All Questions
6,014 questions
9
votes
1
answer
339
views
A topological characterisation of a.e. continuity
We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
- 6,213
3
votes
0
answers
154
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 393
6
votes
1
answer
210
views
Is the Hardy Littlewood “minimal function” comparable to the original function in $L^1$ norm?
Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the minimal function $m_\varepsilon f$ by
$$m_\varepsilon f(x) := \inf_B \frac1{|B|} \int_B |f| ,$$
where the infimum is taken over ...
- 6,213
1
vote
1
answer
264
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 825
2
votes
1
answer
170
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
- 393
2
votes
1
answer
112
views
Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave
I am struggling with the following problem. Let $f$ be a real smooth function:
strictly convex on $(-\infty,0)$,
strictly concave on $(0,\infty)$,
strictly increasing.
For $\sigma>0$, how can one ...
- 393
4
votes
1
answer
296
views
The maximal difference between a function and translates of itself
Note: We view the sphere $S^1$ as $[0,1]$ with the endpoints identified, and equip it with its usual addition structure, and Lebesgue measure.
Question: Does there exist an absolute constant $C > 0$...
- 6,213
1
vote
0
answers
143
views
Analyticity of a function in two complex variables
Let $f$ be a function defined on $\mathbb{C}^2$ given by
$$
f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
- 11
6
votes
1
answer
379
views
An inequality for a concave function $f(x)=x^{p/2}$
Assume that $p\in(1,2]$, $a,b\ge 1$, $b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
- 333
3
votes
4
answers
353
views
What real distributions solve $f'=0$? [closed]
I mean specifically real-valued Schwartz distributions on the real line. That is linear functionals on $C^{\infty}_c(\mathbb{R})$ continuous in the canonical LF topology. My question is, what are ...
- 11.1k
1
vote
0
answers
269
views
Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$
Let $f(x) = \log(\cosh(x))$, and define the kernel density:
$$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}...
- 391
6
votes
1
answer
828
views
Twisted Riemann sums
Let $f(x)$ be a real-valued Riemann integrable function supported in $[0,1]$ with range in $[0,1]$. Let $\alpha$ be irrational. Consider the weighted Riemann sum
$$S_N:=\frac{1}{N}\sum_{k=1}^Nf\left(\...
- 315
3
votes
3
answers
550
views
Solving interval problems without outer measure
Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ?
Problem 1
If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
0
votes
1
answer
118
views
For any smooth function $f$ on $[0,1]$, do we have $\lVert P_N f \lVert_{1} \leq \lVert f \rVert_1$ for $P_N$ defined by Fourier expansion?
Let $C^\infty[0,1]$ be the space of periodic smooth functions on $\mathbb{R}$ with the period $1$. It is well-known to be a Frechet space with the uniform convergence of all derivatives.
Also, $\{ e^{...
- 3,477
1
vote
1
answer
151
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
- 391
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 825
0
votes
1
answer
33
views
Sign Regularity of a Density Kernel with Convexity Properties
(Asking a final time in a new question because the previous version had insufficient conditions, as pointed out in the answer there.)
Define the densities:
$$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\...
- 391
7
votes
0
answers
204
views
Permutations which change the value of a convergent series
I'm interested in the following combinatorial problem: What is a necessary and sufficent condition on a permutation $\sigma : \mathbb{N} \rightarrow \mathbb{N}$, so that there exist a summable ...
- 71
-3
votes
1
answer
638
views
Analysis I, simpler proof of Tao's construction of the integers [closed]
In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
In the language of set theory, what we are doing here is starting with the ...
- 23
1
vote
1
answer
312
views
Showing that the infimum is a minimum
Let $V > 0$ and let $\Phi(\cdot)$ be the standard normal CDF.
Consider the infimum of
$$f(x_1, x_2,x_3, p_1, p_2, p_3) := p_1 \Phi(x_1) + p_2 \Phi(x_2) + p_3 \Phi(x_3)$$
with respect to $x_1, x_2, ...
- 113
1
vote
1
answer
111
views
Monotone likelihood ratio of a family of densities with convexity property
(Asking again in a new question because the previous version had insufficient conditions, as pointed out in the answer there.)
Define the densities:
$$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\theta)\...
- 391
4
votes
3
answers
369
views
Non-negativity of a complicated function
Show that $f(x)\ge 0$ for $0\le x \le 1$, where:
$$f(x) = \arccos(x)^2 -8x(5x^2-2) \sqrt{1-x^2}\arccos(x)+36 x^8-112 x^6+93 x^4-17 x^2$$
The endpoints are $f(0)=\pi^2/4$ and $f(1)=0$. Plotting ...
- 391
3
votes
1
answer
135
views
Recover an $L^1$ integrand by partial differentiation
Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by
$$F(x,y) = ...
- 41
2
votes
0
answers
319
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
- 333
4
votes
1
answer
305
views
Holomorphic extension of the Fourier transform of a measure
If an entire holomorphic function $f(z)$ is given by the analytic continuation of $f(x)=\int_\mathbb{R}e^{-ix\xi}\,d\mu(\xi)$ with a finite Borel measure $\mu$ on $\mathbb{R}$, then $g(x):=\int_\...
- 63
3
votes
1
answer
120
views
How to establish regions of convexity/concavity of a ratio of exponential polynomials?
Problem:
Let $f\colon \mathopen[0,1\mathclose] \to \mathbb{R}$ be defined as
$$
f(x) = \frac{e^{\rho x}-1}{e^{\rho x}-1+e^{\rho (1-\gamma x)}-e^{\rho (1-\gamma) x}}
$$
where $\rho$ and $\gamma$ are ...
- 33
1
vote
1
answer
89
views
Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers
This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
- 573
1
vote
1
answer
133
views
Monotone likelihood ratio of a family of densities with compact support
Define the family of densities:
$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta)\big)\Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}, \...
- 391
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:
$$\sqrt{1-\frac{\alpha }{\kappa }} \tanh \left(y \sqrt{1-\frac{\alpha }{\kappa }}\right)=\tanh \left(y-\frac{\alpha }{2}\...
- 11
0
votes
0
answers
145
views
Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
2
votes
0
answers
63
views
Evaluation of a certain area
I asked a version of this question on Math Stack Exchange 6 days ago, but without any responses: The area of a certain region
I am interested in evaluating the area of the region defined by
$$A_{L_1, ...
- 26.9k
0
votes
1
answer
254
views
Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of ...
- 825
1
vote
2
answers
188
views
Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?
Let $d \in \mathbb N^*,p \in [1, \infty]$ and $T>0$. Let
$$
F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0}), t \mapsto F_t
$$
be measurable. I would like to ask if there is a measurable function ...
- 825
2
votes
1
answer
215
views
Asymptotics for oscillatory integral
Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$
$$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
6
votes
1
answer
608
views
Total positivity, log-concavity and Pólya frequency
I am not familiar with the definition of total positivity. I am not sure about the link between log-concavity and total positivity.
In a paper On Variation-Diminishing Integral Operators of the ...
- 393
1
vote
1
answer
2k
views
Product of Dirac delta function
The following equation may be meaningful, but how can we make it well-defined
$$\delta(x-a)\cdot\delta(x-b)=0$$
Question: How do we defined this equation? Or more broadly define product between ...
- 13
1
vote
0
answers
67
views
Estimating commutator of Fourier integral
Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by
$$ s_n(g)(x) = \frac{nf(x)}{4\pi i} \int_{\mathbb R^2} e^{\frac{in\vert x-y\...
0
votes
0
answers
149
views
Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477
2
votes
0
answers
325
views
Examples of RKHS that are "classical"
Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs?
It is easy to construct example of RKHSs by applying the ...
- 21
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
- 833
1
vote
1
answer
294
views
What is the exact description of the homogeneous Besov space $\smash{\dot{B}}^0_{1,1}(\mathbb{R})$?
The Besov space is defined briefly in Wikipedia and I looked for a number of references to find some information on the homogeneous Besov space $\smash{\dot{B}}^0_{1,1}(\mathbb{R})$.
However, ...
- 3,477
5
votes
0
answers
162
views
Closed formula for series $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$
What can be said about $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$ (for $|x|>1$ and $|y|>1$ and $x\neq y$)?
Is there a kind of closed formula for this?
By comparing to the geometric series, this sum ...
- 139
1
vote
1
answer
175
views
Analyzable functions and accelero-summation
Is there a complete and rigorous, yet concise, definition of what an analyzable function is, along with the related notion of accelero-summation, both in the sense of Écalle? All of the definitions I ...
- 4,985
1
vote
1
answer
258
views
What is the measure of two sets which partition the reals into subsets of positive measure?
This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.
(In ...
- 63
1
vote
1
answer
86
views
The distance between a collection of points and a sequence of sets
Fix $m \geq 2$, and consider a sequence of sets
$$
J_m^{(n)} = \left\{ \frac{2}{mn}+\frac{i-1}{n}\right\}_{i=1}^n.
$$
For any collection of $m-1$ points $x_1,...,x_{m-1} \in (0, 1)\cap \mathbb{Q}$, ...
- 1,087
-3
votes
1
answer
167
views
Is there a simple function similar to exp? [closed]
As far as I know exp have such properties:
$f'(x) >0$
$f''(x) >0$
$\lim_{x \to -\infty}f(x)=0$
$\lim_{x \to +\infty}f(x)=\infty$
$f(x)f(-x)=1$
Let's say f(x) comply such rules.
The closest I ...
- 1
2
votes
2
answers
127
views
Is there a restriction on the structure of the set of points where all derivatives of a $C^\infty$ real function are 0? [duplicate]
Let $f$ be an infinitely differentiable real function and let $Z(f)$ denote the set of points on which all derivatives of $f$ vanish. It is not hard to describe an $f$ such that $Z(f)$ is any ...
- 481
2
votes
1
answer
294
views
Are the jumps of a càdlàg function "summable"?
This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen ...
- 708
2
votes
1
answer
475
views
A continuous injection from $[0,1]$ to $\mathbb{R}^2$
Consider the continuous and injective mapping
\begin{eqnarray*}
\varphi:[0,1] &\rightarrow& \mathbb{R}^2, \\
t &\mapsto& (x(t),y(t)),
\end{eqnarray*}
such that $x(0)<x(1)$, and
\...
- 105
6
votes
2
answers
254
views
Eigenvalues of polynomials of two matrices
In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting.
Let $\chi(M)$ denote the characteristic ...
- 37.7k