Questions tagged [real-analysis]

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

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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{...
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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 ...
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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
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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 ...
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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
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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.$...
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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}^{...
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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 ...
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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
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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 ...
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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
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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
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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 ...
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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
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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
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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 ...
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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 $C_{\lambda}(f) = \sum\limits_{i=1}^{n}(f(p_i)-a_i)^2 + \|f\|_{L^2}^2 + \lambda\|\nabla^kf\|_{L^2}^2$ ...
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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
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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
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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
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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 ...
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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
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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
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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
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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_{\...
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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-...
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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
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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
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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
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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
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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
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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)])...
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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
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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 $f$,...

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