Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
4,851
questions
0
votes
0
answers
23
views
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 ...
- 4,881
-4
votes
0
answers
40
views
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’...
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 ...
- 218
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\...
- 63
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 ...
- 39
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)=...
- 479
3
votes
2
answers
80
views
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 ...
- 33
8
votes
1
answer
508
views
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$.
...
- 2,574
2
votes
1
answer
355
views
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 ...
- 227
1
vote
1
answer
76
views
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 ...
- 277
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. ...
- 39.4k
0
votes
0
answers
41
views
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 ...
- 1,803
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)} |...
- 2,574
3
votes
0
answers
97
views
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 < \...
- 2,574
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....
- 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 "...
- 103
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 \...
- 39
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 ...
- 6,469
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 ...
- 555
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,...
- 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 $C_\frac{1}{2}$...
- 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 $d\times d$ ...
- 98.3k
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 ...
- 508
8
votes
1
answer
579
views
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 ...
- 3,366
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:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
- 3,863
3
votes
1
answer
356
views
An exercise on log-concave random variable on the real line
Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...
- 235
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)\...
0
votes
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?
- 1
0
votes
0
answers
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 ...
- 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
$$
\begin{cases}
f(x) = 0 \\
\nabla f (x) \neq 0
\end{cases}\text{ on }\...
- 11
9
votes
2
answers
522
views
Proving the simple form of a function from statistical mechanics
Suppose we have a function $f_0:{\mathbb R}^3\rightarrow {\mathbb R}_+$ that satisfies the following property
\begin{equation}
\begin{split}
&\mathbf{v}_1^2 + \mathbf{v}_2^2 = \mathbf{v}_1'^2 + \...
0
votes
0
answers
62
views
Sum power series not continuous unit circle
This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
- 6,756
2
votes
0
answers
65
views
Inequality for log-likelihood ratio
Let $ p, q $ be two probability densities on $ [0,1] $, strictly positive over $ (0,1) $. Let $ P $ be the cumulative function of $ p $, i.e., $ P(x) = \int_0^x p(x') \, \mathrm{d}x' $, $ x \in [0,1] $...
- 445
2
votes
0
answers
50
views
Representation of Baire 1 functions
Upper semi-continuous functions on the reals are Baire 1, which is readily observed by considering
$$ f_{n}(x):= \sup_{y\in [0,1]}(f(y)- n |x-y| ) \qquad (A).$$
Indeed $f_n$ as in (A) is continuous ...
- 2,877
1
vote
1
answer
54
views
Pair of functions that vary in the same direction
Say we have 2 functions $f$ and $g$ such that:
$f(a)<f(b) \Leftrightarrow g(a)<g(b)\;\; \forall a,b \in \mathbb{R}^n$
Is there an accepted name for a couple of functions like these?
Is there a ...
- 13
0
votes
1
answer
132
views
Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 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
$$ \|u\|^2_{L'^1_\delta(\Omega)}= \int_{\Omega} |x|...
- 3,863
11
votes
1
answer
1k
views
New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
- 45
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 ...
- 4,416
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
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
- 347
0
votes
0
answers
85
views
Sobolev estimates on domain with boundary
Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...
- 11
2
votes
0
answers
51
views
Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
- 973
0
votes
0
answers
102
views
$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 42.5k
8
votes
3
answers
479
views
Regularity of Newtonian potential along smooth boundary
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...
- 1,320
0
votes
0
answers
45
views
Representation of concave point-to-set maps
Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...
- 53
4
votes
0
answers
81
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,385
4
votes
1
answer
151
views
Finding a real-analytic diffeomorphism
Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
- 3,863
5
votes
2
answers
522
views
Stone-Weierstrass without the "subalgebra" condition
Suppose I consider $C_0(\mathbb{N})$ consisting of function on the natural numbers vanishing at $\infty$. For an irrational $1<\alpha<2$, let $p_{m\alpha}(\cdot)$ be the function $p_{m\alpha}(n)=...
- 131
2
votes
0
answers
39
views
K functional of $L^{1,\infty}$ and $L^\infty$ real interpolation
I want to know where can I find how to compute, given a function $f$ and $t>0$
$$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 ...
- 21
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 ...
- 4,309