Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
3,095
questions
-1
votes
0answers
11 views
When is weighted average decreasing?
Suppose we have sequences $\{w_n(x)\}_{n\in\mathbb{N}}$ and $\{f_n(x)\}_{n\in\mathbb{N}}$ that are both strictly decreasing in $x$. Further, we have $w_n(x)> 0,f_n(x)\geq 1\quad \forall n\in\mathbb{...
2
votes
1answer
76 views
Duality form of $L^q$ norm, without assumption that $\int fg$ defined?
The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden.
Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space.
Let ...
5
votes
0answers
64 views
Points where singular sum is small
We consider $x_1,..,x_N$ points in the plane $\mathbb{R}^2.$
We define the sum
$$F(x):=\frac{1}{N^2}\sum_{i=1}^N \sum_{j \neq i} \vert x_i-x_j \vert^{-2}.$$
I am looking for a statement of the ...
4
votes
1answer
110 views
Elliptic estimates for self-adjoint operators
Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$
Let $T$ be a densely defined and closed operator ...
8
votes
2answers
203 views
Recognizing Lipschitz functions up to change of target metric
Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.
Is ...
3
votes
1answer
121 views
Isoperimetric inequality for analytic functions on an annulus
Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
2
votes
1answer
104 views
Proof of a discrete isoperimetric inequality
The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions:
$$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...
-1
votes
0answers
19 views
norm inflation for cubic NLS
I'm trying to understand the norm inflation result for the cubic NLS with initial data in negative Sobolev spaces, specifically, Theorem 1.1 in this paper. I'm have the trouble in verifying one ...
-1
votes
1answer
58 views
Minimal covering sets of continuous endomorphisms
For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ covers $\text{End}(X)$ if for every $f\in \...
2
votes
1answer
69 views
Hardy-Littlewood-Sobolev for “componentwise product” of Riesz kernels
Let $d\in\mathbb N$ and $0<\alpha<d$. Define the Riesz kernel $K_\alpha(x):=|x|^{\alpha-d}$, and the associated convolution operator
$$K_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$
The ...
1
vote
0answers
36 views
A Bessel-like integral
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
3
votes
0answers
74 views
Building strong topologies on the space of continuous functions
Let $\{K_k\}_{k=1}^{\infty}$ be a compact exhaustion on $\mathbb{R}^n$ (ie $\bigcup_{k=1}^{\infty} K_k = \mathbb{R}^n$). Equipe each $X_k:=\{f \in C(\mathbb{R}^n):\, \operatorname{supp}(f)\subseteq ...
20
votes
1answer
336 views
Is every function $f: \mathbb R \to \mathbb R$ differentiable at at least one point when restricted to some everywhere dense subset of $\mathbb R$?
I was doing some fairly simple research a few hours ago and I almost asked a similar question with the word continuous instead of differentiable in the title, but then I found this question asked by ...
1
vote
0answers
26 views
Independence of variables in curvilinear coordinate systems
Let $U$ be a connected open subset of $\Bbb{R}^n$, and let $(\xi_1,\dots,\xi_n)$ be a curvilinear smooth ($C^\infty$) coordinate system on $U$. Suppose $1\leq k<n$. A smooth function $f:U\...
6
votes
1answer
94 views
Equivalence of antiderivative in L1 sense and in the usual sense
We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that:
$$
\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0
$$
where $\Vert \cdot \Vert_1$ is the $L_1$ ...
3
votes
0answers
62 views
Approximation of Lipschitz functions
Let $X$ be an open set in $\mathbb{R}^N$ and call $C^{1,1}(X)$ the set of functions $X\rightarrow \mathbb{R}$ that are $C^1$ with Lipschitz first derivatives.
I have a $C^{1,1}$ function $f$ and a ...
0
votes
0answers
254 views
+50
A function fitting method: can we recover binary step functions?
A function fitting method
Data $(p_i,a_i) \in \mathbb{T}^m\times\{-1,1\}$, $i = 1,2,...n$.
Let $C_{\lambda}(f) = \sum\limits_{i=1}^{n}(f(p_i)-a_i)^2 + \|f\|_{L^2}^2 + \lambda\|\nabla^kf\|_{L^2}^2$ ...
1
vote
0answers
54 views
Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$
I'm currently trying to work with the non-local operator given by
$$
(-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy,
$$
where $f :\mathbb ...
7
votes
1answer
135 views
When is the cut-locus normal coordinate collared
Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold, $p \in M$ be fixed and let $C_p$ be the cut-locus of $p$.
Other than when $M$ is non-positively curved (in which $C_p=
\emptyset$ by ...
0
votes
1answer
46 views
Estimate difference $f(x)\,H(f(x)-M) - g(x)\, H(g(x)-M)$, with $H$ the Heaviside function and $M>0$ fixed, in terms of the difference $|f - g|$? [closed]
Let $f,g: \mathbb R \to \mathbb R$ such that $f,g \in L^\infty(\mathbb R)$.
Fix $M>0$ and let $H$ denote the Heaviside function. How can we estimate the difference
$$f(x)\,H(f(x)-M) - g(x)\, H(g(...
1
vote
1answer
55 views
Integral average near a point of dispersion
Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...
-1
votes
0answers
31 views
Hessian matrix of the function defined with implicit function theorem [closed]
Let $x=(x_1,...,x_n) \in \mathbb{R}^n, y\in \mathbb{R}$ and let $F(x,y)=F(x_1,...,x_n,y) \in C^2(\mathbb{R}^{n+1})$.
Suppose we have all the hypothesis for the existence of the function $f(x)=y$ ...
0
votes
1answer
64 views
Distance function and its approximation
An easy and quick question:
Consider a function $u\in C(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$.
Define a function $Q$ that measures the distance of a point $(x,y) \in\mathbb{...
1
vote
1answer
110 views
Injectivity of an integral operator
Consider the operator
$$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$
with $k\in L^2((0,1)\times(0,1)).$
I want to know under what assumption the kernel is reduced to zero. i....
4
votes
0answers
116 views
A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic
I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...
-1
votes
1answer
72 views
Sum of $\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}$ [closed]
I want to calculute or estimate of order $O(n^{2-\varepsilon})$, where $\varepsilon>0$, of the following sum for $0<\alpha<1$
$$\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}.$$
2
votes
0answers
47 views
Measure of the convex hull of a ball and a point
I need to prove the following statement:
Let $B_s(z)$ be a ball centered at $z$ of radius $s$ s.t. $0\not\in B_s(z)$. Moreover let $K_s(z)$ the convex hull of $\{0\}\cup B_s(z)$.
Then
$$ \...
3
votes
0answers
91 views
Second derivative estimates
I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates.
In one of his papers, Lin proves the following result:
Let's consider a ...
3
votes
0answers
67 views
Implicit function theorem for subdifferentiable convex functions
I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
4
votes
1answer
87 views
Every convex set is of locally finite perimeter
I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter.
$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...
0
votes
0answers
86 views
Equivalence of weak/strong uniform Glivenko-Cantelli classes
Let $(S, \sum, P)$ be a probability space and $H$ a class of real valued measurable functions on $S$. Then we say that $H$ is a weak uniform Glivenko-Cantelli class if for any $\epsilon > 0$
$$ \...
0
votes
3answers
182 views
Does the generalised directional derivative satisfy any version of the chain rule?
Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient.
The generalised directional derivative ...
0
votes
1answer
111 views
Finding the conjugate of a function
I know that the Fenchel conjugate of a function is
$$f^*(x^*) = \sup_x\{\langle x, x^*\rangle - f(x)\}.$$
However, how do I find the Fenchel conjugate of the function
$$f(x) = \frac{1}{p}\sum\limits_{...
2
votes
1answer
57 views
Convergence of semi convex functions
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
2
votes
0answers
38 views
Second derivative estimates for a subsolution of linear elliptic equation
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
-2
votes
1answer
46 views
Does having the derivative in the limit suffice to solve the function at the limit? [closed]
Suppose that I have a function $f(x, \epsilon)$ and I know that
$$
\lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x).
$$
Now let $g(x)$ be the function whose derivative appears above. How can I ...
0
votes
0answers
65 views
Existence of the inverse Fourier transform, Carr Madan
I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
2
votes
0answers
64 views
First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
4
votes
0answers
196 views
An inequality in harmonic analysis with the BMO flavour
I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes).
Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in
...
2
votes
0answers
202 views
Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
1
vote
0answers
58 views
Intuition from Hopf lemma (boundary point lemma )
Consider the classical boundary point lemma:
Let $L$ be an elliptic operator.
Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
1
vote
1answer
91 views
Understanding a family of Sobolev-type inequalities
I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:
Denote the following inequality as $S_{r,s}^{\theta}$: $\...
3
votes
1answer
171 views
Aleksandrov maximum principle for semi-convex function
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
0
votes
0answers
83 views
Are periodic real-analytic functions dense in the Frechet space $C^\infty(S^1)$?
The question is as above:
Are periodic real-analytic functions dense in the Frechet space $C^\infty(S^1)$?
Here I gave $C^\infty(S^1)$ the metric topology in which the convergence is the uniform ...
1
vote
0answers
32 views
Example of periodic semidifferentiable function without absolutely convergent Fourier series
Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but
with a non-absolutely convergent Fourier ...
4
votes
0answers
79 views
Asymptotic of a functional as $x\rightarrow \infty$
Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
0
votes
0answers
109 views
Making area/volume calculations that use SIA rigorous
There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples:
A proof that $\sin'(0) = 1$.
A proof that the surface area of a cone is ...
2
votes
2answers
92 views
Convergence of fraction of expectation values
Let $X_1,...,X_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...
1
vote
0answers
35 views
Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
2
votes
1answer
49 views
Estimates on divergence-type operator for the matrix
Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form
$$
{\rm div}(Av)=f
$$
where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
