Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,269
questions
2
votes
1
answer
373
views
$C^1$ extension with compact support
Knowing that $\omega\Subset\Omega\subset\mathbb{R}^2$ (compactly included) are two open and bounded sets with $C^2$ boundary, is it true that for any function $\phi_0:\overline{\omega}\to\mathbb{R},\ \...
0
votes
0
answers
132
views
Harmonic measure of a punctured disc
Let $D$ be a disc in $\mathbb{C}\cong\mathbb{R}^2 $ and $z_0$ a fixed point of $D$. Is the harmonic measure for $V=D\setminus\{z_0\}$ known? Any reference would also be welcome.
3
votes
1
answer
323
views
Main utility of the monotonicity formula for generalized surfaces
I hope not to be too simplistic.
I read about this monotonicity formula A question on the monotonicity formula for minimal submanifolds
I noticed that the monotonicity formula is often used in ...
1
vote
0
answers
179
views
A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$
Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
3
votes
1
answer
185
views
Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
5
votes
2
answers
347
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
7
votes
1
answer
337
views
A property of $C^2$ functions
Let $f\in C^2(\Bbb R^m), f\geq 0$, Hessian matrix of $f$ is upper bounded by some constant $C$. Do we have $|\nabla f|\leq \alpha \sqrt{f}$ for some $\alpha$, even if the Hessian matrix is degenerate?
3
votes
1
answer
354
views
What do convergent sequences of rational functions look like?
Let us consider the projective line over $\mathbb C$ equipped with a nice metric $\eta$ (like the Fubini-Study metric). We can define a metric $\mu$ on rational functions $f: \mathbb P^1 \to \mathbb P^...
7
votes
1
answer
996
views
An equivalent condition for differentiability almost everywhere?
Given a function $f \in L^1 (\mathbb R)$, define the roughness $R_f$ of $f$ at $x \in \mathbb R$ by
$$\DeclareMathOperator{\esssup}{\operatorname{esssup}}
R_f (x) := \limsup_{r \to 0+}\dfrac{r \...
10
votes
2
answers
972
views
On equibounded sequences in $L^\infty$
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup_{n \in \mathbb N} \|f_n\|_{L_\...
6
votes
1
answer
565
views
Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?
Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via
$$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
2
votes
0
answers
57
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
8
votes
1
answer
359
views
A dichotomy for the quadratic variation of differentiable functions?
For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula
$$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$
where the $\limsup$ is taken over all "partitions" ...
0
votes
0
answers
48
views
Surjectivity of the limiting operator
Consider the operator
\begin{eqnarray*}
K_{n} &:&L^{2}(0,1)\longrightarrow L^{2}(0,1)^{n}, \\
u(x) &\mapsto &A_{n}U_{n}(x)=A_{n}(u(\frac{x}{n}),u(\frac{x+1}{n}),...,u(%
\frac{x+n-1}{n})...
5
votes
3
answers
2k
views
Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
6
votes
1
answer
460
views
Can I find a bump function $\psi$ such that $\nabla\log\psi$ vanishes too?
Consider a bump function supported in the ball of radius $1$, that is $\psi:\mathbb R^n\to\mathbb R$ such that
$\ \psi(x)>0$ for $|x|<1$
$\ \psi(x)=0$ for $|x|\geq 1$
$\ \psi\in C^\infty$.
...
5
votes
1
answer
466
views
Upper bound an integral with exponential function
I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...
2
votes
1
answer
161
views
A question on subharmonic functions on the unit disc
I have the following question:
Let $u$ be a smooth subharmonic function on the unit disc $\mathbb{D}:=\left\{ z\in\mathbb{C}:\left|z\right|<1\right\} $.
Assume that $u=0$ on the boundary of $\...
10
votes
1
answer
846
views
Pointwise convergence imples uniform convergence in an infinite subset
I came upon this statement in a stack answer.
Statement :
If $f_n$ is a sequence of real valued functions (not necessarily continuous or measurable) on $[0,1]$ such that $f_n$ converges point-wise to $...
6
votes
1
answer
246
views
Decomposition of non negative Radon measure into $L^1$ and $H^{-1}$ functions
What is a reference for the following result (which appears to be well-known in measure theory)?
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely continuous ...
4
votes
1
answer
329
views
Irrationality of this trigonometric function
I'd like to prove the following conjecture.
Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$).
Then
$f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$
is irrational if ...
6
votes
1
answer
265
views
How to solve the following ODE with a parameter?
I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim_{|y|\to \infty}u(y) = 0.
\end{split}
\end{...
2
votes
1
answer
368
views
continuous function on the space of probability measures
Let $X$ be a compact space. Let $\mathcal{M}(X)$ be the space of all probability measures on $X$. Denote by $C(X)$ and $C(\mathcal{M}(X))$ the real continuous function on $X$ and $\mathcal{M}(X)$ ...
0
votes
1
answer
102
views
Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...
6
votes
2
answers
488
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 ...
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 $\Omega$}...
5
votes
2
answers
197
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
-1
votes
1
answer
160
views
Searching the roots of a self-consistent transcendental equation
I have the equation
$$M = c_1 + c_2M - c_3T\ln\left(\left|\frac{e^{(c_4M + c_5)/T}-1}{e^{(c_6M + c_5)/T}-1}\right|\right)$$
where $c_1, \dots, c_6$ are constants.
I am interested in the roots of $$M\...
6
votes
2
answers
493
views
When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?
If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
1
vote
1
answer
126
views
A subharmonic function with a growth property
Let $B=\left\{ \left(x,y\right)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\right\} $
be the unit ball in $\mathbb{R}^{2}.$
Can we construct a subharmonic
function $f:B\rightarrow\left[-\infty,0\right]$ such ...
2
votes
0
answers
100
views
Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?
I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...
0
votes
0
answers
104
views
Extension of super harmonic functions
The following is stated as an exercise in the "Classical Potential Theory" of Armitage and Gardiner (pg 195). Let $K$ be a compact of $\mathbb{R}^m$ ($m\geq2$) and $\Omega$ an open set ...
8
votes
1
answer
311
views
Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?
Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function.
If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in ...
0
votes
1
answer
175
views
Prove that $f(0+)=f(0)$ if $f \in R(\beta_1)$ [closed]
Let $\beta_1$ be a function defined by $$\beta_1(x)= \begin{cases} 0 & x \le 0\\ 1 & x >0 \end{cases} $$
Now we define $f(x)$ which is a bounded function on $[-1,1]$.
We need to how that $ ...
8
votes
1
answer
2k
views
What is the intuition behind the Kantorovich potential in optimal transport?
From what I currently understand, under certain conditions one may turn the usual Kantorovich problem - a minimisation problem in terms of measures into a maximisation problem in terms of functions. ...
1
vote
0
answers
95
views
$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]
For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is
$$
\lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
5
votes
0
answers
98
views
What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?
Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence.
Question: What is a minimal topological condition on $X$ ...
3
votes
0
answers
54
views
On Sobolev's inequality for weakly conformal maps
Suppose $u\in W^{2,p}(B^2,\mathbb{R}^n)$, $1<p<2$, is weakly conformal, that is
$$|u_x|=|u_y|,\quad u_x\cdot u_y=0$$
for almost every $(x,y)\in B^2$. Here $B^2$ is the unit open ball in $\mathbb{...
2
votes
1
answer
163
views
Random sequence with positive Lyapunov exponent?
Consider the following self-adjoint matrix
$A_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix},$ where $i$ is the imaginary unit and $X$ is a uniformly distributed random variable on some ...
22
votes
2
answers
790
views
Do equal integrals of $1/(1+x^a)$ imply equal measure?
Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$ with the property that $\int_{0}^{1} \frac{1}{1+x^a} ~d\mu(x) = \int_{0}^{1} \frac{1}{1+x^a} ~d\nu(x)$ holds for every exponent $a > ...
2
votes
0
answers
65
views
A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
0
votes
0
answers
58
views
The solutions of a system of differential equations
Let $P(x,y) = \frac{x}{y}^{\frac{x^2}{y-x}}$ for $x \neq y$ and using the proper limits $P(x,y)=e^{-x}$ for $x=y $, $P(x,y)=0$ for $x\neq0, y=0,$ and $P(x,y)=1$ for $x=0, y\neq0.$
Consider this system ...
5
votes
1
answer
259
views
Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$
Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
4
votes
2
answers
243
views
Tangent cone of null sets
Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that
$$ \liminf_{r \to 0+} r^{-1}...
1
vote
0
answers
139
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 ...
4
votes
1
answer
419
views
Finiteness of Hausdorff measure of balls
Let $(X,d)$ be an arbitrary metric space and let $\Bbb B(x,r)$ denote the closed ball with center $x \in X$ and radius $r>0$. For $p\geq 0$, let $H^p$ denote the $p$- dimensional Hausdorff measure. ...
0
votes
3
answers
163
views
Hausdorff convergence in bounded set preserves the volume
I was wondering if Hausdorff convergence relates to the volume of the converging sets. In particular, let $(C_n)$ be a sequence of closed sets contained in a bounded, closed set $Q$. Assume that $|C_n|...
3
votes
0
answers
139
views
Chebyshev Equioscillation Theorem in presence of extra conditions
Let $P_\ell$ be polynomials of degree $\ell$. For $f \in C[0,1]$, define the minimax error $E_\ell(f) = \min_{p \in P_\ell} \max_{x \in [0,1]} |f(x) - p(x)|$. We know that for the above scenario the ...
0
votes
1
answer
341
views
What's the condition to prove the equicontinuity?
Let $K: I\times I\rightarrow \mathbb{R}$ be a scalar kernel, where $I=[0,1]$, and $a: I \rightarrow (0,+\infty)$ an $L^1(I)$ function.
For $t_1,t_2\in I$, define
$$I_{t_1,t_2}=\int_{0}^{1} \left |\...
0
votes
0
answers
77
views
A property of the Hilbert transform involving the cotangent function
A lemma of a paper by T. Elgindi and I.-J. Jeong (Arch. Rational Mech. Anal. 235 (2020) 1763–1817, Lemma 2.2) states the following:
Let $g(z)=\operatorname{sgn}(z)k(|z|^\alpha)$ with $k$ smooth and $k(...