Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
3,579
questions
2
votes
0answers
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 ...
0
votes
0answers
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
votes
0answers
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
0answers
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
$$
\alpha(H,\epsilon) \triangleq \inf\left\{
\|H(x)-H(y)\| :\, \|...
1
vote
0answers
62 views
Fourier transform of a Sobolev function dependent on a “parameter”
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
1
vote
1answer
170 views
Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$
Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...
-1
votes
1answer
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 ...
1
vote
0answers
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
1answer
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
$$
\lim_{x\to\infty}\frac{\...
0
votes
1answer
66 views
Can the identity function be approximated by compositions of a “uniformly monotone-and-convex” set of functions?
Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties?
For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$.
There exist $0&...
1
vote
0answers
46 views
wave equation with $H^{-1}$ source
Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation
$$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$
with $u|_{(0,T)\times ...
1
vote
0answers
102 views
Fractional Laplacian extension problem and uniqueness question
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
0
votes
0answers
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
votes
0answers
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
votes
0answers
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
0answers
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
vote
0answers
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
0answers
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
1answer
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
1answer
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
votes
1answer
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
1answer
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:
...
1
vote
0answers
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
1answer
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
1answer
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 $\...
0
votes
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
votes
0answers
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
0answers
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
5answers
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
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
4
votes
1answer
133 views
Decay estimate on wave equation
In this paper here it is claimed in (1.3) that it is classical and immediate from the explicit solution of the wave equation in 3D
$$(\partial_t^2 -\Delta )u(t,x)=0$$
with $u(0,x)=0$ and $u_t(0,x)=g(x)...
3
votes
1answer
66 views
wave equation with vanishing trace at infinity
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
7
votes
1answer
251 views
An elementary inequality for three complex numbers
The following problem arose in asymptotic analysis of difference equations.
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
0
votes
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
$$P(n) ...
0
votes
1answer
75 views
Kolmogorov entropy of a subset of $L^1$
How can we estimate the Kolmogorov $\epsilon$-entropy
$$H_\epsilon (A,L^1(\mathbb R))$$
where
$
A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\}
$?
2
votes
0answers
86 views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\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
1answer
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
1answer
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 $\...