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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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Representation of continuous, monotone, concave functions

Is there a characterization of all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying: $f(0)=0$ $f$ is monotonically increasing $f$ is concave My intuition is that $f$ should admit ...
ABIM's user avatar
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-4 votes
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Proof that the infimum of a set is incorrect [closed]

So we will say that we have a set S⊆R so that Inf(S)=p then n∈S ∃ε>0∈R s.t. n>ε+p <—-> inf(S)≠p. (Is this a good proof, please comment if there are any mistakes or improvements to make, I’...
IHateMath's user avatar
2 votes
0 answers
33 views

Uniformly closed ideals of smooth/real analytic functions

Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
Thomas Kurbach's user avatar
2 votes
1 answer
55 views

Smooth approximation of nonnegative, nondecreasing, concave functions

Let $f\colon [0, \infty)\to\mathbb{R}$ be nonnegative, nondecreasing, and concave. Prove the following claim or give a counter example: There is a sequence of functions $f_n\colon [0, \infty)\to\...
Froomfondel's user avatar
2 votes
1 answer
151 views

The number of roots of the sum of radicals

Let $n\in \mathbb{N}$ and $$-\infty < a_1 < b_1 < a_2 < b_2 < a_3 < b_3<\cdots<a_n<b_n<+\infty$$ and $k_i\in \mathbb{R}, i=1,2,\ldots,n$. Is there any information about ...
eN.meshok's user avatar
3 votes
0 answers
336 views

Surprisingly difficult limit of a sequence

Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$? Of course $|a_n| \to \infty$, but we have $$ \operatorname{Re}(a_n)=...
J.Mayol's user avatar
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3 votes
2 answers
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Subdifferential of a convex function admits a continuous selection

Let $F$ be a continuous convex function on $\mathbb{R}^n$. If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...
Aimar's user avatar
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8 votes
1 answer
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Does the family of fat Cantor sets contain a measurable rectangle?

Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$. ...
Nate River's user avatar
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2 votes
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Stone-Weierstrass theorem: coefficients of approximating sequence bounded?

Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$. The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points ...
fsp-b's user avatar
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1 answer
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Submodularity of a particular function derived from a convex function?

Consider a convex function $f : \mathbb{R}^d \to \mathbb{R}$. Define now the set-input function $g_f : 2^{[d]} \to \mathbb{R}$ as follows, \begin{align} g_f(S) = \min \left\{ f(x) : x \in \mathbb{R}^d ...
Television's user avatar
7 votes
0 answers
212 views

Can you identify this irrational number?

There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
Gerald Edgar's user avatar
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On $[0,1]$ with the Lesbegue measure, is it possible to have $\lVert \cdot \rVert^{1/n}_p \leq \lVert \cdot \rVert_q$ for $p>q$ and $n$ large?

The question is as in the above. In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$. (I deleted the last question ...
Isaac's user avatar
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4 votes
1 answer
203 views

If a function $f$ is $\varepsilon$-times Lebesgue differentiable, is $f$ continuous?

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Given an $\varepsilon > 0$, we say $f$ is $\varepsilon$-times Lebesgue differentiable if $$\lim_{r \to 0} \frac{\int_{B_r (x)} |...
Nate River's user avatar
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3 votes
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Does “on average” Hölder continuity imply Hölder continuity?

Let $\Omega$ be a smooth, bounded, connected open subset of $\mathbb R^n$. A function $f: \Omega \to \mathbb R$ is said to be strongly Hölder continuous on average of order $\alpha$, for $0 < \...
Nate River's user avatar
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3 votes
0 answers
112 views

If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?

The question is as in the title. Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
Isaac's user avatar
  • 1,803
3 votes
2 answers
384 views

Proof of the inequality $\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x}$ when $x,y \in (0,1]$

I am trying to prove the following inequality: $$\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x} \quad \forall x,y \in (0,1]$$ This inequality appears in the paper "...
good bandit's user avatar
1 vote
0 answers
46 views

Variation of the fractional derivatives

$\DeclareMathOperator\AC{AC}\DeclareMathOperator\Lip{Lip}$Suppose we have $f\in L^1(\mathbb{R})\cap \AC(\mathbb{R})\cap \Lip(\mathbb{R})$ and $f$ piecewise linear function, bounded and $|f|\leqslant \...
eN.meshok's user avatar
3 votes
2 answers
161 views

Recovering a set from its projections in varying coordinate systems - a projection hull?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
M.G.'s user avatar
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0 votes
0 answers
94 views

Explicit bounds on derivatives of moments related to Bernstein polynomials

Background This question relates to finding explicit bounds for the derivatives of moments related to Bernstein polynomials. Answering it will help me find explicit bounds for polynomials that ...
Peter O.'s user avatar
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1 vote
1 answer
84 views

A generalized form of the approximation to identity?

This question is an extension of the one I posted on ME: https://math.stackexchange.com/questions/4701500/if-alpha-nx-int-lvert-x-y-rvert-leq-1-n-lvert-x-y-rvert2-d-muy It might be elementary for here,...
Isaac's user avatar
  • 1,803
7 votes
0 answers
365 views

Does the intersection of middle third and middle half Cantor sets contain an irrational number?

Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$. Likewise let $C_\frac{1}{2}$...
Dmitry K's user avatar
  • 1,414
14 votes
1 answer
311 views

Lipschitz property of the determinant

$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
Iosif Pinelis's user avatar
2 votes
0 answers
120 views

Banach space of vector measures

Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...
user72829's user avatar
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8 votes
1 answer
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Measure without measurable sets

This question is a little on the softer and speculative side, so bear with me. Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
Amir Sagiv's user avatar
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1 vote
1 answer
137 views

On an integral equation

Let $B: C^{\infty}([0,1]^3)$ satisfy $$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$ Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation: $$ \int_0^1 f(t,x)\,dx + \int_0^t\...
Ali's user avatar
  • 3,863
3 votes
1 answer
356 views

An exercise on log-concave random variable on the real line

Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$. Show that there is a universal (independent of $X$) constant $c>0$ such that: $$P(X\in[-1/2;0])\...
Gericault's user avatar
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7 votes
1 answer
260 views

Log-convexity of determinant

Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
António Borges Santos's user avatar
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0 answers
45 views

Schwarz symmetrizaton and rank one symmetric spaces

does the symmetrization argument (Schwarz symmetrizaton ) works well in the setting of all non compact symmetric spaces of rank one?
jacob's user avatar
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88 views

Existence of a smooth extension

In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface $$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$ Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...
Ali's user avatar
  • 3,863
0 votes
0 answers
58 views

Morse functions on subset $\bar \Omega$ of $\mathbb {R}^d$ and its level sets

Let $f$ be a $C^{2}(\bar{\Omega})$ Morse function, where $\Omega$ is a bounded open set of $\mathbb{R}^d$: this means that $$ \begin{cases} f(x) = 0 \\ \nabla f (x) \neq 0 \end{cases}\text{ on }\...
L19's user avatar
  • 11
9 votes
2 answers
522 views

Proving the simple form of a function from statistical mechanics

Suppose we have a function $f_0:{\mathbb R}^3\rightarrow {\mathbb R}_+$ that satisfies the following property \begin{equation} \begin{split} &\mathbf{v}_1^2 + \mathbf{v}_2^2 = \mathbf{v}_1'^2 + \...
LuckyJollyMoments's user avatar
0 votes
0 answers
62 views

Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there. Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
Libli's user avatar
  • 6,756
2 votes
0 answers
65 views

Inequality for log-likelihood ratio

Let $ p, q $ be two probability densities on $ [0,1] $, strictly positive over $ (0,1) $. Let $ P $ be the cumulative function of $ p $, i.e., $ P(x) = \int_0^x p(x') \, \mathrm{d}x' $, $ x \in [0,1] $...
aleph's user avatar
  • 445
2 votes
0 answers
50 views

Representation of Baire 1 functions

Upper semi-continuous functions on the reals are Baire 1, which is readily observed by considering $$ f_{n}(x):= \sup_{y\in [0,1]}(f(y)- n |x-y| ) \qquad (A).$$ Indeed $f_n$ as in (A) is continuous ...
Sam Sanders's user avatar
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1 vote
1 answer
54 views

Pair of functions that vary in the same direction

Say we have 2 functions $f$ and $g$ such that: $f(a)<f(b) \Leftrightarrow g(a)<g(b)\;\; \forall a,b \in \mathbb{R}^n$ Is there an accepted name for a couple of functions like these? Is there a ...
BGoodman's user avatar
0 votes
1 answer
132 views

Finite dimensionality of a subspace

Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds: $$ \...
Ali's user avatar
  • 3,863
0 votes
0 answers
41 views

Compact embedding of homogeneous weighted Sobolev spaces

Let $n\geq 2$ and let $\Omega$ be the open unit ball with the origin removed. For each $\delta>0$ and each $u\in C^{\infty}(\Omega)$ let us define $$ \|u\|^2_{L'^1_\delta(\Omega)}= \int_{\Omega} |x|...
Ali's user avatar
  • 3,863
11 votes
1 answer
1k views

New method to compute square roots [closed]

In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds: $$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
polygamma's user avatar
2 votes
2 answers
89 views

What are the bounds of $xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a$ for $0 \le x \le 1$ and $a > 0$?

Posting from MSE since it was unanswered in MSE. Let $0 \le x,y \le 1$ and $a$ be a real and let $$ f(x,y,a) = xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \tag 1 $$ For a fixed $a$, the graph of the ...
Nilotpal Kanti Sinha's user avatar
2 votes
0 answers
175 views

Schrödinger representation of the Heisenberg group

Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have $$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
zoran  Vicovic's user avatar
0 votes
0 answers
85 views

Sobolev estimates on domain with boundary

Could someone point me to a reference for the proof of the following Sobolev estimate $$ \|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)}) $$ for ...
L19's user avatar
  • 11
2 votes
0 answers
51 views

Inequality for a weighted bilinear form in Fourier variables

Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$. Consider the ...
Guy Fsone's user avatar
  • 973
0 votes
0 answers
102 views

$\log$-classes of irrationals

Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
Dominic van der Zypen's user avatar
8 votes
3 answers
479 views

Regularity of Newtonian potential along smooth boundary

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define $$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$ Is it true that $V(z) \in C^{\infty}(\partial \Omega)$? ...
student's user avatar
  • 1,320
0 votes
0 answers
45 views

Representation of concave point-to-set maps

Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...
Ded's user avatar
  • 53
4 votes
0 answers
81 views

Find at least one square-boxed subcontinuum

Recall that a plane continuum is a closed, bounded, connected subset of the plane. It is non-degenerate if it contains at least two points. (We may sometimes just say "continuum" even if we ...
Mirko's user avatar
  • 1,385
4 votes
1 answer
151 views

Finding a real-analytic diffeomorphism

Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
Ali's user avatar
  • 3,863
5 votes
2 answers
522 views

Stone-Weierstrass without the "subalgebra" condition

Suppose I consider $C_0(\mathbb{N})$ consisting of function on the natural numbers vanishing at $\infty$. For an irrational $1<\alpha<2$, let $p_{m\alpha}(\cdot)$ be the function $p_{m\alpha}(n)=...
F J's user avatar
  • 131
2 votes
0 answers
39 views

K functional of $L^{1,\infty}$ and $L^\infty$ real interpolation

I want to know where can I find how to compute, given a function $f$ and $t>0$ $$K(t,f;L^{1,\infty};L^\infty)$$ where $L^{1,\infty}=\sup_{t>0}t f^*(t)$ and $K$ is the Petree K-functional ...
User 2234's user avatar
3 votes
0 answers
77 views

A variant of the Laplace principle

$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
leo monsaingeon's user avatar

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