All Questions
5,857 questions
8
votes
1
answer
694
views
A generalization of Jensen's Inequality
Jensen's inequality is well known as
$$E\big[f(X)\big]\le f\big(E[X]\big)$$
where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also http://en.wikipedia.org/...
- 1,835
4
votes
1
answer
241
views
Is a function $u\in \mathrm{SBV}(\Omega)$ with these additional properties essentially bounded?
Some related earlier discussion can be found here.
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary, $\mathcal H^{N-1}(\partial\Omega)<\infty$ and $u\in SBV(\Omega)$. Then
$$
...
- 679
1
vote
0
answers
78
views
Potential for a Monotone Operator
[Cross-posted from math.stackexchange]
I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
- 291
1
vote
1
answer
175
views
Stochastic operator on $\ell^1$ has dense range
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I ...
- 329
6
votes
1
answer
2k
views
Jackson's theorem for partial sum of Fourier series
There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - ...
- 2,217
1
vote
0
answers
71
views
Continuous injection of metric ball into Euclidean ball
This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...
- 5,405
6
votes
0
answers
2k
views
Interchange of integral and infimum
Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ U\...
- 61
1
vote
2
answers
371
views
Weak convergence in vector-valued Hilbert space
Let $V$ be a separable Hilbert space and define $X=L^2(0,T;V)$. Then $u_m\to u$ weakly in $X$ means
for every $v\in X'=L^2(0,T;V')$
$$
\int_0^T\langle v(t),u_m(t)\rangle\ dt\to\int_0^T\langle v(...
user14319
2
votes
2
answers
287
views
How can we obtain the $-\frac{4\pi}3\mu(x)$ term?
Given the expression
$$K_{ik} := \frac{\partial}{\partial x_k} \int_{\mathscr X} \frac{y_i-x_i}{|y-x|^3} \mu(y) dy,$$
where $\mathscr X=\mathbb R^3$, how does one derive the expression
\begin{align}
...
- 183
0
votes
1
answer
186
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
4
votes
2
answers
4k
views
Embedding of $BV$ and $L^p$ spaces
An elementary question about Sobolev spaces:
Is there some explicit theorem about embedding relation between spaces $BV(\Omega)$ and $L^p(\Omega)$?
Formulated otherwise: is $BV$ a subset of $L^2$ (i....
- 53
4
votes
1
answer
1k
views
For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
- 1,649
5
votes
1
answer
249
views
If $\mathcal R_j f\in L^1$ then $\widehat{\mathcal R_j f}=-i\frac{\xi_j}{|\xi|}\widehat{f}(\xi)$
For any $f\in L^1(\mathbb{R}^n)$ and $1\le j\le n$, recall that the Riesz transform $\mathcal{R}_jf\in L^{1,\infty}(\mathbb{R}^n)$ is given by
$$ \mathcal{R}_jf:=c_n\lim_{\epsilon\to 0}\left(\frac{x_j}...
- 3,146
1
vote
1
answer
106
views
Inequality satisfied for $t=1/2$ and Measurability implies Log-Concavity
Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that
$$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$
Why must $f$ be log-concave? (That is, why must
$$\...
- 276
6
votes
1
answer
314
views
Generators of a convex cone defined by a differential inequality
Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_{++})$ and $f\colon \mathbb R_{++}\to\mathbb R_{++}$) that satisfy
\begin{...
- 61
0
votes
1
answer
87
views
Differentiablity of certain composite function
Let $I_1$ and $I_2$ be two closed bounded intervals.
Suppose $W(x,y)$ is a smooth function whose support is contained inside $I_1 \times I_2$.
Suppose I have $\Phi= (\Phi_1(x,y), \Phi_2(x,y)) : \...
- 3,625
21
votes
2
answers
924
views
Codimension of Measurable Sets
I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...
- 8,493
0
votes
0
answers
112
views
On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
- 577
3
votes
1
answer
113
views
maximum likelihood estimation of X is better than that of f(X)?
Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
- 482
3
votes
1
answer
233
views
Bounds on the positive roots of a bivariate polynomial
It is well known that various real root isolation methods are based on computing, first, the bounds on the values of the positive real roots of a polynomial equation. For the univariate case such ...
- 33
-1
votes
1
answer
136
views
An elementary question about integration by parts! [closed]
Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
0
votes
1
answer
563
views
Continuous Sobolev embedding
I have a question about Sobolev spaces.
In the following, we assume $d \ge 2$.
Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. Note that $D$ is not ...
- 721
1
vote
1
answer
258
views
How to prove or disprove a type of states form an overcomplete basis in the Hilbert space?
I am a PhD student in Physics. Let us consider a vector in an infinite dimensional Hilbert space as
\begin{equation}
|f\rangle\equiv
\begin{bmatrix}
1 \\
z \\
z^2 \\
\vdots
\end{...
- 145
9
votes
1
answer
460
views
Why should the map $-\Delta^{-1}$ be continuous?
I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$
in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
- 91
2
votes
1
answer
389
views
An irresistible inequality
The following occurred while working on some research project. Since the methods of proof I used were lengthy, I wish to see a skillful or insightful approach (perhaps even conceptual). Anyhow, here ...
- 43.2k
1
vote
1
answer
604
views
Partition of Real Number [closed]
Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
- 275
1
vote
1
answer
99
views
Equivalent conditions for a real function to have antiderivates
Lebesgue theorem says that a bounded function $f$ is Riemann Integerable if and only if $f$ continuous almost everywhere.
Unfortunately, we know a function has antiderivative has no relation to ...
- 35
0
votes
1
answer
629
views
Fourier Transform of sub-Gaussian distributions
The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-...
- 445
3
votes
3
answers
546
views
Determining Roots of a Polynomial with Interval Estimates of Coefficients
Let $f$ be a monic univariate polynomial with real coefficients:
$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$
The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as $B=(b_{n-1},...,...
- 103
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 113
5
votes
2
answers
429
views
Does the truncated Hausdorff moment problem admit absolutely continuous solutions?
Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
- 311
1
vote
0
answers
214
views
Restriction of a Sobolev function to a straight line
I have been asked the following question, and I have to admit that I have no idea about the answer.
Assume that $f \colon (a,b) \to \mathbb{R}$ is a function. Assume also that there exists a ...
- 459
5
votes
1
answer
101
views
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 ...
- 53
6
votes
1
answer
791
views
Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable.
Suppose that a continuous function $f$ on the line and satisfies
$$
|f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1]
$$
...
- 111
4
votes
0
answers
261
views
Is the following integral positive or not?
Let $n$ be a given even positive integer. We have the following integral
\begin{eqnarray}
&&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
- 1,997
3
votes
1
answer
594
views
What is the rate of convergence? [closed]
How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
- 11.3k
5
votes
2
answers
271
views
Continuous map from connected subset of R^n to one of the real zero of an odd degree polynomial whose coefficients are polynoms of the variables
Let take a real multivariate polynomial $P(x_1, \ldots, x_n, y)$ such as the degree of P relatively to the variable $y$ is odd. Thus, for each $X = x_1,\ldots,x_n \in\mathbb{R}^n$, the univariate ...
- 103
8
votes
1
answer
527
views
Interpolation between L^1 and Sobolev Space
Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that
$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ||f||_{L^...
- 101
1
vote
1
answer
201
views
Existence of a certain norm on space of measurable functions
Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can ...
- 13
1
vote
0
answers
148
views
Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
3
votes
0
answers
1k
views
Concentration of Sub-exponential random Vectors
I was wondering if there is a similar definition of multivariate sub-exponential distribution as the sub-Gaussian case.
Specifically, a random vector $X \in \mathbf{R}^d$ is sub-Gaussian if
\begin{...
- 1,127
8
votes
3
answers
637
views
Method to compute fundamental solutions which are distributions
The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
3
votes
0
answers
189
views
Discussing results that follow from : $\sum_{n=1}^{\infty}\frac{1}{x_n}$ converges if and only if $\sum_{n=1}^{\infty}\frac{1}{1+x_n}$ converges [closed]
The following result is a simple result but I think it reveals some interesting results as consequences.
Given a sequence $\{x_n\}$ of positive integers then the $\sum_{n=1}^{\infty}\frac{1}{x_n}$ ...
- 850
1
vote
1
answer
338
views
Is this a superharmonic function?
Hi everyone: Let $ \Omega $ be a bounded open set of $ \mathbb{R}^{N} $, $ N\geq2 $, and $ F\subset \Omega $ with empty interior. Suppose there exists a superharmonic function $ u $ on $ \Omega\...
- 411
2
votes
1
answer
153
views
Function in $B(\mathbb{R})$
Denote by $B(\mathbb{R})$ the set of all functions on $\mathbb{R}$ which are representable in the form $f(x)=\int_{\mathbb{R}}e^{itx}d\mu(t)$, where $\mu$ is a finite complex-valued Borel measure.
...
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
- 13.4k
3
votes
1
answer
496
views
Prove that these two definitions of "natural" integration constant coincide when both converge
These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...
- 10.1k
1
vote
0
answers
136
views
Reference for Existence and uniqueness of an Integro-Differential Equation
I have an Integro-Differential Equation (IDE) of the following form:
$$
x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds,
$$
I have found this classical reference, but the IDEs considered therein ...
- 121
4
votes
1
answer
484
views
Question about normalization factors in the direct integral of operators
So the original question I wanted to ask was this one:
I'm currently a bit puzzled about the normalization for the Gelfand transform $U$:
So if we have a periodic Schrödinger operator $H$, then we ...
- 95