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13 votes
2 answers
1k views

Is the exponential function the sole solution to these equations?

Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
user avatar
4 votes
0 answers
672 views

Proofs of the second fundamental theorem of calculus

I am referring to the following version of the theorem, in the setting of the Lebesgue integral. Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
coudy's user avatar
  • 18.7k
27 votes
3 answers
2k views

Kasteleyn's formula for domino tilings generalized?

It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$. Kasteleyn's ...
T. Amdeberhan's user avatar
3 votes
1 answer
940 views

What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?

Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
Henry.L's user avatar
  • 8,071
5 votes
1 answer
248 views

Approximation of monotone Sobolev functions

Let $f\in W_{loc}^{1,2}(\mathbb R^2)$ be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of $f$ in a precompact open set are attained at the boundary)....
Dimitrios Ntalampekos's user avatar
3 votes
0 answers
228 views

Sub-multiplicative function in expectation or pointwise? [closed]

Consider the function that satisfies $$ \mathbb{E}[f(X)f(Y)]\leq \mathbb{E}[f(XY)],$$ where $X\in\mathbb{R}$ and $Y\in\mathbb{R}$ are Gaussian random variables with mean $0$ and variance $1$, and ...
Richard Simmons's user avatar
1 vote
1 answer
133 views

Every $W^{1,p}$ has a representative in ACL

Let $\Omega:=(0,1)^n$ and define $ACL_i(\Omega)$ as the set of all Borel functions $u:\Omega\to\mathbb{R}$ such that $$ t\mapsto u(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n) $$ is $AC$ for a.e. $(x_1,\...
Mizar's user avatar
  • 3,146
1 vote
0 answers
161 views

level sets portrait near a critical point

Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an isolated critical point of $f$. I am looking at the local level sets diagram near $O$ from topological ...
user94090's user avatar
5 votes
0 answers
431 views

Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
user avatar
2 votes
0 answers
67 views

On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$ \begin{equation} (\...
5th decile's user avatar
  • 1,461
3 votes
2 answers
435 views

A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again. Take the vector space of infinitely ...
Delio Mugnolo's user avatar
5 votes
1 answer
654 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
Bumblebee's user avatar
  • 1,093
4 votes
1 answer
197 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
Bobby Grizzard's user avatar
1 vote
0 answers
120 views

Interpolation functional for BV spaces?

Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
João Ramos's user avatar
1 vote
0 answers
69 views

Norm-averaging reference request

(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...
charlestoncrabb's user avatar
10 votes
2 answers
352 views

Two elementary inequalities for real-valued polynomials

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different,...
Andrés E. Caicedo's user avatar
1 vote
1 answer
321 views

A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
godelian's user avatar
  • 5,902
3 votes
0 answers
165 views

Extreme derivatives in Baire class 1

In the 1994 volume of "Differentiation of Real Functions" A. Bruckner poses the following problem (p.41): "Find necessary and sufficient conditions on a continuous function $F$ that its Dini ...
Damian Reding's user avatar
5 votes
1 answer
680 views

When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$ I don't know of any references or methods for this -- not even for $x=1$, for which the ...
Clark Kimberling's user avatar
1 vote
1 answer
168 views

Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$. If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$, $$ A_wf(y)=\int_y^1\frac{f(x)w(x,y)}{\sqrt{x^2-...
Joonas Ilmavirta's user avatar
2 votes
1 answer
268 views

Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\...
Iosif Pinelis's user avatar
3 votes
2 answers
431 views

A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality: $$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$ I have just ...
Włodzimierz Holsztyński's user avatar
-1 votes
1 answer
226 views

separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
Rajesh D's user avatar
  • 698
3 votes
0 answers
373 views

An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as $$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
Iosif Pinelis's user avatar
7 votes
2 answers
1k views

Intermediate value for a vector-valued function

Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f_1(x), ..., f_n(x)\}$ with $x\in[0,1]^n$, where the $f_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the ...
MthQ's user avatar
  • 41
0 votes
1 answer
96 views

Optimal covering with finite subcollection of open sets

This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for ...
user82538's user avatar
2 votes
2 answers
411 views

Asymptotics of the derivatives of analytic functions

Are there sources that treat questions like the following ones? Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x\...
Iosif Pinelis's user avatar
33 votes
4 answers
2k views

Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
Jochen Wengenroth's user avatar
5 votes
2 answers
957 views

Dirichlet's approximation only using prime power as denominator

I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...
Subhajit Jana's user avatar
7 votes
3 answers
905 views

A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it? $$\frac{d^n}{dx^n}f(x) = \lim_{h \...
Halbort's user avatar
  • 1,129
4 votes
1 answer
559 views

How to construct i.i.d. standard normal random variables on $\Omega = [0, 1]$ with the Lebesgue measure

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Then we can find independent random variables $X_1, X_2, X_3, \dots$ defined on $(\Omega, \...
user avatar
3 votes
0 answers
237 views

Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has: (i) the ...
Salvo Tringali's user avatar
6 votes
1 answer
1k views

About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
Henry.L's user avatar
  • 8,071
12 votes
1 answer
239 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}...
Stephan Kulla's user avatar
3 votes
1 answer
265 views

Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants. Let $Q(R) := Q(R^2,R) = B(0, R) \times [-R^...
Juhana Siljander's user avatar
3 votes
0 answers
166 views

Monotone version of one-dimensional Whitney extension theorem

Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...
Igor Belegradek's user avatar
2 votes
2 answers
232 views

Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation $$ u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1} $$ for a Hölder continuous coefficient $a(x,t)$ satisfying $$ 0<C_o \le a(x,...
Juhana Siljander's user avatar
1 vote
0 answers
42 views

Error bounds for approximation with dyadic sums of polynomials

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables? In the 2-dimensional case the ...
Manfred Weis's user avatar
  • 13.2k
5 votes
0 answers
364 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
Name's user avatar
  • 51
7 votes
1 answer
501 views

Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and $$ u_B := \frac1{|B|}\int_B u \, dx. $$ The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
Juhana Siljander's user avatar
7 votes
1 answer
609 views

$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
Guo's user avatar
  • 71
6 votes
1 answer
489 views

Henstock, Differentiation under the integral sign

Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...
green113's user avatar
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
  • 452
5 votes
0 answers
252 views

Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential $$ I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy. $$ The classical Hardy-Littlewood-Sobolev ...
Juhana Siljander's user avatar
4 votes
1 answer
383 views

Horizontal Sobolev space on Carnot group

This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and ...
Nikita Evseev's user avatar
8 votes
0 answers
433 views

Heisenberg group: function without vertical derivative

Let $\mathbb H$ be Heisenberg group with vector fields $$ X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t $$ and $U\subset\mathbb H$ is an open set. I am ...
Nikita Evseev's user avatar
10 votes
1 answer
700 views

Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by $$ I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ By the classical Hardy-Littlewood-Sobolev theorem ...
Juhana Siljander's user avatar
4 votes
1 answer
1k views

Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $...
Hugo Chapdelaine's user avatar
3 votes
4 answers
934 views

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...
Mirko's user avatar
  • 1,375
26 votes
5 answers
8k views

Proof that no differentiable space-filling curve exists

Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists? Or piecewise differentiable? Must every continuous space-filling curve be nowhere ...
Joseph O'Rourke's user avatar

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