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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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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0answers
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 ...
4
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0answers
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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1answer
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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0answers
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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0answers
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 "...
4
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1answer
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 ...
5
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1answer
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 ...
5
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0answers
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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1answer
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 ...
2
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1answer
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 $(l, r)$...
5
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1answer
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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1answer
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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0answers
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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0answers
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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1answer
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 ...
2
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0answers
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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2answers
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 $x_1,\dots,x_n \ge 0$? ...
5
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1answer
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 ...
5
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0answers
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) := \...
1
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0answers
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 ...
2
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0answers
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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0answers
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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0answers
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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0answers
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 $f(x)$ ...
2
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3answers
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 $1 \le p <2$, ...
1
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1answer
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, ...
3
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1answer
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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1answer
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 $...
2
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0answers
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 ...
1
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0answers
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 ...
6
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1answer
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, ...
1
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0answers
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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0answers
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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0answers
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 ...
-4
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1answer
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^...
4
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1answer
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 $\mathbb{R}^n$...
3
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1answer
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 ...
3
votes
3answers
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 ...
1
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1answer
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 ...
4
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2answers
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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0answers
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 ...
0
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0answers
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:\...
0
votes
1answer
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
1answer
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=...
3
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0answers
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
2answers
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 ...

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