# 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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votes

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111 views

### Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...

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46 views

### Convergence of fractional laplacian

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

**2**

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**0**answers

35 views

### A lower-bound on matrix-function with vector product

I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...

**1**

vote

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63 views

### Reconciling notions of moduli of continuity

Let $H:K\rightarrow K$ be a self-homeomorphism of a compact subset $K\subseteq \mathbb{R}^n$. For $\epsilon>0$, define the function
$$
\alpha(H,\epsilon) \triangleq \inf\left\{
\|H(x)-H(y)\| :\, \|...

**1**

vote

**0**answers

62 views

### Fourier transform of a Sobolev function dependent on a “parameter”

Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...

**1**

vote

**1**answer

170 views

### Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$

Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...

**-1**

votes

**1**answer

198 views

### Square root of a continuous function

Is it square root of a real $\alpha-$Holder continuous function $f$ defined on $[0,1]$ a $\alpha/2$ Holder continuous, provided $\sqrt{f(x)}$ it exists and is continuous, i.e. whether $|f(x)-f(y)|\le ...

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**0**answers

31 views

### Extension problem of fractional laplacian and Fourier transform of $L^1_\text{loc}$ function?

I have understand the proof oh the lemma 4.1.9 "SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY" by C.Bocur, there is a link. In this paper, we define, for $u\in L^1_\text{loc}(\...

**4**

votes

**1**answer

246 views

### Mertens formulas aren't enough for prime number theorem

For the primes it's true that
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x)
$$
where, $M$ is suitable constant, and, moreover, the prime number theorem gives that
$$
\lim_{x\to\infty}\frac{\...

**0**

votes

**1**answer

66 views

### Can the identity function be approximated by compositions of a “uniformly monotone-and-convex” set of functions?

Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties?
For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$.
There exist $0&...

**1**

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**0**answers

46 views

### wave equation with $H^{-1}$ source

Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation
$$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$
with $u|_{(0,T)\times ...

**1**

vote

**0**answers

102 views

### Fractional Laplacian extension problem and uniqueness question

I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...

**0**

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**0**answers

45 views

### Passing to the limit under integral sign and fractional laplacian

I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. Suppose $f\in\mathcal{S}(\mathbb{R}^n)$, at page 5, the authors ...

**2**

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**0**answers

84 views

### Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable

Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...

**0**

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**0**answers

85 views

### When does a potential function with given partial derivatives exist

I am looking for the answer to the following question:
Consider an integrable function $f:X\rightarrow X$ with $X$ being a compact subset of $\mathbb{R}^n$. What are the conditions on $f$ so that a ...

**0**

votes

**0**answers

35 views

### Analyticity of $ \int\limits_x {\int\limits_y {Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}d{F_x}d{F_y}}}$

I need to show that the following function is analytic on the bounded complex plane. Lest define the function,
$f = \int\limits_x {\int\limits_y {Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x +...

**1**

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**0**answers

32 views

### A question about extension problem related to fractional laplacian

I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...

**0**

votes

**0**answers

87 views

### Approximating vanishing at infinity function by smooth compactly supported functions

Let $f$ be a function in $L^1_{\text{loc}}(\mathbb{R}^n)$, with weak derivative $Df \in L^p(\mathbb{R}^n)$, that vanishes at infinity in the following sense: for any $\epsilon > 0 $ the set $\{x: |...

**3**

votes

**1**answer

65 views

### Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane?

Let I have the following function,
$f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$
Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y)...

**2**

votes

**1**answer

75 views

### Eigenfunctions of the fractional Laplacian are smooth?

Let $\Omega\subset\mathbb{R}^n$ open, bounded with smooth boundary, let $s\in(0,1)$. I know that the fractional Laplacian has a sequence of eigenfunctions $\{e_k\}_{k\in\mathbb{N}}\subset H^s(\mathbb{...

**0**

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**1**answer

86 views

### Tight upper bound on $\sum_{1\leq k\leq m} \sqrt{(n/k)+b}$

I apologise if this is obvious or off-topic.
Let $n$ be large and fixed, $b>0,$ and $m \in [\log n,n),$ say. This sum seems to be hard to evaluate/upper bound analytically (in closed form). ...

**0**

votes

**1**answer

37 views

### Bounding the ratio of the $\ell_1$-norms of two real-valued $n$-vectors as a linear combination of their $n$ element-wise ratios

Let $a_1, a_2, \ldots a_n$ and $b_1, b_2, \ldots b_n$ be two sequences of $n\gg 1$ real numbers such that, for all $1\le i\le n$, we have $0<a_i \le b_i\le 1$. Let the ratio $R$ defined as follows:
...

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38 views

### Projection of a real analytic manifold onto subspace is union of real analytic submanifolds

Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...

**9**

votes

**1**answer

225 views

### Does ODE uniqueness unconditionally implies the flow continuity?

Suppose we have a (say compactly supported) $C^0$-vector field $X:\mathbb R^n\to\mathbb R^n$ such that for every $x\in\mathbb R^n$ there is a unique $C^1$-curve $\gamma:\mathbb R\to\mathbb R^n$ ...

**4**

votes

**1**answer

167 views

### “Simple” condition that would prove a function transcendental

I've already asked the question on MSE but there are still no answers, so I'm going to ask it here.
I conjectured that for every algebraic function $f(x)$ that is differentiable on $\mathbb{R}$, its $\...

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**0**answers

49 views

### Integral equality involving fractional laplacian

Let $s\in(0,1)$, let $u\in H^s(\mathbb{R}^n)$. For all $\psi\in\mathcal{S}(\mathbb{R}^n)$, let:
$$ (-\Delta)^s\psi(x)=c(n,s)\lim_{\epsilon\to0^+}
\int_{\mathbb{R}^n\setminus B_\epsilon(0)}\frac{\psi(...

**2**

votes

**0**answers

24 views

### Generalized Hardy operator and Lorentz gamma spaces

I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...

**2**

votes

**0**answers

53 views

### Set of discontinuities for Thomae's function in $\mathbb{R}^2$

For this question, label:
Part A: Is $f(x,y)$ integrable? Question $3$-$7$ from Spivak's Calculus on Manifolds
Part B: Prove $f:[0,1]\times[0,1]→\mathbb{R}$ is integrable.
For part A and B question, ...

**1**

vote

**1**answer

117 views

### Solving an equation with fractional laplacian [closed]

Let $s\in (0,1)$, how i can solve the equation:
$$ (-\Delta)^su=1,\quad\text{in}\quad(-1,1)?$$
I have no idea, any help would be appreciated.

**0**

votes

**1**answer

159 views

### The first eigenfunction of fractional laplacian

Let $\Omega$ be bounded and smooth domain in $\mathbb{R}^n$, $s\in(0,1)$, $e_1\in \mathbb{H}^s(\Omega)$ the first eigenfunction of fractional laplacian $(-\Delta)^s$ with eigenvalue $\lambda_1>0$, ...

**3**

votes

**2**answers

195 views

### Baire class 1 and discontinuities

Is it true that a bounded real function $f:[0,1]\to[0,1]$ with only countably many discontinuities has to be of Baire class 1, that is pointwise limit of a sequence of continuous functions? Is there a ...

**0**

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**0**answers

22 views

### Hermite interpolation with nested functions

Question:
provided the existence of a unique solution, how to go about solving the following Hermite interpolation problem:
given $f:y\in\mathbb{R}\mapsto z\in\mathbb{R},\ \boldsymbol{g}:x\in\mathbb{...

**0**

votes

**0**answers

76 views

### A question about Fourier transform of a function defined by an integral

I have the function:
$$ G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta}, $$
for all $x\in\mathbb{R}^n$ and $k>0$....

**5**

votes

**5**answers

464 views

### Elementary inhomogeneous inequality for three non-negative reals

I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...

**4**

votes

**1**answer

133 views

### Decay estimate on wave equation

In this paper here it is claimed in (1.3) that it is classical and immediate from the explicit solution of the wave equation in 3D
$$(\partial_t^2 -\Delta )u(t,x)=0$$
with $u(0,x)=0$ and $u_t(0,x)=g(x)...

**3**

votes

**1**answer

66 views

### wave equation with vanishing trace at infinity

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...

**7**

votes

**1**answer

251 views

### An elementary inequality for three complex numbers

The following problem arose in asymptotic analysis of difference equations.
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...

**0**

votes

**1**answer

84 views

### Expressing the measure of a set in terms of the characteristic function of the measure

Let $\mu$ be a discrete, finitely supported probability measure in $\mathbb{R}^d$ and denote by $\phi$ be the characteristic function of $\mu$, i.e. $\phi(t)=\mathbb{E}e^{i<t,X>}$, where $X$ is ...

**1**

vote

**1**answer

37 views

### Convergence of the sum of a family of real-valued functions

Let $\phi_1,...,\phi_n,...$ be a sequence of real-valued functions so that $\phi_j:[0,1)\to[0,1)$, $\phi_j(0)=0$, and $\phi_j(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\...

**0**

votes

**0**answers

41 views

### Optimal approximation of circles with sum of logarithms

By playing around, I found that
$$\left\|\frac{\log{(a\cdot(x+1)+1)}+\log{(1+(1-x)a)}-\log{(2a+1)}}{2\log{(a+1)}-\log{(2a+1)}}- \sqrt{1-x^2}\right\|_\infty\lt 0.12$$
indicating that the fraction quite ...

**1**

vote

**1**answer

105 views

### Is a locally invertible weak limit of injective maps injective almost everywhere?

This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...

**0**

votes

**1**answer

34 views

### Upper bound of a uniformly converging sequence of polynomials

Let $k\geq 2$, and let $P_k$ be a sequence of polynomials, such that:
$P_k=\sum_{n=2}^{k+1}a_{n,k}X^n \in \mathbb{Q}[X]$, $a_{2,k}\neq 0$, $\deg P_k \leq k+1$, and consider $P_k :[0,1]\rightarrow \...

**0**

votes

**0**answers

42 views

### Decomposition of a probability measure into positively and negatively supported ones

Consider the class $\mathcal{M}$ of probability measures $\mu$ on $\mathbb{R}$ whose moment generating function $\int e^{tx} d\mu(x)$ is finite for $t$ in a neighborhood of zero. Let $\mathcal{M}_{+}$ ...

**8**

votes

**1**answer

101 views

### Interpolation theory and $C^k$-spaces

Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...

**15**

votes

**2**answers

350 views

### Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets

Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets.
Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...

**2**

votes

**1**answer

99 views

### “Approximating” linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients

In a lecture I once attended, I remember the speaker using a result of the following nature:
$``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form
$$P(n) ...

**0**

votes

**1**answer

75 views

### Kolmogorov entropy of a subset of $L^1$

How can we estimate the Kolmogorov $\epsilon$-entropy
$$H_\epsilon (A,L^1(\mathbb R))$$
where
$
A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\}
$?

**2**

votes

**0**answers

86 views

### Lower bound on iterated matrix application

Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...

**5**

votes

**1**answer

152 views

### Optimal constant in Sobolev embedding

It is well-known that the Sobolev space $H^1(0,s)$ embeds continuously in the space of continuous functions $C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is $\sqrt{\...

**1**

vote

**1**answer

112 views

### Decomposition of the sum of nonnegative random variables [closed]

Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...