All Questions
5,856 questions
2
votes
0
answers
124
views
Error term for a Fourier integral
There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable,
$$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$
So it should be that
...
- 3,799
1
vote
1
answer
527
views
An Integral Functional Equation
Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function $...
- 2,239
3
votes
1
answer
643
views
Is a Cauchy principal value invariant under a "change of variables"?
Let $f \in C^{\gamma}_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere ...
- 33
2
votes
1
answer
403
views
The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 1,788
4
votes
1
answer
2k
views
Determinant and symmetric power
Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
- 1,510
2
votes
3
answers
947
views
How can I measure the Morse index in infinite dimensions?
Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
- 54.6k
3
votes
0
answers
314
views
Is a particular set of polynomials dense in a set of functions?
Let us consider the set $\mathcal{F}$ of strictly increasing continuous functions from $[0;1]$ on $[0,1]$ that cancel in $0$ and are equal to $1$ in $1$. So, if $f\in \mathcal{F}$ one has $f(0)=0$ and ...
- 31
2
votes
0
answers
108
views
Distributive law
I was wondering whether there is any reference that deals with the distributive law for infinitely many elements, i.e.
$$
\prod_{i\in \mathbb N} \sum_{k\in \mathbb N} \alpha_{i,k} = \sum_{(k_i)_{i\in ...
1
vote
1
answer
164
views
Estimates on evolution operator
Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert_{...
- 13
0
votes
1
answer
96
views
Optimal covering with finite subcollection of open sets
This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for ...
4
votes
2
answers
323
views
Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?
Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that
$$\lim_{n} ~a_{nn} = L_1, \qquad
\lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$
exist. Is there a sufficient criteria to conclude ...
- 3,245
1
vote
0
answers
41
views
Necessary additive and multiplicative properties to characterize a mildly growing function
Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...
- 13.9k
0
votes
0
answers
428
views
Given an even function how to obtain the most close odd function and vise versa?
Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference $|...
1
vote
1
answer
270
views
Non-continuous higher differentiability, II
In a comment on this question, Tom Goodwillie proposed a notion of higher differentiability that I elaborate to something like the following:
Let $f:\mathbb{R}^n \to \mathbb{R}$. Let's say that $f$ ...
- 66.8k
2
votes
1
answer
255
views
Quotients of perfect powers separated by an integer
Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
- 3,595
1
vote
0
answers
53
views
Given a fixed convex domain $\Omega$ in 3D, for what value $c$ the function $f(c) := \int_{\partial \Omega} |x-c| d \sigma_x$ gets its minimum?
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^3$, then consider the following minimization problem:
$$\inf_{c \in \overline{\Omega}} f(c), \quad f(c) := \int_{\partial \Omega} |x-c| ...
- 1,350
1
vote
1
answer
487
views
New differintegral formula: how is it related to other differintegral formulas?
Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x)= \lim_{t\to s} \...
- 10.1k
1
vote
0
answers
90
views
Expansion of a power series as integral of cosine functions
Suppose $$f(x)=\sum_{k=0}^\infty (-1)^k x^{2k} \int_0^1 p(\xi_{2k})\int_0^{\xi_{2k}}q(\xi_{2k-1})\cdots\int_0^{\xi_3}p(\xi_2)\int_0^{\xi_2}q(\xi_1) d\xi_1 \;d\xi_2\cdots d\xi_{2k-1}\;d\xi_{2k},$$
...
- 587
0
votes
1
answer
905
views
Hölder continuity of uniform limit of piecewise constant functions
Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...
- 81
1
vote
0
answers
92
views
Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
14
votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
- 24.7k
3
votes
0
answers
295
views
Density of function spaces
Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an ...
- 83
9
votes
2
answers
791
views
Asymptotic difference between a function and its "binomial average"
(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{...
- 3,283
1
vote
1
answer
1k
views
How to verify the weak convergence?
Given a finite measure on a compact, take $f_n\in L^1$ with norms $\leq 1$ and suppose that $\int f_n g$ tends to a limit for all continuous $g$. Is it true that then $\int f_n g$ converge for any $g\...
- 11
2
votes
0
answers
84
views
limit multiple integral
I want to know if $\lim_{T-> \infty}$ of this integral
$$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\
\times \int\limits_{[0,T]^{4}}e^{\theta(t_{1}-s_{1})}e^{\theta(t_{2}-s_{2})}\left\...
- 31
2
votes
1
answer
579
views
Does the Border (Boundary) Points of a convex body make a concave function?
Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis ...
- 1,421
0
votes
0
answers
173
views
Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
- 679
1
vote
1
answer
370
views
A question which belongs to a class of Zygmund functions
Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, \,\,\,\...
- 197
2
votes
1
answer
689
views
Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
- 2,358
2
votes
1
answer
447
views
Original source for a well-known result of convergence in measure and almost everywhere
A well-known result in measure theory states that given a sequence $(f_n)_{n=1}^\infty$ of measurable functions from a $\sigma$-finite measure space $(X,\mathcal{A},\mu)$ to $\mathbb{R}$ then the ...
- 729
1
vote
1
answer
741
views
Some infinite products related to prime numbers.
Let $P$ be the set of all odd prime numbers. I am looking for all $s\in(1,\infty)$ for them
$
A=\prod_{p\in P} (1+\frac{1}{(p-1)^s})^{p-1}
$
exists (i.e. is finite). I know that it should be ...
6
votes
2
answers
425
views
orderings of the field R((x, y))
I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the ...
- 620
7
votes
1
answer
1k
views
Can a continuous, nowhere differentiable function have specified "shape" at every point?
I'm a bit embarrassed to admit that:
a) This is a rather unmotivated question.
b) I can't remember whether or not I've asked this before, but searching doesn't seem to turn anything up so ...
...
- 793
1
vote
0
answers
71
views
The jump set of $SBV$ function for different value of parameter in image denoising problem
The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...
- 679
3
votes
1
answer
153
views
Separability of $R_+\times\mathcal{C}(R_+)$
Let $\mathcal{C}(R_+)$ be the space of continuous functions $f$ defined on $[0,+\infty)$ with $f(0)=0$. Denote by $\Omega$ the product of $R_+$ and $\mathcal{C}(R_+)$. Now endow $\Omega$ with the ...
- 1,835
6
votes
1
answer
634
views
Arbitrary small positive lower semi continuous functions
This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...
- 1,788
2
votes
2
answers
518
views
Lower bounds on derivative around zero set of a positive smooth function
As part of a different problem, I came across the following simplified question, for which I cannot exhibit a proof nor a counterexample. Note that the assumptions of smoothness and strict positivity ...
- 83
1
vote
2
answers
292
views
specific improper integral involving erf
I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:
$$
\int_{1}^{\infty} \left[\mathrm{erf}\...
- 21
1
vote
0
answers
150
views
Positivity of alternating series
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then
$$
f(x)=\sum_{n\geq 0}a_n x^n
$$
converges absolutely for all $x$. Under ...
- 1,329
2
votes
2
answers
3k
views
Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 7,442
2
votes
2
answers
283
views
A general inequality about spherical mean of a function
suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...
- 21
2
votes
0
answers
60
views
A question about Kolmogorov Superpositions
D.A. Sprecher showed (https://www.researchgate.net/profile/David_Sprecher2/publication/243052898_A_Representation_Theorem_for_Continuous_Functions_of_Several_Variables/links/554929f20cf2ebfd8e3ad956....
- 371
1
vote
1
answer
110
views
Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$
Let $\Omega
\subset
\mathbb{R}^{N}$
be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Caratheodory function such that $g(x,t)=0$
for $t\leq0$
. Suppose that ...
- 135
4
votes
0
answers
428
views
Inverse of matrix-valued function
Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by
\begin{equation}
\gamma_c(\Omega)=\frac1{\sqrt{(2\pi)^{k}|\Omega|}}\int_{\mathbb{R}^k}\{(-...
- 133
5
votes
2
answers
917
views
Is the inclusion of Lebesgue spaces compact?
[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-...
- 39.1k
1
vote
1
answer
114
views
question about $TGV^2$ space
Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and
$$
TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(...
- 679
1
vote
2
answers
654
views
Limit with theorem of dominated convergence
Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$)
I have to calculate this limit
$$\lim_{|x-y|\to 0}\int_{\...
- 25
5
votes
0
answers
195
views
Characterizations of an exotic measure on the open sets in the circle $S^{1}$
Suppose that $U\subseteq S^{1}$ is open where $S^{1}=\{z\in\mathbb{Z}:|z|=1\}$. Then define $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
-2
votes
1
answer
80
views
Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [closed]
Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative $\lim_{\epsilon->0}\frac{h(x+\epsilon)-2h(x)+h(x-\epsilon)}{\epsilon^...
- 39
0
votes
0
answers
152
views
The Lebesgue measure of the low level sets of the two-dimmension Fourier transform of a compactly supported function
Let $f\in {{L}^{1}}\left( {{\mathbb{R}}^{2}} \right)$ be a density function with the support $\operatorname{supp}\left( f \right)\subset \left[ a,b \right]\times \left[ c,d \right]$. Denoted by $\hat{...
- 141