Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
0
votes
0answers
5 views
Definition of $C^{m,k}$-capacity of a point
I have come across the following notation and a new term $C^{m,k}$-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.
-4
votes
0answers
27 views
Nth root of a being greater than 1 when a is greater than 1 [on hold]
In a part of a bigger proof, we need to prove that when $a > 1$ then $\sqrt[n]{a} > 1$. We were told to do it using the binomial theorem, so I tried to do it like this. If $a > 1$ then $a = 1 ...
-5
votes
0answers
35 views
Do closed bounded intervals with a common end point non overlap each other? [on hold]
Source: Introduction to Real Analysis, Fourth Edition, Robert G. Bartle, Donald R. Sherbert, Page 199
In the above extracted text, intervals $I_1, I_2, …, I_n$ are said to be non-overlapping, but ...
-4
votes
0answers
56 views
the limits of series [on hold]
let $S_n=\sum^n_{i=1} \frac{3 i^2}{4^i-1}$
$$\frac{3 i^2}{4^i-1} \leq \frac{3 i^2}{3^i}=\frac{i^2}{3^{i-1}}$$
put $A_i=\ln \frac{i^4}{3^{i-1}}$
$$A_i=4 \ln i- (i-1)\ln 3 $$
for $i>1$ $A_i=\frac{1}{...
-4
votes
0answers
34 views
Finding bounding in a formula [on hold]
Here's my solution, where did I make a mistake?
The formula:
My solution:
-4
votes
0answers
46 views
Can we apply L'Hopital's rule with respect parameter $u$? [on hold]
If we know that $\int_1^\infty g(x,t_o)dx=0$ and $\int_1^\infty h(x,t_o)dx=0$
$ t $ is real variable here
$$f(x,t_o)=\lim_{t \rightarrow t_o} \frac{\int_1^\infty g(x,t)dx}{\int_1^\infty h(x,t)dx}= ...
3
votes
1answer
100 views
A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?
Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
8
votes
0answers
69 views
Borel-Ecalle re-summation and resurgence: Criteria and Results
This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\...
0
votes
0answers
51 views
$L^{\infty}-L^{\infty}$ estimates on the Schrödinger evolution
A classical estimate for solutions to Schrödinger equations are $L^1-L^{\infty}$ estimates, also known as dispersive estimates.
I wonder whether there are also $L^{\infty}-L^{\infty}$ estimates?
...
2
votes
1answer
137 views
Counter example about blow-up solution of DEs
Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
3
votes
1answer
223 views
Constant “periodization” of a function
Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that
$$ \sum_{n \in \mathbb{Z}} w(x+n) = 0$$
for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume ...
1
vote
0answers
34 views
Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball
Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [...
5
votes
2answers
197 views
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances?
Formal version of question. If $X$ is a set, let $[X]...
9
votes
1answer
174 views
The discrete Hardy-Littlewood-Sobolev inequality
Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...
5
votes
0answers
133 views
Function of two sets intersection
Let $U$ be the set of all subsets of $[0,1]$ that are a union of finitely many closed intervals (not allowing intervals that are single points). Does there exist a function $f:U\times U\rightarrow U$ ...
23
votes
4answers
888 views
show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$
I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....
-2
votes
1answer
97 views
Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence?
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
5
votes
2answers
213 views
Bounded deformation vs bounded variation
Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...
3
votes
1answer
217 views
Is there any Menelaus-type theorem for polynomials?
Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
10
votes
3answers
1k views
Adventure with infinite series, a curiosity
It is easily verifiable that
$$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
...
0
votes
0answers
48 views
Extension of a derivation on w [duplicate]
Let I be a closed left ideal of a Banach algebra A such that A has a bounded approximate identity and I just has a right approximate identity. let $D:I\to I^*$ be a derivation. Does $D$ extend to a ...
3
votes
1answer
103 views
Log concavity of the maximum of dependent Gaussians
Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
2
votes
0answers
61 views
The Poisson equation
I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates
$$\triangle u=f,in \> B_2 \>(1)$$
Lemma 7: There is a constant $N_1$ so that for any $ε > ...
2
votes
0answers
39 views
Convergence to the probability generating function of a Poisson process
I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that
$\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, ...
1
vote
2answers
146 views
A min-max approximation
Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$.
My question is : Is it true that
$$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
2
votes
1answer
87 views
Removable set for Sobolev space
It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...
8
votes
4answers
644 views
How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
1
vote
0answers
34 views
Defining Boundary Conditions for Spline Interpolation via the Euler–Maclaurin Formula
The Euler–Maclaurin formula states an interdependency between
\begin{align}
I\quad:=&\quad\int_m^nf(x) \, dx;\ m,n\in\mathbb{Z}\\[6pt]
S\quad:=&\quad\sum_{k=m}^n f(k) \\[6pt]
D\quad:=&\...
2
votes
0answers
75 views
Lebesgue density theorem for “doubling uniformly covering collections of subsets”
I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...
0
votes
1answer
50 views
Does differentiating an integro-differential equation results in equivalent stability of the solution?
I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
...
0
votes
1answer
41 views
Strict positive type function on hypersurface also of positive type in neighborhood?
Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
4
votes
1answer
173 views
Bounded growth of functions vs bounded growth of functions on countable sets
I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $...
-1
votes
0answers
78 views
Does this sequence have a convergence subsequence?
Let $w_n\in C([0,\tau];L^2(\Omega))$, and $\Omega$ be an open bounded set of $\mathbb{R}^2$. For every $t\in [0,\tau]$, $w_n(t)$ has a convergent subsequence in $L^2(\Omega)$. Does the sequence $$\...
3
votes
2answers
213 views
Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$
This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question
Let tangents be drawn to the curve $y=\sin x$ from ...
2
votes
0answers
96 views
Are $C^1$ immersions dense in $C^1$?
Let $M$ be a closed compact manifold.
Is the space of all $C^1$ immersions from $M$ to $\mathbb{R}^m$ ($m> \dim M$) dense in $C^1(M; \mathbb{R}^m)$ (in the $C^1$ topology)?
0
votes
0answers
27 views
Existence of solutions to NLS: Local existence and boundedness
I was wondering when the following argument is valid:
Consider a nonlinear Schrödinger equation
$$i \partial_t \varphi = -\Delta \varphi+ N(\varphi)$$
where $N$ is a nonlinearity.
Often it is ...
1
vote
1answer
80 views
Optimal estimate in trace norm
Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...
5
votes
2answers
591 views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
1
vote
0answers
34 views
On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
4
votes
1answer
64 views
Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
-2
votes
1answer
161 views
Strong estimates for the zeta function on natural numbers
Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
be the Riemann zeta function (here we just consider real $s$).
We do have a description given by
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
0
votes
1answer
196 views
Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]
In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-...
1
vote
1answer
86 views
Quotient with positive second derivative in the limit?
I am studying the quotient of
$$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$
and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$
for some $\...
4
votes
0answers
87 views
Injectivity of product functions on natural number sequences
Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...
3
votes
1answer
158 views
Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
3
votes
0answers
186 views
Eudoxus real numbers
I recently remembered the eudoxus construction of the real numbers.
Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction?
Clearification: The ...
2
votes
1answer
129 views
Chain rules for Dini Derivative
Could someone provides some references for the chain rule concerning Dini derivatives. For example, let $f(\cdot) \in \mathcal{C}^1\left( \mathbb{R} ; \mathbb{R}\right)$, and $g(\cdot) \in \mathcal{C}\...
3
votes
0answers
66 views
Anderson Localization and Homogenization theory
I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here.
The question is mostly related to homogenization theory in mathematical physics.
$\textbf{...
2
votes
1answer
285 views
Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$
I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?
This seems to be a well-known result, but I can ...
6
votes
2answers
443 views
Summing Bernoulli numbers
Consider the Bernoulli numbers denoted by $B_n$, which are rational numbers.
It is known that the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$ are not integers once $n>1$.
I am curious about the ...
