# Questions tagged [real-analysis]

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

3,352
questions

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16 views

### Non-square multiplication operator matrix

Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...

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56 views

### How to give an analytic set which is not a Borel set but whose projection is Borel?

I wonder that how Suslin got the idea "analytic set" and the advanced knowledge in this field.
Can anyone explain this simply or recommend some reference?
All discussions and comments are ...

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109 views

### How to compute $\lim_{n\to \infty}a_1b_n+a_2b_{n-1}+\cdots+a_nb_{1}$

As is well known, Stolz–Cesàro theorem is the following:
Let $\displaystyle {(a_{n})_{n\geq 1}}$ and ${\displaystyle (b_{n})_{n\geq 1}}$be two sequences of real numbers. Assume that ${\displaystyle (...

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votes

**1**answer

84 views

### Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$

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37 views

### Characterizing a set of functions

Let $\mathscr C$ be the space of measurable functions $f$ on the interval $[0,1]$ that can be written in the form
$$ f(t)=\sum_{k=0}^\infty c_k e^{-kt},$$
for some square summable series $\{c_k\}_{k=1}...

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30 views

### Non constant delay differential equations

Given $\varphi:[0,1] \to [0,1]$ a continuous function, let $(E)$ be the delay differential equation (I am not sure about the terminology, as the delay is non constant): $y'(t) = y(\varphi(t))$. It is ...

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71 views

+50

### Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$

Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...

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**1**answer

129 views

### Is a specific product function orthogonal to all harmonic functions

Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...

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138 views

### Is every Baire metric space a complete metric space in disguise?

I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...

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**1**answer

144 views

### First order PDE in complex variables?

Consider the equation
$$f'(x)+ g(x)f(x)=0$$
This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$
Similarly, we can look at complex variables and consider the equation and ...

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78 views

### Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?

While solving a problem in calculus of variations, I came to the following question:
Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)=...

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66 views

### Measure of zero sets of functions on lower-dimensional subsets

Let $X\subset\mathbb{R}^n$ be a manifold$^1$ of dimension $<n$, and $p=(p_1,\ldots,p_k)$ be a system of polynomials in $n$ variables. Let $Z$ be the zero set of $p$ and assume $Z\subset X$. Is it ...

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74 views

### What is the convergence rate of the minimum separation distance?

Let $\Omega\subset \mathbb{R}^m$ be a bounded Lipschitz domain. Let $D$ be a countable dense subset of $\Omega$, denoted as $D = \{p_1,p_2,p_3\ldots \}$. Define the minimum seperation distance among ...

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278 views

### Asymptotic behaviour of an integral. How should I proceed?

Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $b, \sigma: (l, r)\to\mathbb{R}$, $−\infty \leq l < r \leq \infty$ bounded functions on compact intervals of $(l, r)$...

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165 views

### An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space

By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says:
“If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not ...

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**1**answer

73 views

### A simple clarification on Riesz decomposition theorem

Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions&...

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41 views

### Harmonic measure for unbounded domains?

Harmonic measure in the literature is defined for bounded domaines. It is also clear that it cannot be defined on an unbounded domaine in $\mathbb{R}^2$, since there the harmonic measure coincides ...

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41 views

### Hermitian decomposition via Trace norm

I am doing some quantum information. I face some problems related to linear algebra. The problem is somewhat long, so I will provide the reference and detail problem that I am dealing with.
We first ...

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**1**answer

141 views

### Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$

The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$
Where $$u = [u_1,u_2,\ldots u_n]^T$$
Now I want to rewrite these same equations but with a new ...

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48 views

### Prove integral inequality for divergence-free vector fields

Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?
$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...

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317 views

### Determining if a quadratic form is non-negative if variables are non-negative

Let $f(x_1,\dots,x_n) = \sum_{1 \le i \le j \le n} c_{i,j}x_ix_j$ be a homogeneous quadratic form. Is there a quick-ish way to determine whether $f(x_1,\dots,x_n) \ge 0$ for all $x_1,\dots,x_n \ge 0$?
...

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**1**answer

163 views

### When is a function on symmetric positive definite matrices an expectation of Gaussian?

Is there some characterization of real-valued functions of the form $\Phi(C)=\mathbb{E}F(X)$, where $X$ has the Gaussian $N(0,C)$ distribution, on the space of symmetric positive semidefinite $n\times ...

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154 views

### Lipschitz constant of the differential of the matrix exponential $\mathfrak{so}(3)\to \mathrm{SO}(3)$

I'm interested in numerical methods on $\mathrm{SO}(3)$ manifold, and working on a particular problem using the exponential coordinates. These can be computed using the Rodrigues' formula:
$$
R(u) := \...

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74 views

### A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...

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111 views

### Mixed partial derivatives of planar functions converging to delta distribution

Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...

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43 views

### Approximating step functions using step functions

Let $f$ be any BV function over $\mathbb{T}$. Let the Fourier series partial sum be $S_n$ which is constructed using the first $n$ Fourier series coefficients. We know that $s_n \to f$ pointwise at ...

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34 views

### A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...

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232 views

### Transforming a continuous function into a differentiable function

Given a continuous function $f(x)$ when does there exist a non-constant continuous function $g(x)$ such that $f(g(x))$ is differentiable what about $g(f(x))$?
Does there exist any examples of $f(x)$ ...

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**3**answers

218 views

### How to show continuity and monotonicity of solutions to this parametrized equation?

Let $1 \le p <2$ be a parameter. Consider the equation
$$
\frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1}
$$
I am rather certain that for each $1 \le p <2$, ...

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**1**answer

65 views

### Terminology: Co-completion of Met?

In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, ...

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120 views

### Convergent of improper integral [closed]

Let $f \in C^1[0,\infty)$ be an increasing function with $f(0)>0$, suppose $\int_0^\infty \frac{1}{f(x)+f'(x)} < \infty$, prove that $\int_0^\infty \frac{1}{f(x)} < \infty$.
I find it weird ...

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**1**answer

133 views

### Is this recurrent sequence decreasing?

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...

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43 views

### Hilbert transform on a Besov space

Consider the usual Hilbert transform of periodic functions
$$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$
We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...

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29 views

### Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...

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101 views

### Existence of smooth function that characterizes boundary and interior of set

It is well known that every closed set $A \subset \mathbb{R}^{n}$ is the zero level set of some smooth function. It follows that every closed set is also the zero sublevel set of some smooth function, ...

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22 views

### Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; \boldsymbol{x}(t) \in \mathbb{R}^n, \;\; t \geq 0, \; \...

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55 views

### Log-concave sequences and triangular arrays

A non-negative sequence $\{a_i\}$ is said to be log-concave if $a_i^2 \geq a_{i+1}\,a_{i-1}$ for all $i\geq 1$.
I'm interested in investigating triangular arrays $\{a(n,k)\}_{n,k\geq 0}$ such that $\{\...

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53 views

### Solving $\int_0^\infty N(1-F(t))^{N-1}tf(t)dt$ when the expected value is known

Suppose that $f:\mathbb R_{\geq 0} \rightarrow \mathbb R_{\geq 0}$ is a probability density function, and $F$ is a cumulative distribution function (i.e. $F(t)=\int_0^t kf(k)dk$). Also, assume that ...

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**1**answer

72 views

### roots of combined e-function [closed]

I need to determine the roots of the following function analytically:
$f(x) = 1 - x - x*e^{-2x}$
This is my try on it:
$0 = 1 - x - x*e^{-2x}\quad |-1$
$-1 = - x - x*e^{-2x}\quad |+ x$
$-1 + x = - x*e^...

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**1**answer

168 views

### Finding super(sub)-harmonic functions for an elliptic operator

I am looking for a super(sub) harmonic function for an elliptic operator.
Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$...

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**1**answer

202 views

### Simple closed forms for sums such as $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{qk - p}$ and related integrals

My goal here is to get a simple expression for $\zeta(3)$. This is a follow up to my previous question posted here. Any Taylor-like expansion from everything I tried won't make it. So this is my last ...

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103 views

### Detecting slow growth in a finite number of queries

The following question was asked at Can you solve this problem using a finite number of queries?
:
Let $g:[0,1]\to[0,1]$ be a continuous monotonically-increasing function. You can access $g$ using ...

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vote

**1**answer

123 views

### Function series of normal lower semi-continuous functions

For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is
defined as follows:
$
f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in
U\right\} :U\in ...

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votes

**2**answers

315 views

### Can you solve this problem using a finite number of queries?

Let $g:[0,1]\to[0,1]$ be a continuous monotonically-increasing function. You can access $g$ using queries of two kinds:
Given $x\in[0,1]$, return $g(x)$.
Given $y\in[0,1]$, return $g^{-1}(y)$.
Given ...

**4**

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**0**answers

90 views

### What is the completion of $L^\infty$ in the dual of BV?

Every $f \in L^\infty([0,1])$ induces a continuous linear functional on BV via $g \mapsto \int f g \mathrm{d}x$. I believe $L^\infty([0,1])$ is also separable in BV$^\ast$, while BV$^\ast$ is not ...

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46 views

### Integral inequality with Fractional Laplacian

Is the following inequality true
$$
\int_{B_1(0)} f(x) (-\Delta)^\alpha u(x) dx - \frac{1}{|B_1(0)|}\int_{B_1(0)}(-\Delta)^\alpha u(x) dx \cdot \int_{B_1(0)}f(x) dx \ge 0
$$
for a strictly convex $f:\...

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votes

**1**answer

89 views

### Average over spheres finite

Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$
I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...

**2**

votes

**1**answer

41 views

### Lower semi-continuity of length-dependent functional

Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional
$$
\begin{aligned}
F_f:&\ell^1 \rightarrow [0,\infty]\\
(x_n)_{n=0}^{\infty} &\to \sum_{n=...

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64 views

### Constructing a norm inequality for a given functions [closed]

Let $F_1(x)=x\log^2(2+x)$ and $F_2(x)=x\log(2+x)$ be such that
$$
F_2^{-1}\left(\int_0^{\infty} F_2(c/x \int_0^{t}g(s) ds)\right)\leq F_1^{-1}\left(\int_0^{\infty}F_1(g(x))dx\right).
$$
I have been ...

**6**

votes

**2**answers

241 views

### Non-sequential spaces in the wild

TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...