# Questions tagged [real-analysis]

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

3,579
questions

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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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**0**answers

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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votes

**1**answer

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 \...

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votes

**1**answer

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 ...

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**0**answers

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 ...

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votes

**2**answers

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 $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...

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vote

**1**answer

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 ...

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vote

**2**answers

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}\...

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**0**answers

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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**2**answers

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**

vote

**1**answer

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 ...

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**0**answers

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. ...

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votes

**2**answers

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((...

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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**

votes

**1**answer

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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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 $F$ ...

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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 ...

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votes

**1**answer

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**

votes

**1**answer

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 $f$, ...

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votes

**1**answer

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

**1**answer

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,\...

**-4**

votes

**1**answer

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 $n\in\mathbb{N}$, $n>1$, ...

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votes

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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

**1**answer

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}}(...

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votes

**1**answer

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 $f_j(t)\in L_t^{\infty}((0,1),l_j^1(\mathbb{N}))$. ...

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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 ...

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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....

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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\...

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**1**answer

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 ...

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votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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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**1**answer

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} = [...

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votes

**1**answer

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 ...

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**1**answer

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_{\...

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vote

**0**answers

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**

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**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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**0**answers

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 ...

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**0**answers

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 ...

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**0**answers

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

**2**answers

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

**0**answers

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

**4**answers

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

**0**answers

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

**1**answer

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

**1**answer

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 ...