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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7 votes
2 answers
783 views

Mapping exponentiation onto addition

I was inspired by Does there exist a function which converts exponentiation into addition? to think about mapping exponentiation onto addition. The question asks whether there exists $f:\mathbb{R}\...
1 vote
1 answer
246 views

Intersecting points of increasing convex functions

Can two increasing and differentiable convex functions agree exactly on a countable set of cardinality greater than two?
2 votes
2 answers
270 views

The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$

How to prove the following inequality $$\forall t>0,\quad\int^\infty_0 \sin(rt)\frac{r^3}{\sinh^2(r)} dr\leq c \big(te^{-At}\big)$$ for some constants $A>0,c>0$
1 vote
1 answer
304 views

An inequality in four variables

Let $f(x,y)=\frac{10xy-(x+y)+1}{8xy-2(x+y)+5}$ and $g(x,y)=\frac{1}{4}\left[1+\frac{1}{3}(4x-1)(4y-1)\right]$. I want to prove that for any $0.5\le a\le b\le 1$ and $0.7\le c\le d\le 1$, it holds that ...
0 votes
0 answers
117 views

From convergence pointwise to convergence of the supremum for semicontinuous functions

Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
5 votes
1 answer
181 views

Writing a class of functions in terms of positive semidefinite matrix-valued functions

Consider a continuously differentiable function $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ such that $f(0) = 0$ and $\langle f(x), x \rangle \ge 0$ for all $x \in \mathbb{R}^{n}$. Does there exist a ...
4 votes
1 answer
239 views

How to unperiodise a function

We know that given a sufficiently regular function $f: \mathbb{R} \to \mathbb{R}$, then its periodisation (say to period $1$) is given by $$ \begin{align} F(x) := \sum_{n\in\mathbb{Z}} f(x + n).\tag{$...
3 votes
2 answers
182 views

Bounding integral expression with total variation of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$ for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
1 vote
1 answer
78 views

Determine $\alpha \in (0,1)$ such that $J_{\alpha}(\phi):=\int \psi/\phi^{\alpha}$ exists?

Fix $\alpha \in (0,1)$ and $\psi\in C^{\infty}_{c}(\mathbb{R}\to \mathbb{R})$. For a smooth function $\phi\geq 0$ define the integral $$J_{\alpha}(\phi):=\int \frac{\psi}{\phi^{\alpha}}.$$ If $|\phi^{...
2 votes
0 answers
121 views

Uniqueness of a moment style problem

This is a leftover from this question (and I've modified slightly to make the question more natural in the new setting). It's maybe not a very fascinating question by itself, but it seems this is what ...
1 vote
0 answers
42 views

Notation for dominating (or uniformly bounded) function

While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function. A situation like this. For some true function $f:\mathbb{R} \to \...
1 vote
0 answers
53 views

Differentiability of functions given as integral of some singular kernel

Let $A: \mathbb R_+\to [0,1]$ be $1/2$-Holder continuous and $k: \{(s,t): 0\le s\le t\}\to\mathbb R$ be continuous. Define $f:\mathbb R_+\to\mathbb R$ by $$f(t):=\int_0^t\frac{k(s,t)}{\sqrt{t-s}}\big(...
2 votes
1 answer
346 views

If function and derivative extend continuously to boundary of domain, can one extend as $C^1$ function?

Let $D \subseteq \mathbb{R}^n$ be open, connected, bounded. Let $f : D \to \mathbb{R}$ be $C^1$ and assume that $f$ as well as $\partial_i f$ extend as continuous functions to the closure $\overline{D}...
1 vote
1 answer
176 views

Does pointwise convergence yield the convergence under Skorokhod topology?

Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
0 votes
1 answer
97 views

Existence of a particular positive definite and radially unbounded function

Let $A \subset \mathbb{R}^{n}$ be a closed set. Does there exist an open set $O$ containing $A$, and a smooth function $f : O \to \mathbb{R} $ such that $f(x) = 0$ for all $x \in A$, $f(x) > 0$ and ...
6 votes
0 answers
193 views

Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
1 vote
0 answers
121 views

Tiling a rectangle with squares

Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle: The task was to determine the area of the rectangle tiled with $8$ squares, of which the ...
2 votes
0 answers
95 views

An inequality in Huisken paper

I am reading a paper is written by Gerhard Huisken 'Flow by mean curvature of convex surfaces into sphere' and I want to show the following statement Let $p\ge 2$ then for $\eta>0$ and any $0\le \...
1 vote
1 answer
82 views

What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$

What is $\left\| f \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$. I'm reading a paper Chern-Yamabe flow which said ...
2 votes
0 answers
111 views

Bounding integral expression with BV norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
0 votes
1 answer
116 views

Bounding integral expression with Sobolev norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
3 votes
1 answer
205 views

The energy of a semilinear ODE

I'm currently reading Caffarelli, Gidas, Spruck's paper "Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth". For some background, we ...
4 votes
1 answer
191 views

Harmonic functions as limits of harmonic functions on graphs?

I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this ...
0 votes
1 answer
154 views

Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicative up to an factor?

Related to this question. For $x_+ \in (0,\infty)$, $a \in \mathbb{R}$ let $F\colon[0,x_+] \to [a,\infty)$ be a twice continuous differentiable (in $(0,x_+)$) function with $f := F'$, $f(x) > 0$, ...
1 vote
1 answer
99 views

Positive, monotone decreasing function, with limit in 0 equal to ∞ submultiplicative up to an factor?

For $x_+ \in (0,\infty)$ let $f\colon(0,x_+] \to (0,\infty)$ be a continous differentiable function with $f(x) > 0$ and $f'(x) < 0$ for all $x \in (0,x_+]$. Moreover, we assume that $$\lim_{x \...
4 votes
2 answers
318 views

Chain rule for $e^f$, where $f$ has bounded variation

Let $f : \mathbb{R} \to \mathbb{R}$ be a function of normalized bounded variation (NBV), meaning that $f$ is of bounded variation, $f$ is right continuous, and $f(x) \to 0$ and $x \to -\infty$. As ...
16 votes
1 answer
871 views

Kakeya crossed-needles problem

The Kakeya needle problem asks for the minimum area planar region in which one can completely turn around a line segment through a series of translations and rotations. There is no minimum: There are &...
5 votes
1 answer
192 views

The Lipschitz constant of convex sphere in $\mathbb{R}^3$

Is every convex sphere (in the sense of Alexandorff, which is the boundary of some convex body in $\mathbb{R}^3$) with Alexandorff curvature $\geq 1$, admitting a bijective map to the unit round ...
3 votes
2 answers
187 views

Generalization of stationary points to stationary lines (valleys)?

For suitably smooth functions $f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}$, the concept of stationary points, in particular local minima, can as easily be mathematically defined as it is intuitively ...
5 votes
1 answer
371 views

Why is it valid to take uncountable infimum of one dimension of a multivariate function of random variables?

let $\xi,\eta: \Omega \to \mathbb R$ be i.i.d. random variables on a measurable space $(\Omega , \mathcal F,\mathbb P)$, and let $f: \mathbb R^2 \to \mathbb R$ be a bivariate measurable function (say ...
3 votes
1 answer
561 views

A constant ratio of integrals? Part II

This question is a follow up on my latest MO post which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context ...
4 votes
1 answer
362 views

A constant ratio of integrals? Part I

Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$. For $0<r\leq1$, consider the average of its Dirichlet integral $$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
22 votes
3 answers
2k views

The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known $...
1 vote
0 answers
73 views

Straightening a function supported on a strip

Given a positive smooth function $b:\mathbb{R}^{N+M}\to \mathbb{R}$ which vanishes exactly outisde of some $U\times\mathbb{R}^M$ ($U\subset \mathbb{R}^N$ open), is there another positive smooth ...
9 votes
2 answers
377 views

Rearrangement, conditional convergence, and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
8 votes
1 answer
427 views

Pseudo differentiable functions

Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. Let $\mathcal L$ be the set of linear functions $\mathbb R \to \mathbb R$. Define the roughness $\mathcal Rf(x)$ of $f$ at $x \in \mathbb R^...
13 votes
1 answer
792 views

“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
0 votes
1 answer
111 views

Existence of an eigenpair for d-bar operator in the unit disck

Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem: $$ \overline{\...
4 votes
1 answer
347 views

Lipschitz-regularity of partition of unity

Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
0 votes
0 answers
46 views

Taming families of rate functions

$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$. Let us say that a family $(r_j)_{j\in J}$ of ...
0 votes
0 answers
54 views

Terminology: maps which are bi-Lipschitz on compact subsets

Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...
1 vote
1 answer
127 views

Is there a uniform bound on the number of solutions to ${\partial p \over \partial x_i} (x_1,...,x_n) = c_i$ outside a set of measure zero?

Suppose $p(x_1,...,x_n)$ is a polynomial in $n$ real variables whose Hessian is not identically zero. Can we say that there is a set $Z \subset {\mathbb R}^n$ of measure zero and a constant $M$ such ...
2 votes
1 answer
142 views

Borel $\sigma$-algebras on paths of bounded variation

Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$. Let further $B\subset C$ be the subspace of $0$-started ...
2 votes
1 answer
97 views

Is this function positive on a set of large measure?

Throughout, we denote by $\mu$ the Lebesgue measure on $[0, 1]$. Let $f \in L^1([0, 1])$ be a nowhere zero function, that is, the set of all $x \in [0, 1]$ such that $f(x) = 0$ is empty. Suppose there ...
2 votes
0 answers
66 views

measure corresponding to certain orthogonal polynomials

Given a scalar $\lambda\in (0,1)$, consider a sequence of monic polynomials $\{p_n(x)\}_{n\geq 0}$ over real variables satisfying the following recurrence relations: $xp_n(x)=p_{n+1}(x)+\lambda p_{n}(...
2 votes
1 answer
160 views

Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?

Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex. For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that $$ (f'...
0 votes
0 answers
76 views

Linear dependence of the derivatives of a vector valued function

Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function $$ g:\mathbb{R}^5\rightarrow\mathbb{R}^5 $$ given by $$ g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
4 votes
1 answer
180 views

Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory

In Mike Shulman's article Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, the fundamental axiom adopted for his real-cohesive homotopy type theory (axiom $\mathbb{R}\flat$), which ...
2 votes
0 answers
127 views

Second differential of total variation

I am trying to give meaning to the notion of second differential of total variation. For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by $$TV(u)=...
1 vote
1 answer
188 views

Expressing the integral over boundary of a domain as an integral over the domain

Let $\Omega \subset \mathbb{R}^2$ be a domain which is "well behaved" (has all "wishable" properties), so as its boundary. For every $u \in C^\infty(\Omega,\mathbb{R})$, I would ...

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