Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

Filter by
Sorted by
Tagged with
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},\ \...
Bogdan's user avatar
  • 1,330
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.
M. Rahmat's user avatar
  • 411
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 ...
Son Gohan's user avatar
  • 215
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 ...
Alexander Kuleshov's user avatar
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) = ...
Pritam Bemis's user avatar
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 ...
Pritam Bemis's user avatar
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?
zhangwei's user avatar
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^...
Asvin's user avatar
  • 7,648
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 \...
Nate River's user avatar
  • 4,822
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_\...
Nate River's user avatar
  • 4,822
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+\...
Ali's user avatar
  • 4,077
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{...
Ali's user avatar
  • 4,077
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" ...
Iosif Pinelis's user avatar
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})...
Gustave's user avatar
  • 545
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 (...
spaceman's user avatar
  • 575
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 $|x|\geq 1$ $\ \psi\in C^\infty$. ...
tituf's user avatar
  • 311
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 ...
Quicky2357's user avatar
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 $\...
AndewUK's user avatar
  • 23
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 $...
Kr Dpk's user avatar
  • 203
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 ...
user175203's user avatar
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 ...
nervxxx's user avatar
  • 207
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{...
Jacob Lu's user avatar
  • 903
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 $\mathcal{M}(X)$ ...
user119197's user avatar
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)...
Harish's user avatar
  • 251
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 ...
Vu Thanh Tung's user avatar
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 $\Omega$}...
student's user avatar
  • 1,320
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<...
Iosif Pinelis's user avatar
-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\...
Essa Ibrahim's user avatar
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(\...
apanpapan3's user avatar
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 ...
Hana_a_student's user avatar
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 ...
David Manolis's user avatar
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 ...
M. Rahmat's user avatar
  • 411
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 ...
Louis Deaett's user avatar
  • 1,513
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 $ ...
ThirstForMaths's user avatar
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. ...
Nate River's user avatar
  • 4,822
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)}...
Ad_M's user avatar
  • 11
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 $X$ ...
user141240's user avatar
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{...
MathPhys's user avatar
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 ...
Kung Yao's user avatar
  • 192
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 > ...
Xiaosheng Mu's user avatar
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 $...
M. Rahmat's user avatar
  • 411
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 ...
moonlight's user avatar
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 ...
Keefer Rowan's user avatar
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}...
Longyearbyen's user avatar
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 ...
kodlu's user avatar
  • 10.1k
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. ...
John D's user avatar
  • 185
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|...
Daniele's user avatar
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 ...
Rahul Sarkar's user avatar
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 |\...
Motaka's user avatar
  • 291
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(...
Jesús A. Álvarez López's user avatar

1
27 28
29
30 31
106