All Questions
746 questions
3
votes
2
answers
203
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 ...
- 7,127
3
votes
1
answer
274
views
Function square-integrable
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...
3
votes
1
answer
1k
views
Reference request: interpolation of Hölder spaces
On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space ...
- 1,263
2
votes
1
answer
143
views
Roots of rational function
Sorry, I asked a similar question yesterday which contained a mistake in the question posed, here is the real question.
Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property ...
- 73
2
votes
1
answer
206
views
Bound for zero-crossings of heat equation
I am considering the following problem.
Let $\mathcal{P}$ the classical heat-diffusion problem:
$$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
- 393
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
1
answer
186
views
Local equality of functions implies global equality?
The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
- 1,117
2
votes
1
answer
154
views
Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$
Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
- 835
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
2
votes
2
answers
667
views
Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
- 39
2
votes
1
answer
167
views
On a characterization of inward unit normal vector
Let $D$ be a smooth domain of $\mathbb{R}^d$. Let $\partial D$ denote the boundary of $D$. We denote by $B(x,r)=\{y \in \mathbb{R}^d \mid |y-x|<r\}$ the Euclidean ball centered at $x$ with radius $...
- 721
2
votes
1
answer
301
views
A question about pushforward measures and continuous Borel isomorphisms
It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
- 33
2
votes
1
answer
150
views
When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?
First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
- 850
2
votes
1
answer
315
views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions?
I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 311
2
votes
1
answer
255
views
On the infimal convolution of two norms on $\mathbb R^n$
$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation*}
K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1),
\end{equation*}
\begin{equation*}
M:=M_{n,...
- 128k
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 835
2
votes
1
answer
307
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 459
2
votes
1
answer
105
views
Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define
$$
\begin{split}
K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\
M_n(a,t) &:= ...
- 6,853
2
votes
0
answers
470
views
Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?
If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
- 10.1k
2
votes
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
2
votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
2
votes
1
answer
142
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
- 23
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
1
answer
157
views
Inequality with decreasing rearrangement and non-decreasing function
This question is a continuation of the question here.
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$...
- 459
2
votes
2
answers
1k
views
Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions
Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum_{i=1}^m g_i(x) h_i (y)$ where $g_i, h_i \in C^{\infty}(\mathbb{R})$ are ...
- 33
2
votes
1
answer
450
views
Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
- 167
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
2
votes
0
answers
232
views
Is an orthogonal projection of a Lipschitz domain still a Lipschitz domain?
Let $\mathcal{X}\subseteq\mathbf{R}^n$ be a Lipschitz domain, i.e., for each $x\in\partial\mathcal{X}$, there exists a radius $r_x>0$ and a Lipschitz continuous function $F^x:\mathbf{R}^{n-1}\to\...
- 21
2
votes
0
answers
197
views
Orthogonality relation in $L^2$ implying periodicity
Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...
- 405
2
votes
1
answer
437
views
If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set
If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
- 167
2
votes
1
answer
104
views
Limit of biggest share of the pie
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest ...
- 48.9k
2
votes
3
answers
1k
views
on the set of numbers generated by integer linear combination of two real numbers.
Let $b > a > 0$ be two real numbers. I am interested in the set of numbers
$X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$.
What ...
- 29
2
votes
1
answer
328
views
Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
Question 1.
How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$?
Question ...
- 839
2
votes
2
answers
255
views
Do we have a name for this space?
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Consider the class
$$
\mathcal{F}=\{f\in L^{1}(\Omega):\exists C>0 \text{ s.t. } \int_{U}|f|\leq C\sqrt{|U|},\text{ for any }U\subset \Omega.\...
- 21
2
votes
2
answers
952
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 201
2
votes
1
answer
389
views
Intersections of algebraic surfaces with hypercubes of a $d$-dimensional grid
This is a follow-up question, to a question I asked earlier.
See Algebraic curve intersecting square-grid.
Consider $n^d$ unit hypercubes in $d$-dimensional Euclidean space
tightly packed in the ...
- 479
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
1
answer
193
views
A question on the partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Is the following statement true ?
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^na_{ij} >0
$$
Any reference or comment on this is ...
- 171
2
votes
1
answer
289
views
On semi-discrete Wasserstein distance
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$, where $\nu$ has a bounded support. Consider the $2-$Wasserstein distance below:
$$...
- 345
2
votes
1
answer
433
views
bounding the absolute value of a trigonometric polynomial
Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
\...
- 859
2
votes
1
answer
113
views
Continuous inclusion of metric spaces of smaller capacity
If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is
$$
\kappa_X(\epsilon)\leq \...
- 5,405
2
votes
2
answers
190
views
One-Sided Analyticity Condition Guarantees Analytic Function?
Let $f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying:
$f$ is differentiable infinitely many times in $(0,\infty)$, and has a right-derivative of any order at $0$.
$f$ satifsfies the ...
- 403
2
votes
2
answers
4k
views
a limit of the laplace transform and its derivative
If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
2
votes
1
answer
130
views
Uniformly Converging Metrization of Uniform Structure
This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...
- 12.5k
2
votes
2
answers
2k
views
convergence of the infima of convex functions
Can one give a reference to a result like this:
If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then $\displaystyle\inf_{\mathbb{R}...
- 128k
2
votes
3
answers
3k
views
dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
- 1,835
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 167
2
votes
1
answer
162
views
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
- 1,209
2
votes
1
answer
324
views
Uniform estimation of an integral involving a Hölder-continuous function
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\...
- 339