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

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2
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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 ...
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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 ...
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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)\| :\, \|...
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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 ...
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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{...
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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 ...
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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}(\...
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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{\...
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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&...
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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 ...
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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, $...
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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 ...
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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) \|_{\...
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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 ...
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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 +...
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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 ...
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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: |...
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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)...
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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{...
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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). ...
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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: ...
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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 ...
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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
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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 $\...
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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(...
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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 ...
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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, ...
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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.
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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 $\lambda_1>0$, ...
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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 ...
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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{...
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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 $k>0$....
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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 ...
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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
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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 \...
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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)...
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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 ...
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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\...
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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 ...
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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 ...
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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 \...
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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
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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 ...
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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}...
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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) ...
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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 $TV(f) \le M$}\} $?
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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
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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{\...
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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 $\...

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