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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. ...
Amr's user avatar
  • 1,117
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-...
OverLordGoldDragon's user avatar
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 ...
Isaac's user avatar
  • 3,477
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 ...
ebg's user avatar
  • 33
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 ...
Isaac's user avatar
  • 3,477
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 $$...
Shaq155's user avatar
  • 459
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 ...
Guido Li's user avatar
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} \...
RotemBZ's user avatar
  • 23
2 votes
1 answer
260 views

Squaring a semi-convergent series

Let $S=\sum_{n=1}^\infty a_n$, be a semi-convergent series with $T=\sum_{n=1}^\infty a_n^2 < \infty$ and $\sum_{n=1}^\infty |a_n|=\infty$. Under which conditions are the following formulas valid? ...
Vincent Granville's user avatar
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}...
Iosif Pinelis's user avatar
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\...
inoc's user avatar
  • 339
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 \...
ABIM's user avatar
  • 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 ...
co.sine's user avatar
  • 403
2 votes
1 answer
273 views

Is it always possible to partition $[a,b]\times[c,d]$ into disjoint blocks $D_{ij}$ s.t. $\left.f\right|_{D_{ij}}$ is bijective?

Consider the function given by $f:[a,b]\times[c,d]\to[0,1]^{2}$ such that $0\leq a < b \leq 1$, $0 \leq c < d \leq 1$. Moreover, we do also have that $f\in C^{1}([a,b]\times[c,d],[0,1]^{2})$ and ...
user avatar
2 votes
0 answers
144 views

Does this geometric PDE have a solution?

Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes. Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
Asaf Shachar's user avatar
  • 6,741
2 votes
1 answer
677 views

Lipschitz continuity of an implicit function

Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ and $z=F(x,y)$ is Lipschitz continuous. Assume that for any $x\in\mathbb R^d$, there is a unique $y$ such that $F(x,y)=0$....
zbh2047's user avatar
  • 601
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: $$...
user111097's user avatar
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})} \...
mohi's user avatar
  • 859
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 ...
Leonardo's user avatar
  • 405
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\...
MTP's user avatar
  • 21
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 $\...
user avatar
2 votes
1 answer
144 views

Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. $A_{m,1}...
BCLC's user avatar
  • 247
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 ...
Till's user avatar
  • 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 (...
ABIM's user avatar
  • 5,405
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.\...
Ahmed Tori's user avatar
2 votes
1 answer
689 views

Partitions of an interval

This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there. Specifically, consider "partitions" ...
Emilio Pisanty's user avatar
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 ...
James E Hanson's user avatar
2 votes
1 answer
101 views

Convergence of energy of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by \begin{equation} f_r(x):= \begin{cases} ...
user84068's user avatar
  • 169
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 ...
Skarr's user avatar
  • 29
2 votes
2 answers
218 views

Convergence for a non-linear second order difference equation

In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation $$ u_0,u_1>0, \qquad \forall n\in \mathbb N, \; u_{n+2}=a\ln(1+u_n)+b\ln(1+u_{n+1}) $$ with ...
Paul's user avatar
  • 1,503
2 votes
1 answer
154 views

Is the optimum of this problem convex in the constraint parameter?

Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that $|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
Asaf Shachar's user avatar
  • 6,741
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 ...
Riku's user avatar
  • 839
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, ...
0xbadf00d's user avatar
  • 167
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\...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
190 views

What is the smallest dimension that allows finding $n$ points at distances $|x_i-x_j|^{\delta/2}$, where $0<\delta<1$, and $x \in \mathbb{R}^n$?

Let $x_1,\cdots,x_n \in \mathbb{R}$, are there $\xi_1,\cdots,\xi_n \in \mathbb{R}^s$, such that $|x_i-x_j|^{\delta}=||\xi_i-\xi_j||^2$, $0<\delta<1$, what is the smallest $s$ to guarantee the ...
Tanya Vladi's user avatar
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 ...
user521337's user avatar
  • 1,209
2 votes
1 answer
157 views

Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?

Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?
user avatar
2 votes
2 answers
494 views

Polynomial approximation (Weierstrass theorem) with bounds

Consider the closed interval $[0,1]$ and let $f \in C[0,1]$. Let $g$ be a real valued function on $[0,1]$ such that $g \leq f$. Suppose $g = f$ at atmost finitely many points. Does there exist a ...
Rahul Sarkar's user avatar
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 ...
user118240's user avatar
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 $...
sharpe's user avatar
  • 721
2 votes
1 answer
107 views

Lower bounds on translates of a function over a compact set

Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. Let $K\subset\mathbb{R}$ be a compact set. I would like to compute (or at least lower bound) the following: $$ \inf_{\theta\ne\theta'\...
tim622's user avatar
  • 45
2 votes
1 answer
154 views

Smooth conditional expectation with nonsmooth "reverse"

I am looking for a concrete example of the following: $(X,Y)$ are real-valued random variables such that: $E[Y|X]$ is smooth $E[X|Y]$ is discontinuous Even better, I'd like to see an example where ...
user19200's user avatar
2 votes
1 answer
497 views

Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as $$W(\sigma,...
Vincent Granville's user avatar
1 vote
1 answer
300 views

Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
NancyBoy's user avatar
  • 393
1 vote
0 answers
120 views

Natural candidates for super-half-exponential which limit to half-exponential function from above

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However super-half-exponentials (functions whose composition grows ...
VS.'s user avatar
  • 1,826
1 vote
0 answers
92 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user avatar
1 vote
0 answers
102 views

Proving that a quantity is positive (Gaussian density and Gaussian CFD)

$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{1/2}}$ Hi everyone, I am interested in the following problem: Let consider the heat equation problem: $$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
NancyBoy's user avatar
  • 393
1 vote
1 answer
110 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
user109584's user avatar
1 vote
0 answers
71 views

Continuous injection of metric ball into Euclidean ball

This is a follow-up to this post. Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \...
ABIM's user avatar
  • 5,405
1 vote
0 answers
416 views

When does a proper Zariski closed set have measure zero with respect to a conditional measure?

Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure. Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
Ron's user avatar
  • 61