# Tagged Questions

The real-analysis tag has no wiki summary.

**4**

votes

**2**answers

105 views

### How to find an ODE with prescribed terminal values?

Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...

**7**

votes

**0**answers

79 views

### A monoid-structure on pairs of interlacing polynomials

Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace ...

**1**

vote

**0**answers

23 views

### Writing a function as a sum of functions of bounded diameter

This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial.
Fix $D>0$. A function $f:\mathbb ...

**6**

votes

**0**answers

226 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 ...

**3**

votes

**1**answer

80 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 ...

**5**

votes

**2**answers

163 views

### Continuous map from connected subset of R^n to one of the real zero of an odd degree polynomial whose coefficients are polynoms of the variables

Let take a real multivariate polynomial $P(x_1, \ldots, x_n, y)$ such as the degree of P relatively to the variable $y$ is odd. Thus, for each $X = x_1,\ldots,x_n \in\mathbb{R}^n$, the univariate ...

**8**

votes

**1**answer

120 views

### Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?
For ...

**6**

votes

**1**answer

270 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 ...

**0**

votes

**0**answers

26 views

### Bounds on the moments of truncated sub-gaussian random variables

If $X$ is a centered sub-gaussian random variable, then there exists a constant $c$ such that
$$
\mathbb{P}[|X|>t] \leq \exp(1-ct^2)
$$
for all $t\geq 0$. Moreover, we know that the normalized ...

**4**

votes

**2**answers

361 views

### Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds.
(Cf. discussion on p. 45.)
Definition
Let $E$ and $F$ be two Banach spaces together with a plain subset ...

**8**

votes

**1**answer

578 views

+100

### Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$

The following question has a 500 points bounty on MSE that soon comes to an end, and no answer
as expected was given yet. How would a professional solve the problem? Wish you succcess.
...

**2**

votes

**1**answer

606 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**-3**

votes

**0**answers

59 views

### Question 7F From S. Willard, *General Topology* [closed]

Can someone help me with 7F from Willard? In part two :
$\mathbf{7}$F. Functions to and from the plane.
The facts presented here for the plane will be proved in more generality for ...

**0**

votes

**0**answers

25 views

### Nonzero partial derivative and injectivity [closed]

Let $p,q,x,y:(a,b)\to\mathbb{R}$ be $C^1 ((a,b))$ functions. Knowing that the function:
$F_v:(a,b)\times (a,b)\to\mathbb{R};\ v\in (0,1),\ F_v ...

**65**

votes

**9**answers

18k views

### solving f(f(x))=g(x)

This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...

**0**

votes

**0**answers

97 views

### What are some good mathematics books for a recent graduate who is considering revisiting PhD research in the future? [closed]

Basically, I've decided to take a break from academic mathematics after starting a PhD and withdrawing for personal reasons. I am considering maybe revisiting doing a PhD in future years but at the ...

**5**

votes

**0**answers

153 views

### Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...

**6**

votes

**1**answer

160 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 ...

**-1**

votes

**0**answers

19 views

### Unknown convergence of series of non-negative terms [duplicate]

It's not hard to construct a series of non-negative terms whose convergence is not known if we base its construction on conjectures (the twin-prime conjecture for instance). For pure curiosity, I'm ...

**0**

votes

**0**answers

14 views

### Convergence in $L^1$ for characteristic functions [migrated]

Consider $f\in L^1(I,I)$ where $I=[0,1]$ and $ \langle f, g\rangle =\int fg $. For any given $\frac{m}{n}\chi_{A}$ where $\frac{m}{n}$ rational and $A$ an subinterval in $I$, how would I show ...

**2**

votes

**1**answer

448 views

### A result attributed to Whitney

One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...

**3**

votes

**0**answers

74 views

### Regularity in PDE theory

I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not?
Let $U$ be connected,open and bounded in $\mathbb{R}^n$ ...

**1**

vote

**2**answers

349 views

### Can we extend an a.e. Lipschitz map defined on a closed subset of R^N to the whole space so that it is still a.e. Lipschitz?

I have the following question. Let $A$ be a metrically oriented $n$-dimensional subset of $\mathbb{R}^N$ and $f$ a continuous map from $A$ to $\mathbb{R}^M$. We know that $\operatorname{Lip} f < ...

**0**

votes

**0**answers

79 views

### The Lebesgue measure of the low level sets of the two-dimmension Fourier transform of a compactly supported function

Let $f\in {{L}^{1}}\left( {{\mathbb{R}}^{2}} \right)$ be a density function with the support $\operatorname{supp}\left( f \right)\subset \left[ a,b \right]\times \left[ c,d \right]$. Denoted by ...

**-1**

votes

**1**answer

41 views

### Glueing smooth functions give a smooth function if reparametrized [closed]

Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and
$$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 ...

**8**

votes

**3**answers

448 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**0**

votes

**0**answers

48 views

### A multivariate polynomial question

Split $\{0,1\}^n$ into $S_0[n],S_1[n]$ with $S_0[n]\cup S_1[n]=\{0,1\}^n$ while $S_0[n]\cap S_1[n]=\emptyset$.
For every $n$, let $f(x_1,\dots,x_n),g(x_1,\dots,x_n)\in\Bbb R[x_1,\dots,x_n]$ be ...

**1**

vote

**1**answer

99 views

### Interesting property of analytic functions

Let $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$, be an analytic application, such that: $f(t)=0\Longleftrightarrow\ t=t_0$.
Is it true that there is an analytic function $g:(t_0-\varepsilon, ...

**5**

votes

**0**answers

105 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 ...

**2**

votes

**1**answer

36 views

### Convergence in an infinite matrix

Let $\omega$ be the first infinite ordinal, and let $A$ be a real $(\omega+1)\times(\omega+1)$-matrix, that is $A$ is a map $A:(\omega+1)\times(\omega+1) \to \mathbb{R}$.
Suppose that $A$ has the ...

**2**

votes

**0**answers

48 views

### Weak Morrey Spaces

As is well known, Morrey spaces are widely used to
investigate the local behavior of solutions to second order elliptic partial differential
equations. Recall that the classical Morrey spaces ...

**13**

votes

**1**answer

344 views

### Positive roots of a polynomial

Let $a_i>0$, $i=1,\dots,n$, and put $\overline{a}:=\frac{1}{n}\sum_{i=1}^n a_i$. Assuming not all $a_i$'s are equal, take
$$
p(x):=\sum_{i=1}^n a_i (a_i-\overline{a})\prod_{k=1,\dots,n\;k\neq i} ...

**6**

votes

**1**answer

250 views

### how wiggly is a generic level set?

Typical level sets of smooth real-valued functions are manifolds, so they cannot be fractals. If we coarse grain a bit though, sometimes we get space-filling behavior, eg. every point could be within ...

**0**

votes

**0**answers

63 views

### Orthogonal complements of intersections of closed subspaces

Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$.
$\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...

**0**

votes

**1**answer

155 views

### The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads.
Let's rewrite the product in a way ...

**0**

votes

**1**answer

82 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**-1**

votes

**0**answers

76 views

### Interpolation between strongly convex functions

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^d$ and $t\in [0,1]$ it follows
$$ f(tx+(1-t)y) \leq t f(x) + (1-t) f(y) ...

**3**

votes

**0**answers

56 views

### Maximizing the discrepancy in Jensen's inequality for a certain function

Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter.
Define
$$
D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}}
...

**10**

votes

**1**answer

323 views

### Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $j'_{n+1/2,1}$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$.
...

**9**

votes

**1**answer

144 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 ...

**3**

votes

**1**answer

70 views

### random odes adapted solution

Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode
$$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$
Where $b$ is a bounded continuous function (not ...

**5**

votes

**3**answers

314 views

### Introductory texts to mathematics [closed]

I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would ...

**8**

votes

**1**answer

134 views

### Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...

**2**

votes

**1**answer

46 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**54**

votes

**18**answers

6k views

### Proofs of the uncountability of the reals.

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see a variant proof (as it did not use the diagonal ...

**15**

votes

**4**answers

2k views

### Why is there no Borel function mapping every countable set of reals outside itself?

A choice function maps every set (in its domain) to an element of itself. This question concerns existence of an anti-choice function defined on the family of countable sets of reals. In an answer to ...

**2**

votes

**1**answer

243 views

### Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature.
Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...

**4**

votes

**1**answer

69 views

### Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.
Let $a_k >0$ be an increasing sequence ...

**48**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**1**

vote

**1**answer

152 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 ...