All Questions
5,909 questions
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Interpolation between $L^1$ and $L^2$ spaces
I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
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1
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more on "sinc-ing" integrals and sums
This is a follow up on the MO question here. I kept being fascinated and bemused by these functions.
Denote $\text{sinc}(x)=\frac{\sin x}x$. Experiments suggest that
$$\sum_{n=1}^{\infty}\text{sinc}^...
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0
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A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 557
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1
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Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 24.2k
7
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2
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582
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"sinc-ing" integral
Let $a_1,\dots,a_n, b$ be positive real numbers.
*Question.** Is this true?
$$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=\pi.$$
My ...
- 5,624
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3
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383
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convergence of 2nd eigenvalue
Fix $0<h_1<h_2<h_3<1$ reals. All matrices below are $3\times3$ real.
Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric ...
- 109k
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1
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Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
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1
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Isometry between punctured sphere and punctured triangle?
Setup:
Let $C_n$ be a closed $n$-simplex in $\mathbb{R}^n$ and let $r \in (0,R)$ where $R$ is the distance any one of the vertices $\{v_1,\cdots , v_{n+1}\}$ of $C_n$ to the centroid $\frac{v_1+ \...
- 6,101
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Continuous linear combination of continuously varying vectors?
Let ${\bf{e}}_1, {\bf{e}}_2, {\bf{e}}_3:[0,1]\rightarrow \mathbb{R}^3$ be continuous, $\mathbf{0}\neq \mathbf{v}\in \mathbb{R}^3$.
Suppose that the following condition (C) holds:
$$
\exists d>0: ...
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1
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363
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On a derivative involving the Riemann zeta function
Let $n$ be a positive real number. Can the equality
$$\dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\Big(\pi^{-s/2}\Gamma\Big(1+\frac{s}{2}\Big)\Big)\Big]\Bigg|_{s=1} = - \dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\...
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1
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1
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On a weaker condition of summability for Fourier series
The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
- 109k
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6
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What's the use of Malgrange preparation theorem?
The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
- 3,709
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115
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Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
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Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 3,574
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0
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177
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Interesting stipulation about completely monotone functions
This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
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2
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0
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571
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Integrating a product of integrals involving Bessel functions
I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable.
I ...
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0
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263
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Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 63.9k
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0
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58
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in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
- 43.2k
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2
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Properties of matrix exponential without using Jordan normal forms
There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:
All eigenvalues of $A$ have negative ...
38
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4
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3k
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Binomial again, and again
Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, ...
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interchange of infinite intersection and taking convex hull of a set
Let $B(x,\delta)$ be an open ball centered at $x\in R^n$ with radius $\delta>0$. Let $F:R^n\rightarrow R^m$ be a vector-valued function. Then $F(B(x,\delta))$ would be a subset of $R^m$. Let $\...
- 24.8k
10
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2
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886
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An attempt to generalize the previous inequality
In my previous MO question, the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps ...
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1
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Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
3
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0
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232
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When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
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0
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341
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Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 305
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1
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150
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Solutions to Schrödinger equation parameter dependence
This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) =...
- 305
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0
answers
563
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The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
- 19.6k
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4
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3k
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No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
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1
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Does $p$ integrability in n-1 dimensions give higher integrability in $n$ dimensions?
Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that
$$
\int\limits_t\left(\int\limits_x|...
- 39.1k
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7
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Real analysis has no applications?
I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
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3
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What is the set of all "pseudo-rational" numbers (see details)?
Define a “pseudo-rational” number to be a real number $q$ that can be written as
$q=\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$
Where $P(x)$ and $Q(x)$ are fixed integer polynomials (independent of n). ...
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2
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2k
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Where does the Lebesgue differentiation theorem fail?
The Lebesgue differentiation theorem says that for certain metric spaces $X$ (see below), any Borel measure $\mu$ that is finite on bounded sets and any $f: X \rightarrow \mathbb{R}$ locally $\mu$-...
- 18.7k
4
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2
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334
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what is this sum of squares of algebraic functions?
This question is inspired by the MO query here, although it has no direct implications.
Define the family of polynomial functions
$$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$
and ...
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1
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112
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Enforcing an inequality on series
Let $f:[0,1]\to\mathbb{R}_+$ be a convex, strictly increasing function such that $f(0)=0$ (typically, $f$ is very flat at $0$, i.e. increases very slowly). I would like to prove or disprove the ...
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0
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Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
- 66.3k
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2
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232
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Heat equation: impact of the diffusion coefficient on the Harnack constant
Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le a(x,...
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27
votes
3
answers
2k
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Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
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What is the growth of sum of absolute values of Fourier coefficients
For a periodic BV function $f$ which has jump discontinuties, is there any theorem in Fourier analysis which gives like $$\sum_{k=0}^n\left|c_k\right|\sim C\log\left(n\right)$$ where $C$ is a constant ...
CommunityBot
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3
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398
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Countable shifts of closed positive sets
Let $\mu$ be the Lebesgue measure, and $+$ be addition modulo $1$ in the interval $[0,1)$.
Question1: Is there a closed set $C\subset [0,1)$ of positive measure such that for any countable set $D\...
- 109k
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0
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Modulus of continuity of the Dirichlet Laplacian problem
I remember the following statement is correct but I cannot find a reference for that, can anybody help me to give one?
Let $\Omega\subseteq\mathbb{R}^{n}$ be an open, bounded, smooth domain,
$\varphi\...
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computational complexity: do we gain acceleration?
There is a technique we developed for series acceleration using the Wilf-Zeilberger method. Here is a simple Maple code in this regard, you need to download this too.
The idea is you start with a WZ-...
- 43.2k
8
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1
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386
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Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$
Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$.
Question. Can we prove this?
$$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
- 43.2k
5
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1
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101
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Does minimum of an analytic map restricted to analytic curves implies minimum?
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that its restriction to any arbitrary analytic curve $\gamma$ passing through the origin $0\in \mathbb{R}^n$ attains a local ...
- 60.6k
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2
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Is the exponential function the sole solution to these equations?
Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
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2
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does this sum have a limit?
Define the sequence given by the finite sum
$$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$
Questions.
(1) Is $0<a_n<1$?
(2) Does the limit $\lim_{n\rightarrow\infty}...
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0
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Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
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44
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7
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4k
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The missing link: an inequality
I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:
Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then
$$F_n(x)=\...
- 128k
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1
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Distances between probability distributions by the variance of the test functions
Let $P$ and $Q$ be two probability distributions on $\mathbb{R}$. The goal is to obtain a notion of ``distance'' between $P$ and $Q$, e.g., total variation distance, K-L divergence.
Let $f\colon \...
- 29.8k
3
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1
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411
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Continuation of a smooth function, whose every derivative is strictly monotonic
Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
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235
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Is this "differentiation map" uniquely determined by these properties?
Let $A$ be the set of all real-valued functions having their domain a subset of $\Bbb R$ which are at least differentiable on an open set, and for $f \in A$, let $U_f$ be the largest open set on which ...
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