All Questions
5,628 questions
7
votes
3
answers
662
views
Asymptotics for $\int\exp( -x t / \log t)dt$
What is the asymptotic growth rate of $$f(x) = \int_e^\infty e^{ - x t / \log t} dt$$ as $x \to 0$?
As an example of what is meant by "growth rate" consider $$g(x) = \int_e^\infty e^{-x t} ...
- 457
0
votes
1
answer
130
views
Explanation for Tauberian theorems for Laplace transform
I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow :
Let $F : [0,\infty) \to \mathbb{R}$ of ...
- 393
15
votes
1
answer
693
views
Fourier's proof of reality of all roots of Bessel function $J_0(x)$
In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real.
I want to ask if there is a modern version of this proof exist in literature?
If someone ...
- 790
155
votes
4
answers
18k
views
Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
- 39.1k
10
votes
1
answer
961
views
Ruling out the existence of a strange polynomial II
This is a refinement of my question asked earlier, which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same ...
- 26.9k
5
votes
0
answers
214
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
8
votes
2
answers
655
views
An extension problem
Let $\Omega$ be an open subset of $\mathbb R^n$ for $n \geq 2$, and $p \in \Omega$.
Let $k$ be a positive integer. Suppose that $f: \Omega \setminus \{p\} \to \mathbb R$ is in $C^k$, and $\lim_{x \to ...
- 6,223
1
vote
1
answer
120
views
Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
- 227
3
votes
2
answers
191
views
Is the inequality $\sum_{i} p_i \frac{a_i}{b_i} \leq \frac{\sum_{i} p_i a_i}{\sum_{i} p_i b_i}$ true?
Let $p_i \in [0,1]$ and $\sum_{i} p_i = 1$, and furthermore let $a_i$ and $b_i$ be positive real numbers. Is the inequality
$$
\sum_{i} p_i \frac{a_i}{b_i} \leq \frac{\sum_{i} p_i a_i}{\sum_{i} p_i ...
- 33
2
votes
2
answers
382
views
Asymptotics of an integral requested
Given an integer $n\geq2$, consider the following integral
$$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
QUESTION. Is this true? It appears to be so.
$$\lim_{n\...
- 43.2k
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
3
votes
1
answer
499
views
Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?
The integral is
$$\DeclareMathOperator{\dm}{d\!}
\int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}\dm x.
$$
Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational ...
- 375
6
votes
2
answers
503
views
Computing a limit on the unit sphere: Riemann Lebesgue?
Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align*}
\lim_{|\xi|\to \infty}
\int_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma_{d-1}(w)
= \int_{\mathbb{S}^{d-1}}u(w)d \sigma_{d-1}(...
- 1,101
1
vote
1
answer
179
views
For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$
Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that
\begin{align*}
0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
- 19
3
votes
1
answer
211
views
Other expansion for positive Taylor expansion
I was thinking of the following problem. Let $f$ be a Taylor expansion and $a_k$ the associated coefficients,
$$\forall x\in\mathbb{R},~f(x)\triangleq\sum_{k=0}^\infty a_kx^k.$$
Let suppose that we ...
- 393
74
votes
15
answers
18k
views
$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential
The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
- 24.7k
2
votes
0
answers
170
views
finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
- 4,143
17
votes
3
answers
2k
views
Is every Schwartz function the product of two Schwartz functions?
A Schwartz function on $\mathbb R^d$ is a $C^\infty$ function, such that all differentials of order $k \ge 0$ decay faster than any polynomial. They include the class $C^\infty_c(\mathbb R^d)$ of ...
- 475
2
votes
1
answer
281
views
Global control of locally approximating polynomial in Stone-Weierstrass?
Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\...
- 463
-2
votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 109
2
votes
1
answer
189
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 75
7
votes
0
answers
150
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.5k
0
votes
0
answers
28
views
Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
0
votes
1
answer
92
views
Continuous selectors of a continuous multifunctin on a compact metric space
I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
...
- 79
14
votes
1
answer
417
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
41
votes
6
answers
9k
views
"Long-standing conjectures in analysis ... often turn out to be false"
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
3
votes
1
answer
128
views
Weaker version of the lemma of K.L. Chung
Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds:
$$...
- 132
0
votes
0
answers
63
views
Computing the eta invariant of a rather contrived operator on the circle
For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...
- 111
5
votes
1
answer
117
views
Between BV and Baire 2
My question is about functions of bounded variation (BV) on the reals.
On one hand, Helly's selection theorem provides (fairly restrictive) conditions under which a sequence of BV-functions has a sub-...
- 4,359
2
votes
0
answers
319
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
- 333
0
votes
2
answers
319
views
Representation of continuous, monotone, concave functions
Is there a characterization of all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying:
$f(0)=0$
$f$ is monotonically increasing
$f$ is concave
My intuition is that $f$ should admit ...
- 5,405
1
vote
1
answer
143
views
$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
- 101
7
votes
2
answers
419
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 217
1
vote
1
answer
157
views
On the additive property of the subdifferential of lower semicontinuous functions
Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by
$$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
- 299
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
2
votes
0
answers
120
views
On the integer of the form p^a q^b closest to a given integer N
If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
- 69
2
votes
0
answers
116
views
Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
- 469
2
votes
0
answers
216
views
When do these ODE have positive solutions?
Consider the ODE \begin{equation} x'' + q(t) x = 0 \end{equation} in the unit interval $(0,1)$, with a potential function $q(t) = 4\pi^2 - \frac{Ct}{(1 - t)^2}$ depending on a positive constant $C >...
- 5,048
6
votes
1
answer
210
views
Is the Hardy Littlewood “minimal function” comparable to the original function in $L^1$ norm?
Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the minimal function $m_\varepsilon f$ by
$$m_\varepsilon f(x) := \inf_B \frac1{|B|} \int_B |f| ,$$
where the infimum is taken over ...
- 6,223
3
votes
3
answers
1k
views
Positivity of a one-variable rational function
Let's consider the $1$-variable rational function
$$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$
Numerical evidence convinces me of the truth of the following.
QUESTION. Can you ...
- 43.2k
3
votes
1
answer
185
views
Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 513
17
votes
1
answer
580
views
Aperiodic monotile in $\mathbb{R}$
Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
- 48.9k
3
votes
3
answers
427
views
Quantitative analytic continuation estimate for a function small on a set of positive measure
The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In ...
- 324
2
votes
0
answers
100
views
Are analytic solutions for the Navier-Stokes equations sufficient?
Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space.
However, I am wondering, whether it is possible to consider just analytic ...
- 749
17
votes
2
answers
2k
views
"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?
Recently, I encountered this problem:
"Given a sequence of positive number $(x_n)$ such that for all $n$,
$$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$
Find the limit $\lim_{n \rightarrow \infty} \...
- 327
38
votes
26
answers
57k
views
Text for an introductory Real Analysis course.
Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
2
votes
0
answers
174
views
Product of marginals absolutely continuous with respect to a Borel probability measure
Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
- 3,599
1
vote
1
answer
191
views
Concentration inequality for square roots
Given a sequence of (not-necessarily-iid) real-valued random variables $X_n$ that converge to $a\in\mathbb{R}$ in probability, suppose we have an exponential concentration inequality of the form
$$
P(|...
- 13
5
votes
2
answers
223
views
Continuous functions on $[0,1]^\omega$ and a product lower bound
I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology).
The map $f:X\to [0, 1]$ given by $(x_i)\mapsto \prod x_i$ is well-defined and Borel but not ...
- 53
1
vote
1
answer
265
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 825