All Questions
13,944 questions
61
votes
1
answer
5k
views
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...
- 12.9k
24
votes
3
answers
4k
views
Self-dual normed spaces which are not Hilbert spaces
Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-...
- 4,461
0
votes
0
answers
38
views
Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
5
votes
0
answers
608
views
What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?
1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
- 6,367
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
0
votes
0
answers
68
views
Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.
If ...
1
vote
0
answers
176
views
If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?
Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$.
Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
- 4,461
2
votes
0
answers
117
views
Existence of a sequence of real numbers
Let
$$g_{c;k}(z):=\frac{2 (c-z-1)^{k+2}}{(k+1) (k+2)}+\frac{1}{2} (-c+z+2)^2 z^k+\frac{-2 c (k+2)+4 k+6}{(k+1) (k+2)}+\frac{2z}{k+1}.$$
Do there exist $c\in(1,3/2)$ and a sequence $(a_k)_{k=0}^\infty$ ...
CommunityBot
- 1
3
votes
1
answer
220
views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 498
2
votes
0
answers
191
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
7
votes
2
answers
841
views
Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?
In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
- 128k
0
votes
1
answer
97
views
Numerically bounding a Exponential-Trigonometric Integral [closed]
I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer.
I have tried decomposing into Riemann sum and ...
- 188k
0
votes
1
answer
57
views
Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 128k
5
votes
0
answers
77
views
What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
2
votes
0
answers
124
views
dimensionality reduction of Markov chains
Suppose that $M$ is a time-homogeneous (and, for simplicity, stationary) Markov chain on $d$ states, which induces the probability measure $P$ on paths of length $n$. I seek a Markov chain $M'$ on $d'&...
- 6,541
3
votes
1
answer
228
views
"Essential values" of a function at a point?
Recall that the essential range $\operatorname{ess.im} f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, ...
- 6,215
1
vote
2
answers
2k
views
Bounding the norm of the Laplacian of the gradient of a function having Lipschitz continuous Hessian
It seems that the following claim is true, but I did not manage to prove it neither to find a reference.
Claim Let $f:\mathbb R^p\to\mathbb R$ be a three times differentiable function such that its ...
CommunityBot
- 1
45
votes
1
answer
2k
views
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...
2
votes
2
answers
154
views
Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 60.5k
1
vote
0
answers
87
views
Supremum of sums of functions in $L^1$ taking random signs
Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$.
Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
- 393
5
votes
1
answer
774
views
Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
CommunityBot
- 1
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,215
2
votes
0
answers
75
views
Pullback by surjective submersion is injective?
Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
- 261
2
votes
1
answer
106
views
Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property
I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
CommunityBot
- 1
4
votes
1
answer
298
views
Oscillation of monotone real-analytic function
Let $f:(a,\infty)\rightarrow \mathbb{R}$ be a real-analytic and strictly monotone function. I have been wondering how much this function can "oscillate". Namely, can we always find a ...
- 128k
2
votes
1
answer
547
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
CommunityBot
- 1
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 24.2k
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
0
votes
0
answers
78
views
What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
1
vote
1
answer
124
views
$d(x,y) = \min\{|x_1−y_1|+|x_2−y_2|, 1−|x_1−y_1|+|x_2−(1−y_2)|\}$ defines a metric on $[0,1)\times[0,1]$? [closed]
For $x,y \in [0,1)\times[0,1]$, let $d(x,y)$ be the minimum of $|x_1−y_1|+|x_2−y_2|$ and $1−|x_1−y_1|+|x_2−(1−y_2)|$. Prove or disprove that $d$ is a metric.
I was unable to find a counterexample to ...
- 109k
7
votes
0
answers
619
views
Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 6,367
5
votes
2
answers
149
views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
- 12.6k
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
5
votes
2
answers
373
views
Weak Archimedean property instead of Archimedean property
We say that a sequence $(z_n)$ of real numbers is a modulated Cauchy sequence, whenever there exists a function $\alpha:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
|z_i-z_j| \le \frac{1}{k} \quad ...
- 6,367
3
votes
0
answers
100
views
How to compute the partial derivatives of this function?
For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
- 1,399
1
vote
0
answers
98
views
Equivalence of Sobolev norms for smooth functions with compact support
Let $f\in C^\infty_c([0,1]^n)$, then we can extend it to a $1$-periodic smooth function $\tilde f$. We define the fourier transform (series) of $f$ ($\tilde f$):$$
\hat f(\xi):=\int e^{2\pi i x\cdot \...
- 435
0
votes
1
answer
231
views
Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 60.5k
6
votes
1
answer
194
views
The most even partition of $\mathbb R$ into measure dense sets
Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every ...
- 6,215
3
votes
1
answer
344
views
Asymptotic behavior of a recursion
Let $x_n(0)=1$,
$$
x_n(N+1) = \frac{1}{N+1}\sum_{k=0}^N \sum_{j=1}^n x_j(k)x_{n+1-j}(N-k) + \frac{10}{N+1} x_{n+1}(N) , \quad\quad N\ge 0 .
$$
So the recursion is on $N$, and at each level, we compute ...
- 128k
1
vote
1
answer
173
views
Taut string algorithm and TV-minimization equivalence
Given real numbers $y_i's$, consider the following convex optimization problem:
$$
\min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|.
$$
The paper A Direct Algorithm for 1D ...
CommunityBot
- 1
1
vote
1
answer
102
views
Does $C^{k,s-k}$ function with lipschitz lower order derivatives give a certain bound on the Taylor remainder?
Let $\Omega \subseteq \mathbb{R}$ be open (not necessarily an interval). Let $ s > 0$ and $k \in \mathbb{N}_0$ be such that $s \in (k, k+1]$. Suppose that
$f \colon \Omega \to \mathbb{R}$ is an ...
- 498
4
votes
0
answers
147
views
Weakly compact sets forced to contain $0$
Let $E$ be an infinite-dimensional real normed space and let $K\subset E$ be a weakly compact set such that, for each $\varphi\in E^*\setminus \{0\}$, there exists a unique $\tilde x\in K$ such that
$$...
- 1,724
2
votes
1
answer
150
views
Proof that superlinearly convergent sequence converges faster than linearly convergent sequence
Given real sequences $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$, both converging to the same limit $A$ and such that $|a_n-A|\neq 0$ and $|b_n-A|\neq 0$ for every $n$ sufficiently large, we ...
- 6,215
3
votes
2
answers
118
views
Does the derivative of the antiderivative of a BV function $f$ agree with $f$ at all but countably many points of differentiability?
Let $f: (a, b) \to \mathbb R$ be a function of bounded variation, and write
$$F(x) := \int_a^x f(t) \, dt$$
for the antiderivative. Is it true that at all but countably points of differentiability of $...
- 60.5k
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
0
votes
0
answers
118
views
Find the maximum of an expression under the logconcave assumption
Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 105k
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
2
votes
1
answer
670
views
Integral on level sets
Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
CommunityBot
- 1
0
votes
0
answers
121
views
How to find the inverse of this linear integral operator?
Let $f(x): \mathbb{R}^d \rightarrow \mathbb{R}$ be a function that decays ``fastly enough'' at infinity.
We can define the following linear operator
$$L[f](x):= \int_{\mathbb{R}^d} d^d y \, \frac{f(y)}...
- 1