All Questions
5,629 questions
0
votes
1
answer
98
views
Estimating the bound of the integral over whole $\mathbb{R}$ of the Taylor remainder term?
Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function which has a smooth inverse and satisfies the estimate
\begin{equation}
\lvert f(x) \rvert \leq \lvert x \rvert.
\end{equation}
Also, let $d\mu$ ...
- 3,477
9
votes
0
answers
165
views
Changing coordinate to smoothen a function
Let $U\subset \mathbb{R}^2$ be an open neighborhood of the origin $0$, and let $f:U\to \mathbb{R}$ be a continuous function which is smooth on $U\setminus\left\{0\right\}$. Let's say that $f$ is ...
- 2,178
0
votes
1
answer
224
views
Computing the limit of a kind of "mean value" iteration
Let $n$ be a positive integer and $x_1,\ldots,x_n$ be real number, and define a kind of "mean value" iteration like
$$
x_i^{(k+1)} = \frac{x_i^{(k)}+x_{i+1}^{(k)}}{2}, i=1,\dotsc,n,\quad x_{...
- 193
4
votes
1
answer
172
views
Existence of a domain with simple Dirichlet eigenvalues
Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \...
- 4,115
1
vote
1
answer
161
views
Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$,
such that for all $c\in \mathbb{R}$ both sets
$$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,676
8
votes
1
answer
374
views
Status of the fundamental theorem of algebra for the locale of real numbers
In constructive mathematics without any choice at all, it is well known that the fundamental theorem of algebra cannot be proven for the Dedekind real numbers. On the other hand, Wim Ruitenberg showed ...
- 2,624
12
votes
1
answer
1k
views
The infimum of a gradient over the whole $\mathbb{R}^d$
Let $\{f_k\}:\mathbb{R}^d\to\mathbb{R}$ be a sequence of $C^1$ functions which converges pointwise to 0. Is it true that
$$\lim_{k\to+\infty}\inf_{x\in\mathbb{R}^d}|\nabla f_k(x)|=0?$$
If $d=1$ I ...
- 213
-1
votes
1
answer
189
views
f a continuous function satisfying $\sqrt{xy}(f(x) + f(y)) \leq 1 \; \forall x,y \in [0\; 1]$ Show that $\int_0^1 f(t) dt \leq \frac{\pi}{2} $ [closed]
Let $f :[0 \; 1] \rightarrow R $ be a continuous function satisfying
$ \sqrt{xy}(f(x) + f(y)) \leq 1 \; \forall x,y \in [0\; 1]$ ....(1)
Show that
$\int_0^1 f(t) dt \leq \frac{\pi}{2} $
.... (...
- 9
3
votes
3
answers
272
views
Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$
I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert ...
1
vote
0
answers
57
views
Monotonicity of root of hyperbolic function
For $\kappa \geq \alpha>0$ and $y \geq 0$, consider the following equation:
$$\sqrt{1-\frac{\alpha }{\kappa }} \tanh \left(y \sqrt{1-\frac{\alpha }{\kappa }}\right)=\tanh \left(y-\frac{\alpha }{2}\...
- 11
2
votes
1
answer
242
views
Modify a random variable to make its range Borel?
Let $X: \Omega\to{\mathbb R}$ be a random variable. Is it always possible to modify it (i.e. change the value of $X$ on a subset of $\Omega$ of zero measure) so that the range of $X$ is a Borel set?
...
- 637
0
votes
1
answer
248
views
Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 141
2
votes
1
answer
307
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 459
6
votes
1
answer
222
views
Maximizing $\prod_{i < j} \sin^2(\alpha_i - \alpha_j)$
For an integer $n \geq 2$, define $f_n(\alpha_0, \alpha_1, \ldots, \alpha_{n-1}) = \prod\limits_{0 \leq i < j < n}\sin^2\left(\alpha_i - \alpha_j\right)$ and $$M_n = \max\limits_{(\alpha_0, \...
- 1,625
5
votes
0
answers
135
views
Quantifying the degree of continuity of a function via perturbations
Let $f: \mathbb R \to \mathbb R$ be a measurable function. Define the perturbation operator $T_f$ on measurable functions $g: \mathbb R \to \mathbb R$ by
$$T_f (g)(x) := f(x + g(x)) - f(x).$$
Observe ...
- 6,323
4
votes
1
answer
525
views
On the definition of a continuous function
I remember once reading that "a continuous function can be loosely described as a function whose graph can be drawn without lifting the pen from the paper". We all know that this is not true....
- 113
0
votes
1
answer
248
views
Integral with inequality
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 573
3
votes
2
answers
206
views
Getting Wasserstein closeness from a derivative estimate
In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate:
$$
|\mathbb{E}_{\mu}(f)-\...
- 662
0
votes
1
answer
154
views
Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 4,115
6
votes
1
answer
405
views
Baire class $1$ functions and Baire's characterization theorem
Kechris in his Classical Descriptive Set Theory book gives the following definition (Definition 24.1) and characterization (Theorem 24.15) of Baire class $1$ functions:
Definition. Let $X,Y$ be ...
- 2,286
0
votes
1
answer
516
views
A problem of Fourier transform and Hölder condition
Suppose that $f$ is continuous on $[0,1]$. Thus, $f\in L^1(\mathbb{R})$ and its Fourier transform exists, as
$$ \hat{f}(\xi) := \int_\mathbb{R} e^{-2\pi i x \xi} f(x)dx, $$
which can also be written ...
- 419
5
votes
4
answers
362
views
Dual norm of a subspace of $\ell_\infty^3$
We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
- 1,879
8
votes
2
answers
432
views
Can an Osgood curve be almost everywhere differentiable?
It is known that you can “reparametrize” Osgood curves to make them almost-everywhere smooth curves (simply compose one after the Cantor function). However doing this breaks injectivity, stopping them ...
- 700
1
vote
1
answer
154
views
Dense properties of weighted Sobolev space define on $\mathbb{R}^n$
Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....
- 561
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
11
votes
2
answers
2k
views
Converse of mean value theorem almost everywhere?
Let $f: \mathbb R \to \mathbb R$ be a $C^1$ function.
We say a point $c \in \mathbb R$ is a mean value point of $f$ if there exists an open interval $(a,b)$ containing $c$ such that $f’(c) = \frac{f(b)...
- 6,323
1
vote
1
answer
60
views
Are there $f,h$ such that $h$ is Lipschitz, $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(h(t), x)$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial_t g(t, x)...
- 657
4
votes
2
answers
191
views
Reference request: "Tangent relation" in metric spaces
Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
- 63.1k
6
votes
2
answers
847
views
An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space
By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says:
“If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not ...
- 193
3
votes
0
answers
151
views
Is there a space of smooth functions dense in the domain of Coulomb-like potentials in dimension two?
Let $V : \mathbb{R}^2 \to \mathbb{R}$ be compactly supported, bounded away from the origin, and obey
$$ |V(x)| \lesssim r^{-\delta_0}, \qquad 0 < |x| \le 1, \qquad r : =|x|,$$
for some $0 < \...
- 481
5
votes
0
answers
417
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
5
votes
1
answer
2k
views
Question on an exercise from Terry Tao's blog
I've been reading Tao's An introduction to measure theory, a draft can be found here. An exercise from it is
Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely ...
- 95
-2
votes
1
answer
175
views
Simple closed form for $\int \lfloor x \rfloor dx$? [closed]
Wolfram Alpha claims there is no closed form in terms of standard funcions
for $\int \lfloor x \rfloor dx$ but we believe we found
simple closed form agreeing with experimental data.
Define $i_1(x)=x -...
- 25.4k
2
votes
0
answers
134
views
Extensions of Euler–Maclaurin formula
There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function $f(x)$ to be continuous, but there are several ways to extend the formula to ...
- 67
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
3
votes
0
answers
118
views
If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?
The question is as in the title.
Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
- 3,477
0
votes
0
answers
52
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 749
14
votes
3
answers
2k
views
How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...
- 1,326
0
votes
0
answers
317
views
What is the "best" good kernel?
A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied:
$\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$,
$...
- 3,062
0
votes
0
answers
122
views
A bound for the Bessel function of the first kind J_0
I have proved the following bound for the Bessel function of the first kind:
$$
J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2}
$$
which is
$$
|J_0(x)|\le \frac1{\sqrt[4]{1+x^2}}
$$
but I ...
- 258
2
votes
1
answer
192
views
Asymptotic analysis of an expression involving a Fox's H function
One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...
1
vote
0
answers
108
views
Existence of a smooth extension
In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface
$$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$
Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...
- 4,115
72
votes
9
answers
16k
views
Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)
Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
- 1,093
1
vote
1
answer
295
views
Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...
- 825
1
vote
0
answers
113
views
Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,115
5
votes
1
answer
319
views
Analytical form for the nuclear norm of an $n \times n$ matrix
I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is:
$$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
- 71
4
votes
1
answer
353
views
Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
- 41
4
votes
1
answer
367
views
Inequality with decreasing rearrangement function
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive ...
- 459
4
votes
1
answer
266
views
The difference between Baire 2 and 'effectively Baire 2'
In short: Baire 2 functions are often assumed to be given by a double sequence of continuous functions, thought this is not the exact definition. Does one need the Axiom of Choice (or related) to ...
- 4,359
10
votes
2
answers
493
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 ...
- 20.5k