All Questions
5,630 questions
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Do they have the same limit?
Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$
$$
\frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
- 87
0
votes
1
answer
705
views
Continuity of a Functional
A certain functional $T$ is defined as:
$$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
The result that above functional is ...
- 10.3k
2
votes
1
answer
888
views
Absolutely continuous functions
it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality
$$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$
for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...
- 1,881
0
votes
0
answers
161
views
Asymptotic analysis of a sum of complex summands using integral
I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, |y|\...
- 93
1
vote
1
answer
759
views
meromorphic extension of a function
Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in \...
- 39.9k
1
vote
0
answers
225
views
Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?
By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the $n!...
- 1,893
-2
votes
2
answers
119
views
Systems of ODEs that fulfill a matrix relationship at steady state [closed]
It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...
- 60.6k
1
vote
1
answer
258
views
Do the sequences with divergent associated $\zeta$-function form a vector space?
Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication.
...
CommunityBot
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1
vote
1
answer
281
views
On the Hölder regularity of an integral function
Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
- 2,715
5
votes
1
answer
185
views
Existence of an equivariant Morse function
Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse ...
- 7,583
0
votes
0
answers
145
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A question about the duality principle
Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by $$\Phi_Kf(y)=...
- 133
-1
votes
1
answer
1k
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A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
CommunityBot
- 1
2
votes
1
answer
135
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Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g.
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
Now define ...
- 48.9k
2
votes
0
answers
343
views
continuity with respect to weak-${\ast}$ topology
Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
- 1,835
2
votes
1
answer
800
views
A question about Skorokhod metric
I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
CommunityBot
- 1
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
$$
|f(x) - ...
- 24.2k
2
votes
0
answers
431
views
What is the purpose of the definition of "metric regularity"/"regularity modulus"?
A set mapping $F:X \rightrightarrows Y$ is said to be metrically regular for $\overline{x}\in X$ and $\overline{y} \in Y$ if there exists a $\kappa\in(0,\infty)$ for which
$$
d(x,F^{-1}(y))\leq \...
- 81
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},...,...
- 13.6k
1
vote
0
answers
92
views
vector space of ternary forms with real rooted property
Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that ...
- 3,031
7
votes
1
answer
1k
views
The closed form of $\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$
The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$
where $\psi(n)$ is digamma function
arose in the evaluation of an integral I posted on MSE, https://...
CommunityBot
- 1
3
votes
1
answer
2k
views
A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm
Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...
- 1,590
5
votes
2
answers
644
views
Exotic Lebesgue Measurable Function
Measurable functions whose graphs are dense in the plane are well known. Examples include, the Conway 13 function, as given in the answer in this link: When is the graph of a function a dense set?
...
CommunityBot
- 1
25
votes
9
answers
6k
views
Function with range equal to whole reals on every open set
There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$.
I want to generalize this in a way to get a function ...
- 13k
3
votes
0
answers
187
views
An upper bound for a average of a function in $L_{p}([0,1))$
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j},~
1\leq j \leq 2^{n} \}
\right\} \right)_{n ...
- 103
3
votes
1
answer
153
views
Separability of $R_+\times\mathcal{C}(R_+)$
Let $\mathcal{C}(R_+)$ be the space of continuous functions $f$ defined on $[0,+\infty)$ with $f(0)=0$. Denote by $\Omega$ the product of $R_+$ and $\mathcal{C}(R_+)$. Now endow $\Omega$ with the ...
- 26
3
votes
4
answers
1k
views
Simple bound for generalized geometric series
Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.
Is there a simple way of proving this ...
- 60.6k
3
votes
3
answers
1k
views
On the existence of compactly supported functions whose its Fourier transform satisfies a given condition
My question is concerned with the existence of compactly supported functions whose its Fourier transform satisfies a given condition: For $\gamma\ge 1$, one can prove that there is no compactly ...
- 91.8k
0
votes
1
answer
164
views
Extending derivations to the superposition closure
Let $X$ be a set and $\mathcal{F}\subseteq {\mathbb{R}^X}$ an arbitrary family of functions.
The superposition closure of $\mathcal{F}$ is defined as
$$
\overline{\mathcal{F}}=\{ H\circ(f_1\times\...
CommunityBot
- 1
1
vote
1
answer
242
views
Can (how) one distinguish germs of continuous functions by a countable set of params?
Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...
CommunityBot
- 1
4
votes
1
answer
216
views
Chances for a cosine polynomial to be positive at a point
Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
- 34.7k
4
votes
1
answer
977
views
Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
CommunityBot
- 1
3
votes
1
answer
216
views
Quantitative stability: Hausdorff distance between subdifferentials
Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
- 12.7k
3
votes
1
answer
383
views
"Nice" functions on infinite-dimensional space of germs of continuous functions at a point
Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
CommunityBot
- 1
-2
votes
1
answer
212
views
A calculus question [closed]
Fix $q>1$. Define the function
$$
f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$
The problem is whether the following is true,
$$
\lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
- 23.2k
0
votes
0
answers
182
views
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ \frac{j}{2^{...
- 1,396
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
1
vote
0
answers
139
views
Can we define log-convex operators?
Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq [...
- 55
0
votes
2
answers
118
views
Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
- 2,239
31
votes
1
answer
3k
views
What did Rolle prove when he proved Rolle's theorem?
Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...
CommunityBot
- 1
3
votes
1
answer
640
views
Relationship between LlogL and Hardy spaces
I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in L^1\,s.t.\,\...
- 452
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
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert_{...
- 24.2k
0
votes
1
answer
394
views
Legendre differential equation with additional term
In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( 8\,...
- 13.4k
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 423
1
vote
1
answer
126
views
Evaluation of the multiple integral [closed]
Would you give me any suggestions or comments on evaluating the following $n$-dimensional
integral? $$ \int_{[0,t]^n} h(x) dx $$
where
$ x=(x_1 ,x_2 , \cdots, x_n ), h(x)= \prod_{k=1}^n min( \bar{...
- 24.8k
0
votes
1
answer
128
views
Most natural smooth interpolation of 1,4=2^2,3^27,4^4^4^4, [closed]
Is there a functional equation for extending this to a smooth real function?
- 24.8k
1
vote
1
answer
56
views
Can we implicitly fit a system of linear ODEs by reduced information?
Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...
- 8,086
2
votes
0
answers
76
views
question about a genralized Skorokhod topology
Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
$$\rho(f,g):=\inf_{\lambda\in\Lambda}\Big\{\max\...
- 1,835
3
votes
0
answers
290
views
Does there exist a supersmooth non-polynomial function?
Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
CommunityBot
- 1
4
votes
0
answers
219
views
Is every supersmooth function a local polynomial?
This question is a follow up question to this question that I recently asked.
A $C^{\infty}$ function $f:(c,d)\rightarrow\mathbb{R}$ shall be called a local polynomial if whenever $f:(c,d)\rightarrow\...
CommunityBot
- 1
2
votes
2
answers
144
views
First order pde with characteristics [closed]
Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...
- 54.2k