All Questions
13,944 questions
7
votes
1
answer
299
views
Intermediate spaces of test functions between $\mathcal{S}$ and $\mathcal{D}$?
On $\mathbb{R}^n$, let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{D}(\mathbb{R}^n)$ be the space of smooth, compactly supported functions.
According to p.145 of the book by Reed &...
- 3,477
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
0
votes
0
answers
55
views
Time regularity vs space regularity for parabolic PDE
Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
- 311
0
votes
0
answers
109
views
A Lipschitz function induced by the infimum of the length of curves
Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of ...
- 69
5
votes
1
answer
282
views
Is there a singular function that is Hölder continuous of every order less than $1$?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
Does there exist a singular function that is Hölder ...
- 6,215
16
votes
4
answers
2k
views
Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with
$f_n \to f$ uniformly for some (necessarily) continuous $f$.
$f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
Is it true ...
- 6,215
5
votes
2
answers
297
views
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with
$f_n \to f$ uniformly for some continuous $f$.
$f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,
where we ...
- 6,215
0
votes
0
answers
95
views
Functions representing all strings somewhere
Do there exist "nice" (maybe analytic?) functions $f_0,f_1:\mathbb R \to \mathbb R$ such that
$\forall n\in\mathbb N,\forall \sigma\in\{0,1\}^n,\exists x\in\mathbb R, \forall \tau\in\{0,1\}^...
- 24.8k
2
votes
1
answer
203
views
Does this maximisation problem admit a finite upper bound?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,399
2
votes
2
answers
192
views
Upper/Lower bounds of real-analytic functions with infinite Taylor series
For example, in 1-D, given some positive increasing polynomial $p(x) = a_1x+\ldots+a_nx^n$, $p(0) = 0$, there exists constants $b_1,b_2$ such that for $x<\delta$, for some $\delta > 0$, we have ...
- 171
0
votes
0
answers
120
views
Equality of two measures on functional spaces
It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
- 1
10
votes
1
answer
518
views
Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions
Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
- 435
0
votes
0
answers
128
views
Lipschitz function approximated by smooth functions with zero a regular value
Consider a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$. Then I want a family of smooth functions $f_\epsilon : \mathbb{R}^n\to\mathbb{R}$, such that $f_\epsilon\to f$ uniformly on compact sets, ...
- 1,090
2
votes
0
answers
259
views
Least number of circles required to cover a continuous function on $[a,b]$
I asked this question on MSE here.
Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of closed circles with fixed radius $r$ required to cover the graph of $f$?
It is ...
- 541
11
votes
2
answers
587
views
Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
- 271
1
vote
2
answers
162
views
Proof that the closed convex hull of a weakly convergent sequence has empty interior using properties of $c_0$?
Let $X$ be an infinite-dimensional Banach space, and $x_n\to 0$ weakly in $X.$ Let $K$ be the closed convex hull of $\{x_n\}.$
I remember a proof that $K$ has empty interior as follows: define a map $...
- 1,755
17
votes
0
answers
677
views
Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
- 37.8k
6
votes
1
answer
309
views
Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $f \in C^1 (\mathbb \Omega)$ for some open, connected, bounded subset $\Omega$ of $\mathbb R^n$. We consider for each $t \...
- 6,215
0
votes
0
answers
48
views
First nonzero derivative bounded below (2 dimensions)
Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real-analytic, have only one zero (at $(0,0)$) and be strictly increasing ...
- 171
0
votes
0
answers
96
views
Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 1
-2
votes
1
answer
93
views
Express the connection between roots [closed]
$\DeclareMathOperator\elim{lim}\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\lmb{lmb}\DeclareMathOperator\Lmb{Lmb}\DeclareMathOperator\mts{mts}$There are two similar functions; they determine the ...
- 55
1
vote
0
answers
105
views
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
- 29
1
vote
1
answer
121
views
Question on complemented subspaces of a product space
Assume that we have closed subspaces $Y_1$ and $Y_2$ of Banach spaces $X_1$ and $X_2$, respectively. If the product $Y_1\times Y_2$ is complemented in $X_1\times X_2$, does it follow that $Y_i$ is ...
- 19
6
votes
2
answers
380
views
Proving convergence of solution of a fixed point equation
I encountered a nasty sequence $(x_n)_{n=1}^\infty $ defined as the smallest positive fixed point of the fixed point equation $ x_n = f_n(x_n) $, where $f_n$ is given by
$$ f_n(x) = \sum_{k=0}^{\...
- 61
3
votes
0
answers
141
views
Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$
Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
- 435
0
votes
1
answer
96
views
Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms
The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products".
In physics, these "time-ordered products" ...
- 3,477
1
vote
0
answers
93
views
Multivariate polynomial approximation
Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$.
Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
- 61
5
votes
1
answer
349
views
Equilateral triangle in a Brownian path
I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory ...
- 393
3
votes
0
answers
68
views
How powerful are sequences of Steiner symmetrizations?
I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
- 171
2
votes
2
answers
173
views
Gronwall-type inequality involving norms of distinct Lebesgue spaces
Let $d \geq 1$, $\Omega \subset \mathbb{R^d}$ be a bounded domain and let $\phi : [0,T]\times \Omega \mapsto \mathbb{R}$ be a measurable and bounded function. Assume that the following differential ...
- 213
5
votes
1
answer
375
views
What is the length of an algebraic curve?
The following question seems to be somewhat standard, but I was unable to find any reference. I would be grateful for any pointers to relevant literature.
We consider a real polynomial $p(x,y)$ of ...
- 53
2
votes
1
answer
194
views
Functions with derivatives growing at rate $r>0$
Fix a non-empty closed subset $\Omega\subset\mathbb{R}$.
Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and such that $\sup_{x\in \Omega}\,|\partial^k f(x)|\lesssim k^r$ for some $r\ge 0$ for all $k\in \...
- 364
1
vote
0
answers
215
views
Computing a closed form representation for a Fourier series summation
I want to compute a closed form representation for the below given summation expession.
$$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
- 698
0
votes
0
answers
29
views
$ \sup_{\theta \in [0,2\pi)}\max_{r\leq \delta}\frac{\log\left(\frac{f(r,\theta)}{f(\delta,\theta)}\right)}{\log(r)}<\infty,$ $f$ real analytic
$\textbf{Conjecture.}$
Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $\delta <1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and suppose that $(0,0)$ is the only ...
- 171
4
votes
2
answers
354
views
Injectivity of a convolution operator
Let $p,\mu,\nu$ be probability density functions on
$\mathbb{R}$ such that
$$
\int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x).
$$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
- 407
0
votes
0
answers
106
views
How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin
Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
- 51
3
votes
0
answers
79
views
Continuity of disintegrations in non locally compact spaces
Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
- 63
0
votes
0
answers
49
views
ODE satisfied by a special function
Posted on MSE
Context
I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica . ...
- 393
0
votes
0
answers
71
views
Minimum Slice of Real Analytic Function in Two Variables
Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and have only one zero, namely $(0,0)$. Moreover, assume that $...
- 171
2
votes
2
answers
191
views
Gronwall's inequality in discretized time
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 825
4
votes
0
answers
256
views
Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation
In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...
15
votes
2
answers
888
views
Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms
$$
\Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
- 35.5k
2
votes
0
answers
261
views
When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?
Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$:
$$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
- 20.2k
1
vote
0
answers
82
views
Counting the number of local minima of a function that is the sum of square roots of cosines
Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows
$$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$
where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
- 21
3
votes
1
answer
208
views
Density beyond Stone–Weierstrass
$\DeclareMathOperator\tr{tr}$I need density assertions for spaces of polynomials which are not (that I know of) algebras. One goes like this: Fix $n\in\mathbb N$ let $S$ denote the set of self-adjoint ...
- 460
3
votes
1
answer
233
views
Analytic solutions to analytic differential equations
Let $U \subseteq \mathbb R^{n+2}$ be an open set for some $n \geq 0$, and let $f: U \to \mathbb R$ be an analytic function. Then we say the equation $f(x,y,y',\ldots,y^{(n)})=0$ is an analytic ...
- 31
0
votes
0
answers
46
views
Projection onto Shift Invariant Subspaces of $H^2$
Every shift invariant subspace of the Hardy space $H^2(\mathbb{D})$ is either $\{0\}$ or is of the form $\varphi H^2$ for some inner function $\varphi$. I know that if $\varphi(0) \neq 0$, then the ...
- 23
3
votes
1
answer
172
views
1-1 map on the $\{0,1\}^k$
Let integer $k>0$ and let $\{0,1\}^k$ denote the set of all $1\times k$-dim vectors whose every coordinate is eithor 0 or 1, for example, $(0,1,1,0,\dots,1,0,0,1)$. For any such vector $\alpha$, ...
- 349
6
votes
1
answer
309
views
Well distributed sets
Note: All integrals are taken with respect to Lebesgue measure. The symbol $\def\avint{\mathop{\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int}\nolimits} \avint$ denotes the average integral.
We say ...
- 6,215
1
vote
0
answers
67
views
Distribution of zeros for arbitrary Bessel functions
Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
