avatar
18 votes
1 answer
1k views

Does summing divergent series using cutoff functions give consistent results?

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\...
not all wrong's user avatar
0 votes
1 answer
98 views

Explanation for Tauberian theorems for Laplace transform

I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow : Let $F : [0,\infty) \to \mathbb{R}$ of ...
NancyBoy's user avatar
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1 answer
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The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$

In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...
Ilovemath's user avatar
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Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$

Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying $$ 0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2 $$ where $\Delta$ denotes the Laplace ...
J. Swail's user avatar
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2 votes
2 answers
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$L^1$ norm for a product of cosines

Let $k$ be an integer and consider the function $$ f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t). $$ I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
Itachi's user avatar
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3 votes
2 answers
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Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
gdoug's user avatar
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2 answers
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Limits of integral series

Suppose we have the series of functions: \begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation} where convergence is uniform. Additionally, consider the partial functions of the series: \...
george andrade's user avatar
0 votes
1 answer
155 views

A self-consistent equation that turns into a differential equation

Suppose the function $f(x,y)$ is defined on a small neighbourhood of $(0,0)$ in $\mathbb{R} \times [0,\infty)$ and satisfies the self consistent equation \begin{align*} & f(x,y) = \frac{1}{1-y} + ...
Tardis's user avatar
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4 votes
2 answers
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Does a function exist which is not Riemann integrable and satisfies the given condition:

I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that $$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...
Lakshmi Priya's user avatar
5 votes
2 answers
353 views

$C^1$ harmonic functions on a dense open set are globally harmonic

In a paper I am studying, at a certain point the authors introduce a function $u\in C^1(B_1,\mathbb{R})$ which is harmonic in a dense open subset $U$ of $B_1$. From this, they seem to conclude that $u$...
No-one's user avatar
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4 votes
2 answers
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Implicit function theorem without uniqueness?

Imagine you are given $f(x,y) := y^2-\sin(x)^2$ and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$. One idea that comes to mind is the ...
Daisy_Duck's user avatar
10 votes
5 answers
2k views

Extracting a common convergent indexing from an uncountable family of sequences

Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space. For each $\alpha \in \mathcal{A}$, let \begin{equation} \{ x_n^{\alpha} \}_{n=1}^\infty \end{equation} ...
Isaac's user avatar
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4 votes
1 answer
162 views

Compact-open Topology for Partial Maps?

I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow. Compact open topology is one of the most common ways of ...
Bumblebee's user avatar
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10 votes
1 answer
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Can a nowhere locally Hölder function be differentiable almost everywhere?

Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$-Hölder continuous - that is, it is not $\alpha$-Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...
Nate River's user avatar
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2 votes
2 answers
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Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$

I already posted a similar question on MO and looked into the references therein. However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form. Let $T \in \...
Isaac's user avatar
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7 votes
1 answer
256 views

High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?

Let $d\geq 1$ be a positive integer. If $\{\vec x_n\}_{n=1}^\infty$ is a sequence of $d$-dimensional vectors satisfying $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...
Feng's user avatar
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3 votes
1 answer
236 views

Is this constraint convex?

I have an optimization problem where the following constraint causes DCP Rule Error. $$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
Mojtaba's user avatar
  • 31
5 votes
1 answer
208 views

Can a solution to this parameterized ODE converge to zero?

Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$? $$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$ We are also given y(0) = 2/3, $...
icecuber's user avatar
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1 answer
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Continuous selectors of a continuous multifunctin on a compact metric space

I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector. ...
Saito's user avatar
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0 answers
174 views

Mean Cauchy sequences

Let $X$ be a $\sigma$-finite measure space. A sequence of functions $f_n \in L^1 (X)$ is said to be Cauchy in mean if $$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N}...
Nate River's user avatar
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3 votes
1 answer
181 views

If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?

Let $a,b\in\mathbb R$ with $a<b$ and $f:[a,b]\to\mathbb R$. Assume that there exists a Riemann integrable function $g:[a,b]\to\mathbb R$ such that $f=g$ almost everywhere. Then we can NOT conclude ...
Fergns Qian's user avatar
10 votes
1 answer
525 views

Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let $$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$ and $$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$ It ...
Fergns Qian's user avatar
2 votes
1 answer
179 views

Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
Nate River's user avatar
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0 votes
1 answer
178 views

Can we approximate a Hölder pdf by higher-order Hölder pdf's?

$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$ Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
Akira's user avatar
  • 815
1 vote
1 answer
173 views

Average distance between points of lower dimensional simplices in $\mathbb R^n$

Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
Nate River's user avatar
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5 votes
1 answer
401 views

On the Riemannian integrability of the bounded derivative

Let $f:[a,b]\to\mathbb R$ be a differentiable function with $f'$ bounded. According to this post, $f'$ is not necessarily Riemann integrable on $[a,b]$, see also Volterra's function. I wonder, if $f'$...
Fergns Qian's user avatar
6 votes
1 answer
374 views

How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]

I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
Euler-Masceroni's user avatar
1 vote
1 answer
119 views

On the additive property of the subdifferential of lower semicontinuous functions

Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by $$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
Fergns Qian's user avatar
25 votes
2 answers
2k views

Writing a function on $\mathbb{R}$ as a sum of two injections

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
Burak's user avatar
  • 4,115
2 votes
0 answers
124 views

A generalised Young integral

Let $f: [0, 1] \to \mathbb R$ be a continuous function. The pointwise Holder exponent $H_f (x)$ of $f$ at $x \in [0, 1]$ is defined to be $$H_f (x) := \sup \left \{ \alpha \in [0, 1] \, \big |\, \...
Nate River's user avatar
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-1 votes
1 answer
126 views

Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?

Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...
Tony James's user avatar
3 votes
1 answer
237 views

Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?

Let $S^1$ be the circle, let us consider a function $f(x,t): S^1 \times [0,\infty) \to \mathbb{R}$ such that \begin{equation} \int_0^T \int_{S^1} \lvert f(x,t) \rvert dxdt <\infty \end{equation} ...
Isaac's user avatar
  • 2,727
1 vote
1 answer
132 views

Is the Boltzmann entropy continuous in the supremum norm?

We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\...
Akira's user avatar
  • 815
11 votes
3 answers
924 views

"Simple" integral equation

Let $H(z)$ be a continuous solution of the problem $$ H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1. $$ Is it true that $H(0)=1-\ln2$? The question ...
AAK's user avatar
  • 283
1 vote
1 answer
101 views

If all mixed partials of a $C^1$ function exist and are continuous, is the function $C^2$? [closed]

For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be a $C^1$ function such that the mixed partial derivatives $\partial_i \partial_j f$ exist and are continuous for all $i \neq j$. Is it true that $f$...
Nate River's user avatar
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2 votes
1 answer
166 views

Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$? [closed]

This seems pretty trivial but I can't seem to figure it out. I think it's obviously true, given an unconstrained convergent sequence we just have to add some filler elements, but I'm having trouble ...
uniform_on_compacts's user avatar
3 votes
0 answers
77 views

Embedding theorems for Dini continuous functions

Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
Delio Mugnolo's user avatar
0 votes
1 answer
124 views

Matrices and vectors of intervals

I'm working on a project and think that matrices and vectors of intervals will be useful. I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
Paul R's user avatar
  • 39
-2 votes
1 answer
158 views

If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]

This seems like it should be true but I was wondering if anyone could prove it. Thanks!
li ang Duan's user avatar
2 votes
0 answers
103 views

Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives

If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$? It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
Jakob Möller's user avatar
4 votes
1 answer
491 views

$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?

In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
No-one's user avatar
  • 1,037
0 votes
1 answer
243 views

Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]

I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
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0 votes
0 answers
79 views

Tangent spaces of Lipschitz sub manifolds

Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
jsb's user avatar
  • 351
1 vote
1 answer
119 views

Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?

Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that $$ \int_0^1 |y-x| f(x) \, dx = 0 $$ for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
anonymous_coward's user avatar
0 votes
1 answer
108 views

Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots

I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots. Based on an old paper (this reference), it has been ...
ABB's user avatar
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