All Questions
Tagged with reference-request real-analysis
419 questions
6
votes
1
answer
728
views
An $L^1$ function but (really) no better?
Question: For a smooth, bounded domain $\Omega\subset \mathbb R^d$, does there exist a function $u\in L^1(\Omega)$ such that
$u\not\in L^\Phi(\Omega)$ for any Orlicz space $\Phi$?
For the definition ...
- 5,380
2
votes
0
answers
150
views
Closeness of a rational approximation
What is
$$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N}
|2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$
where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to ...
- 128k
0
votes
1
answer
191
views
Calculating derivatives of arbitrary-order at an operator's root
Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of ...
- 583
9
votes
2
answers
354
views
Asymptotics of a quadratic recursion
Consider the sequence defined by
\begin{align}
c_0 &{}= 1 \\
c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}.
\end{align}
How can you prove that it has the following asymptotics ...
3
votes
1
answer
155
views
Is it a named result (or consequence thereof) that decreasing functions integrable against $e^{kx}$ decay faster than $e^{-kx}$?
Apologies if this question is too basic for MO.
I think it should be the case that
for any decreasing $f \colon [A,\infty) \to [0,\infty)$ and $k \geq 0$, if $\int_A^\infty f(x) e^{kx} \, dx < \...
- 708
7
votes
0
answers
240
views
Sard's theorem for superharmonic functions: less regularity required?
A function $f:\mathbb{R}^d \to \mathbb{R}$ must be at least $C^d$ in order to guarantee in general that
$$\{\phi\in \mathbb{R}|\,\exists x\in \mathbb{R}^d:\,f(x)=\phi,\,(\nabla f)(x)=0\}$$
is a zero-...
- 1,461
3
votes
0
answers
205
views
Uniform limit of pointwise limits of continuous functions
Let $X$ be topological spaces, $Y$ a metric space and $(f_n)_{n\in\mathbb{N}}$ a sequence of functions, with $f_n:X\rightarrow Y$ pointwise limit of continuous functions for each $n\in\mathbb{N}$. ...
- 2,286
1
vote
1
answer
129
views
Lebesgue Hausdorff Banach theorem for Baire class $1$ functions on $\mathbb{R}^\omega$
A theorem by Lebesgue, Hausdorff and Banach says the following (Kechris' Classical Descriptive Set Theory, p. 192):
Let $X$ be a separable metrizable space and $f: X \rightarrow \mathbb{R}$ be a $\...
- 2,286
4
votes
0
answers
140
views
Is the existence of Banach limits independent of ZF+DC?
Is the existence of Banach limits independent of ZF+DC?
Assuming this is known, where can I find a proof?
- 6,453
1
vote
1
answer
164
views
Two trigonometric integrals: looking for a transformation
I have two integrals of trigonometric functions and I would like to ask:
QUESTION. Is there a transformation rule (or general principle) to show this equality?
$$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
- 43.2k
2
votes
0
answers
98
views
Has this "optimal constrained transport" notion of convergence of measures been named and/or studied?
Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$.
Fix $L \geq 1$. I will say that $\mu_n$ converges in ...
- 708
1
vote
0
answers
110
views
Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
0
votes
0
answers
99
views
Does $\sum_{m=0}^{\infty} \left|c_m g(x)^{2m+1}\right|$ converge absolutely to an integrable function?
Consider the integral
\begin{equation}
\int_{0}^{t}J_1(f(t)-f(s))\mathop{ds}=\int_{0}^{t}\sum_{m=0}^{\infty} c_m (f(t)-f(s))^{2m+1}\mathop{ds},
\end{equation}
such that $J_1$ is the Bessel function of ...
- 79
3
votes
3
answers
196
views
Non convex optimization problem in $W_0^{1,2}$
Let $0< \alpha \ll 1$. I'm trying to minimize $\int_0^\pi |f'|^2 dx$ over the functions $f \in W_0^{1,2}([0,\pi])$ (or at least find "good" lower bound in terms of $\alpha$) such that ...
- 393
2
votes
0
answers
108
views
Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
- 739
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
4
votes
1
answer
246
views
Is $C_n$ infinitely log-convex?
A sequence $a_n$ is called log-convex if $\mathcal{L}(a_n):=a_{n+1}a_{n-1}-a_n^2\geq0$ for all $n$; it is infinitely log-convex provided that all the iterates $\mathcal{L}^k(a_n)$ are still log-convex,...
- 43.2k
2
votes
0
answers
131
views
Taylor series with less than differentiability
I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite
$$
F^1 := \lim_{x\to \infty} x \cdot f^0(x)
$$
Then I consider $f^1(x) := x \cdot ...
- 335
1
vote
0
answers
76
views
Error estimates for orthogonal polynomial approximation
tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials?
There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
- 19
7
votes
1
answer
166
views
Asymptotics of truncated logarithm on a cricle
Consider $f_n(x) = \min_{|z|=x} \Re \sum_{j=1}^{n} \frac{z^j}{j}$, a real function of positive variable $x>0$.
I am interested in lower bounds on $f_n(x)$. Specifically, I ask: what lower bounds ...
- 14.6k
2
votes
1
answer
212
views
Covering the surface below a convex function
Is it possible to find the smallest positive real number $c$ (or at least the smallest positive integer $c$) such that the following result holds for all functions $f$ satisfying some conditions?
Let ...
- 3,153
0
votes
0
answers
177
views
On connectedness of the complement
In the application of Runge type theorems on the approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has ...
- 411
3
votes
0
answers
106
views
The behavior of an integral related to the inward normal vector near a point of the boundary of a domain
Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit
$$
\lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz)
$$
where
$...
- 6,367
1
vote
0
answers
77
views
Divergence between random variables after transformation
Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
- 19
1
vote
1
answer
75
views
Looking for a reference for an extension problem of function
Let $\Omega_1$, $\Omega_2$ be bounded,convex, open domains with smooth boundary in $\mathbb{R}^2$ and $\overline\Omega_1\subset\Omega_2$. Suppose we are given a $C^1$ function $f:\overline\Omega_1\cup(...
- 13
4
votes
1
answer
273
views
How bad can pointwise convergence in $C$ be?
$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let
\begin{equation}
f(u):=\frac{2u^2}{1+u^4},
\end{equation}
so that the function $f\colon\R\to\R$ is continuous, $0\...
- 128k
0
votes
1
answer
171
views
How to compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{[-1,1]^n}\exp[2\pi i(\theta_1 v_1.x+\theta_2v_2.x)]d^nx d\theta_1d\theta_2$
Let $\mathbf{v}_1, \mathbf{v}_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral:
$$\int_{-\infty}^{ \infty} \int_{-\infty}^{\infty}
\int_{[-1,1]^n}
e(\...
- 3,625
29
votes
2
answers
4k
views
Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
- 43.2k
4
votes
1
answer
101
views
A narrower dichotomy for the quadratic variation of differentiable functions?
$\newcommand\P{\mathcal P}$A "partition" $P$ (of the interval $[0,1]$) is a finite sequence $(t_0,\dots,t_n)$ such that $0=t_0<\cdots<t_n=1$; then the mesh of $P$ is $\|P\|:=\max_{1\le ...
- 128k
8
votes
1
answer
388
views
A dichotomy for the quadratic variation of differentiable functions?
For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula
$$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$
where the $\limsup$ is taken over all "partitions" ...
- 128k
6
votes
2
answers
513
views
Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
- 335
1
vote
0
answers
46
views
Boundary estimates for Neumann derivative of solution to Laplacian equation with Dirchlet boundary data
Let $\Omega \subset \mathbb{R}^n$ be a smooth domain. Consider the following Laplacian equation with Dirichlet boundary condition.
\begin{equation}
\begin{cases}
\Delta u=0\quad &\mbox{in $\Omega$}...
- 1,350
7
votes
1
answer
352
views
Tight upper bounds on trigonometric polynomials
According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
- 10.4k
1
vote
0
answers
145
views
Integrability conditions imply existence of potential
I'm looking for a proof of the following well-known theorem:
If $f$ is a continuously differentiable vector field in a simply connected region $G\subset \mathbb{R}^n$ which satisfies the ...
- 2,160
3
votes
0
answers
216
views
integration by parts on a Lipschitz domain as $\epsilon\to 0$
For a fixed, bounded, smooth domain $\Omega\subset \mathbb R^d$ and any $u\in W^{1,1}(\Omega)$ with trace $u|_{\partial\Omega}=g\in L^1(\partial\Omega)$ one can prove that
$$
\lim\limits_{\epsilon\to ...
- 5,380
0
votes
1
answer
306
views
Regularity properties of conditional distributions
Let $(X,Y)\in\mathbb{R}^n\times\mathbb{R}^m$ be a pair of random variables with joint density $p(x,y)$. I am interested in the regularity properties of the conditional densities $p(y|x)$ and $p(x|y)$ (...
- 23
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 128k
1
vote
2
answers
140
views
Variant of Parthasarathy's minimax theorem
Does there exist a variant of Parthasarathy's minimax theorem [1] that relaxes the assumption that the spaces $X$ and $Y$ are $[0,1]$?
[1] https://en.wikipedia.org/wiki/Parthasarathy%27s_theorem
- 119
2
votes
0
answers
65
views
Reference request for type of specific integral equation in two variable:
Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...
- 41
7
votes
1
answer
312
views
Real analyticity of continuous function via restriction to analytic curves
Suppose $X\subset \mathbb R^n$ is an irreducible real analytic sub-variety (i.e. the set of solutions of a system $f_1=\ldots=f_k=0$ with $f_i$ analytic)
Let $x\in X$ be a point and let $F: X\to \...
- 14.3k
5
votes
3
answers
163
views
Reference for a Grünwald–Letnikov-type definition of the $n$-th derivative of a function
Let $U\subset\mathbb R$ be an open set. Let $n\in\mathbb N$ and suppose that $f\in\mathcal C^n(U)$, i.e. that $f$ is $n$-times continuously differentiable on $U$. The $n$-th derivative of $f$, denoted ...
- 1,326
2
votes
0
answers
210
views
Theory of mollifiers on the boundary of a $C^2$ domain
Let $D\subseteq\mathbb{R}^d$ be a nice but not smooth domain, somewhere between Lipschitz and $C^2$. I am looking for a reference on the theory of mollifiers and regularization for functions on $\...
- 171
1
vote
1
answer
224
views
Sum of negative roots of a $5^{th}$ degree monic polynomial
Let $f(x)$ be a $4^{th}$ degree monic polynomial say $f(x) = x^4 + a_1x^3+a_2x^2+a_3x+a_4$ with the property that $a_1<0, a_4>0$ and $a_2<a_3$. They by Descartes' rule of signs we can ...
- 199
9
votes
2
answers
1k
views
A tricky integral to evaluate
I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
- 43.2k
2
votes
0
answers
72
views
Product of Besov and Lorentz functions
Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
17
votes
3
answers
1k
views
Decoupling a double integral
I came across this question while making some calculations.
QUESTION. Can you find some transformation to "decouple" the double integral as follows?
$$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
- 43.2k
0
votes
0
answers
353
views
Inverse of the Riesz potential of a measure
Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$,
$$
I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy.
$$
Assuming $f$ ...
- 4,034
3
votes
0
answers
238
views
How to denote a partial derivative?
This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation .
When there are two ...
- 6,891
11
votes
2
answers
539
views
Reference request: A multidimensional generalization of the fundamental theorem of calculus
$\newcommand\R{\mathbb R}$Let $f\colon\R^p\to\R$ be a continuous function. For $u=(u_1,\dots,u_p)$ and $v=(v_1,\dots,v_p)$ in $\R^p$, let
$[u,v]:=\prod_{r=1}^p[u_r,v_r]$;
$u\wedge v:=\big(\min(u_1,v_1)...
- 128k
1
vote
0
answers
81
views
Hardy maximal function on the torus
A few years ago I asked a reference about the Hardy maximal function on the flat torus. Mateusz Kwaśnicki kindly answered in a comment, and confirmed my conviction that basically everything which is ...
- 3,425