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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2
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0answers
49 views

Hilbert transform on weighted Sobolev spaces

Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
2
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0answers
77 views

For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc

Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, $...
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1answer
54 views

Recurrence results for an “on average” measure preserving transformation

I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$ $$ \lim_{n\rightarrow \...
0
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1answer
44 views

One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$?

In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland ...
2
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0answers
138 views

What are (the different aspects of) harmonic analysis good for?

Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
-1
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2answers
115 views

Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]

Under which condition can it form a Hilbert space? Or what space can it form? You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
1
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1answer
101 views

Inferring the modulus of continuity

Let $f:X\rightarrow Y$, $g:Y\rightarrow Z$ be uniformly continuous functions between metric spaces $X,Y,Z$ with moduli of continuity $\omega_f$ and $\omega_g$, respectively. Suppose that we know that ...
1
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2answers
90 views

Estimate for a simple oscillatory integral

If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that $$ \Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\...
2
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0answers
42 views

Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain

Let $s>0$, $1<p<\infty$ and let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. Set $H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|_{\Omega}=u,(I-\...
4
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2answers
420 views

Find an element with given periodicity

Sorry for all the confusion. I think what I am actually asking is: Can we find an explicit smooth non-zero function on $\mathbb R^2$ that satisfies $$f(x_1,x_2) =e^{-i\pi x_2} f(x_1+1,x_2) \text{ and }...
1
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1answer
127 views

Does the following integral converge?

Let $a\in(-1,1)$, let: $$ P(x,y)=C_{n,a}\frac{y^{1-a}}{(|x|^2+y^2)^{(n+1-a)/2}},\quad\forall (x,y)\in \mathbb{R}^n\times(0,\infty),$$ let $f\in \mathcal{S}(\mathbb{R}^n)$, i.e. $f$ is a Schwartz ...
1
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0answers
35 views

Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?

Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$ with Dirichlet boundary conditions. ...
-2
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2answers
160 views

Does function $f(x)=f(2x)$, $f(x)$ - non const, exist? ($f(x)$ - continuous function on real numbers) [closed]

When I tried solve it I had found just answer "No". I spoke with some people but I cannot understand why the answer is exactly it... Frankly speaking, this function haunts me: $f(x) = abs((...
1
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0answers
52 views

Bounds on the inverse multivariate beta function

Can i find some constants $\mathbf a$ and $\mathbf b$ in $\mathbb{R}_+^d$ such that for all $\mathbf{x} \in \mathbb{R}_{+}^{d}$, the inverse beta function : $$IB(\mathbf x) = \frac{\Gamma\left(\lvert \...
0
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1answer
166 views

Sufficient and necessary conditions for decomposing the sum of random variables

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
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0answers
36 views

To show a analytic map is zero from a property regarding logarithmic integral

Let $F$ be analytic on $\mathbb{H}=\{z\in\mathbb{C}:Im(z)>0\},$ continuous upto $\overline{\mathbb{H}}$ and bounded on each of the half plane $\{Im(z)\geq h>0\}.$ How to show that if $F$ ...
1
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0answers
40 views

Convergence of a certain serie

I came cross the following serie : $$\sum\limits_{\mathbf k \in \mathbb N^d} e^{\langle \mathbf r, \mathbf k \rangle}$$ What would be the conditions on the d-dimensional real vector $\mathbf r$ for ...
0
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1answer
95 views

A “simple” surface-integral over the unit-sphere [closed]

Calculate $$ I=\iint_{x^2+y^2+z^2=1}{e^{x-y} \mathbb{d}S} $$ Parameterization is not helpful: $$ I=\int_0^{2\pi}{\mathbb{d}\varphi\int_0^\pi{e^{\sin\theta(\cos\varphi-\sin\varphi)}\sin\theta\mathbb{d}...
8
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1answer
456 views

Can this inequality be proved using weighted maximal function estimates?

I am trying to understand the following fact: Suppose $\{B_i\}_i$ are disjoint balls in $\mathbb R^n$, and $A_i \subset 100 B_i$ is a subset with $|A_i| \geq c |B_i|$. Then for any nonnegative $f$, ...
4
votes
1answer
150 views

An asymptotic expansion of a infinite sum

I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series $$ \sum_{k\ge 0}e^{-k^{2/n}t} $$ for integer $n>2$ (n=1 follows from Poisson summation formula ...
2
votes
1answer
125 views

Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$

Working with the Slater's inequality (compagnion of Jensen's inequality) I find this statement : Let $f(x)$ be a continuous,twice differentiable function ,convex or concave and non constant on $(0,\...
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1answer
109 views

Periodical functions with $f^{(n)} = f$, but $f^{(k)} \neq f$ for $k\in \{1,\ldots,n-1\}$ [closed]

For any infinitely differentiable function $f: \mathbb{R}\to \mathbb{R}$ and positive integer $k\in\mathbb{N}$, let $f^{(k)}$ denote the $k$-th derivative of $f$. For which $n\in\mathbb{N}$, $n>1$, ...
4
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2answers
111 views

Symmetry of one-sided partial derivatives

Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following one-sided second partial derivatives at $(x,y)=(0,0)$ are symmetric: $$ \partial_x^+ ...
12
votes
1answer
458 views

An inequality about unit vector orthogonal to $(1,1,…,1)$

Does there exist a constant $\alpha>0$ such that the following holds? $$\liminf_{n\to\infty}\inf_{x\in\mathbb{R}^n, \sum_{i=1}^nx_i^2=1, \sum_{i=1}^nx_i=0}\frac{\sum_{i<j, |i-j|\leq\frac{n}{4}}(...
0
votes
1answer
25 views

Finiteness of a bilinear combination

For $j\in\mathbb{N}$, consider continuous functions $f_j:[0,1]\to\mathbb{\mathbb{R}^+}$ such that $$\sup_{t\in[0,1]}\sum_jf_j(t)<+\infty,$$ namely $f_j(t)\in L_t^{\infty}((0,1),l_j^1(\mathbb{N}))$. ...
0
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0answers
30 views

A pointwise inequality for fractional Laplacian (s-harmonic function)

Let $s\in(0,1)$ and consider the class: $$ L_s:=\biggl\{ u\in L^1_{\text{loc}}(\mathbb{R}^n):\int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}\,dx<\infty \biggr\}.$$ At the end of the page 23 of ...
6
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0answers
208 views

Odd Steinhaus problem for finite sets

Call a finite subset $S$ of the plane with an even number of points an odd Jackson set, if there is an $A\subset \mathbb R^2$ such that $A$ meets every congruent copy of $S$ in an odd number of points....
0
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0answers
42 views

Fractional laplacian on $H^s(\mathbb{R}^n)$ and symmetry

Let $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n),$ i define the fractional laplacian of $u$ in the following way: $$(-\Delta)^su(x)=C(n,s)P.V.\int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\quad\...
0
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1answer
87 views

Best bounds on the integral of an increasing function

The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval. Let $F\colon[0,1]\to[0,1]$ be a ...
-2
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1answer
86 views

Definition and properties of the inverse of the flow of an ODE [closed]

At lesson, the teacher considers a flow $\Phi$ given by the solutions of the ode system for $t\in[0, T]$ and $x\in\mathbb R^d$, $$ \begin{cases} y'(s)=b(y(s), s),&s\leq T\\ y(t)=x \end{cases},\...
3
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1answer
118 views

Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ...
0
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1answer
128 views

Solve $(x-a)^{\alpha +1} - \lambda*(b-x)^{\alpha + 1} = C(\frac{a+b}2 - x)^{\alpha}$ over $\mathbb R$ [closed]

I have been having trouble solving the following equation. Any help would be appreciated. Let a,b,C,$\alpha$,$\lambda$ be real numbers with $C < 0$, $0 < \alpha < 1$, $\lambda > 1$. We ...
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1answer
71 views

If we restrict the domain of the $C^{1}$ function $f:[x_{1},x_{2}]\to[0,1]$ to an open or closed interval does it remain $C^{1}$? [closed]

I have a function $f:[x_{1},x_{2}]\to[0,1]$ which is $C^{1}$, $[x_{1},x_{2}]\subset[0,1]$ and $f$ surjective. My first question is: can we always decompose the domain of $f$ into intervals $D_{k} = [...
3
votes
1answer
90 views

If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings converge uniformly on compact sets?

Let $u_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi_\delta$ a family of standard mollifiers with compact support. Hörmander claims in The Analysis of Linear Partial ...
0
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1answer
82 views

Symmetry of fractional laplacian

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that: $$\int_{\...
1
vote
0answers
38 views

Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
0
votes
1answer
69 views

Rate of convergence

Consider a function fixed function $f\in L^1(\mathbb{R})$ such that $$ \int_{\mathbb{R}}f(x)dx=0 $$ Now define the following function: $$ F(y)=\int_{\mathbb{R}} f(x)\mathrm{sech}\Big(\frac{x}{\exp(y)}\...
7
votes
1answer
257 views

Is there a real valued function whose limit exists only on irrational numbers?

I have been trying to find a function $f : \mathbb R \to \mathbb R$ such that $\lim_{x \to c} f(x)$ exists when $c$ is irrational and the limit doesn't exist when $c$ is rational. I tried variations ...
2
votes
1answer
66 views

Uniform estimation of an integral involving a Hölder-continuous function

Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\...
0
votes
1answer
84 views

Uniform estimation of an integral

Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
1
vote
1answer
72 views

Why does $l_0$ appear in this statement of the Furstenberg–Katznelson–Weiss theorem?

In Terence Tao's paper Exploring the toolkit of Jean Bourgain is stated: Theorem 3.1 (Furstenberg–Katznelson–Weiss theorem, qualitative version). Let $A\subset\Bbb R^2$ be a measurable set whose upper ...
1
vote
0answers
41 views

Functions that vanish weakly to $\infty$ and a uniqueness problem

I am reading the article "User’s guide to the fractional Laplacian and the method of semigroups" by P.R. Stinga, there is a link. At page 17, in theorem 7, the author state that, for a given ...
0
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0answers
121 views

Functions that satisfy a reverse triangle inequality: do they have a name?

Let $f : \mathbb{R}^n \to \mathbb{R}_+$ satisfy $$f(a) - f(b) \le C f(a - b)$$ $\forall a, b \in \mathbb{R}^n$ for some $C \ge 0$. Is there a name for such functions? (I would be happy to have a name ...
1
vote
0answers
59 views

The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$

For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$: $$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
3
votes
2answers
129 views

smooth functions on closed intervals with values in infinite-dimensional spaces

There are three ways to define when a ($\mathbb{R}$-valued) function on a closed interval is smooth: $f$ can be extended to a smooth function on $(a - \epsilon, b + \epsilon)$ for some $\epsilon > ...
0
votes
0answers
45 views

The fractional laplacian $(-\Delta)^s\colon \mathcal{S}(\mathbb{R}^n)\to\mathcal{S}_s(\mathbb{R}^n)$ is continuous

Let $s\in(0,1)$, let: $$ \mathcal{S}_s(\mathbb{R}^n)=\biggl\{ f\in C^\infty(\mathbb{R}^n): \sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha(x)|<\infty,\,\forall \alpha\in\mathbb{N}_0^n\biggr\}....
5
votes
4answers
211 views

Looking for a reference on conformal mapping on $\Bbb R^n$

A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e., if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then $$\cos (Tx(t_0),Ty(t_0))= \...
0
votes
0answers
59 views

Making sense of a Fourier transform and proving that a function is differentiable and is in $W^{1,1}_\text{loc}(0,\infty)$

I have a function $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that $V(x,\cdot)\in C^0([0,\infty))$ and $V(x,0)=u(x)$, $\forall x\in\mathbb{R}^n$, where $u\in\mathcal{S}(\mathbb{R}^...
2
votes
1answer
60 views

Inequality for generalized Laguerre polynomials

Please. Does anybody know a proof of this inequality $$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and $L^{\alpha}_n$ ...
0
votes
1answer
135 views

Estimate for computing the $L^2$-norm of a function from its data

Let $f:\mathbb{T}^m \to \mathbb{R}$ is a function of bounded variation(BV). Let $D=\{\boldsymbol{p}_i,i=1,2,3\ldots\}$ be a countable dense subset of $(0,1)^m$. Let $E_n, n = 1,2,3\ldots$ be a ...

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