All Questions
1,583 questions
2
votes
2
answers
741
views
Bochner's theorem for measures of positive type
Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\...
- 195
2
votes
1
answer
773
views
On the continuity of map $\Gamma$
Let $M$ be the space of right continuous functions $\ell: \mathbb R_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>...
- 1,334
2
votes
1
answer
1k
views
Are bounded functions L-1 compact?
Let $(X,\Sigma,\mu)$ be a finite measure space (i.e., $\mu(X) < \infty$). Let $\mathcal{F}$ be the set of $\mu$-measurable functions $f:X \to \mathbb{R}$ that are bounded in $[0,1]$, so that $0 \...
- 1,322
2
votes
1
answer
196
views
Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$
Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem
$$
(1) \quad \begin{cases}
(-\Delta)^s u +\lambda u= 0 & x \in \Omega \\
u = 1 & x \in \mathbb R^n \...
- 161
2
votes
1
answer
1k
views
Minimizing the Perimeter of a polyomino
Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...
- 23
2
votes
0
answers
379
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,199
2
votes
1
answer
1k
views
Mixed (anisotropic) Sobolev spaces
Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...
- 23
2
votes
1
answer
614
views
Distance matrices
We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x_1,\cdots,x_n \in X$ such that $M=[d(x_i,x_j)]_{n\times n}$.
Question. For which ...
- 1,991
2
votes
1
answer
756
views
General strategy for studying the decay of eigenvalues of kernel integral operators
Disclaimer. Please, be patient, I'm here to learn functional analysis...
Let $X$ be the unit sphere in $\mathbb R^n$ and let $\sigma$ be the uniform measure on $X$. Consider a positive definite ...
- 6,853
2
votes
1
answer
151
views
On congruent partitions of planar regions
Given any integer $n$, any rectangular region or any sector of a disc (including the full disk as a boundary case) can be cut into $n$ mutually congruent pieces - by equally spaced parallel lines and ...
- 5,979
2
votes
1
answer
244
views
$R$ is a right multiplier and $R(a)b=a\overset{?}{\implies} A$ is unital
Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that
$$
\exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad
$$
implies $A$ is unital. I know this is true if A is a weak$...
- 1,697
2
votes
2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 201
2
votes
1
answer
198
views
Question $B_5 \equiv B_1$ or $B_5 \ne B_1$?
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1, A_2, A_3, A_4$ be four arbitrary points lie on $(C_1)$; $B_1$ be arbitrary point on $(C_2)$. The ...
- 477
2
votes
1
answer
169
views
How to choose minimisers in a continuous way
Let $\langle X, X' \rangle$ be a dual pair equipped with the weak and weak* topologies.
Let $C$ be a weak* compact subset of $X'$ with nonempty interior. For each $x \in X$, let $M(x)$ be the set of ...
- 869
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 167
2
votes
1
answer
1k
views
Strong maximum principle for heat equation. Positivity of solution
I have a non-negative solution $u \in L^2(0,T;H^1) \cap H^1(0,T;(H^1)')$ of the heat equation
$$u_t-\Delta u =0$$
on bounded $C^1$ domain $\Omega$, with the boundary condition
$$\frac{\partial u(t,x)}{...
2
votes
1
answer
168
views
Regarding representation of an outer function
Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
- 111
2
votes
1
answer
205
views
Estimates for an elliptic PDE
Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply ...
- 469
2
votes
1
answer
162
views
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
- 1,209
2
votes
1
answer
94
views
Density of functions into the circle glueing
Let $\{U_i\}_{i=1}^2$ be an open cover of $S^1$, with $U_i\cong \mathbb{R}$ (for example, $U_1$ is the lower arc of the circle and $U_2$ is the upper part). Let $\iota_i:U_i\hookrightarrow S^1$ be ...
- 5,405
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
0
answers
877
views
Changing the order of integration of double integral: references and theorems
The Fubini's theorem states that if we have $ \int_0^{\infty} \int_0^{\infty} |f(t,x)| dt dx$ well defined (i.e. function is absolutely integrable) then we can interchange order of integration:
$$ \...
- 1,199
1
vote
1
answer
264
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 835
1
vote
2
answers
220
views
A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$.
Note that $A$ is non-empty with a Baire category argument.
I ...
- 356
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
1
vote
1
answer
1k
views
Optimal fitting of spheres in a cylinder
How to find the minimum height and width of a cylinder containing n identical spheres?
- 21
1
vote
1
answer
317
views
The continuous convergence given the a.e. convergence
Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a uniformly bounded sequence (i.e., there exists $C>0$: $|f_n| < C$ for every $n$) such that
$$ f_n \in C^2_x \times C^1_t, $$
...
- 41
1
vote
0
answers
92
views
Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
1
vote
1
answer
72
views
Generalised Lebesgue transform continuous wrt. strict topology?
Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the ...
- 463
1
vote
0
answers
133
views
Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?
This is a question subsequent to the one:
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
There, I received a very helpful answer that the Gaussian poincare inequality for any ...
- 3,477
1
vote
2
answers
275
views
Again, proving that specific preorder on the set of measurable functions is symmetric
This question is followup to the previous similar question. There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of ...
- 105
1
vote
1
answer
162
views
Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
1
vote
1
answer
113
views
Is $I-S$ in my attempt of Fredholm alternative injective?
Let $E$ be a Banach space. Let $\mathcal K(E)$ be the space of all compact (bounded linear) operators from $E$ to $E$. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let ...
- 657
1
vote
2
answers
255
views
Continuous Linear Mappings on Subspaces of $\mathcal{D}(\Omega)$
Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$ and $\mathcal{D}(\Omega)$ the usual space of test functions of distribution theory, with the usual topology $\tau$ of the inductive limit of ...
- 640
1
vote
0
answers
99
views
Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$
I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
- 167
1
vote
1
answer
433
views
Why complex conjugate in definition of the Fourier transform?
Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be
$$\widehat{f}(\chi) = \int_G f(g) \bar{\chi}(g) \mathrm{d} g,$$
...
- 21.3k
1
vote
1
answer
319
views
Is $(f \ast K)'' \in L^1(\mathbb R)$ for $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$?
Is it possible to deduce that $$(f \ast K)'' \in L^1(\mathbb R)$$ if $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$? What I can prove is that $(f \ast K)' \in L^1 \cap L^\infty$. Is ...
- 131
1
vote
1
answer
786
views
A question on the Riesz-Markov theorem about dual space of $C_0(X)$
I need to use the version of this theorem about the dual space of $C_0(X)$, which is the set of continuous functions on X which vanish at infinity.
I found in Wikipedia that:
Here the space $X$ is ...
- 43
1
vote
1
answer
128
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$
Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition
$$\oplus_{...
- 407
1
vote
2
answers
153
views
Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
- 5,979
1
vote
1
answer
320
views
A formula for the area of bicentric quadrilateral
Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...
- 2,661
1
vote
0
answers
177
views
A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 839
1
vote
0
answers
164
views
How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
1
vote
1
answer
223
views
A linear algebraic q-difference equation [SOLVED]
I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...
- 133
1
vote
1
answer
90
views
When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?
EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
- 21.8k
1
vote
1
answer
392
views
Frustrating the number of possible common edges between two connected components composed of square Penrose tiles
Imagine I have two bags of square and planar unit square tiles, with Penrose-like "nodules" on their edges s.t. two tiles can only be placed together if their edges are flush (i.e. if the two vertices ...
- 13
1
vote
1
answer
205
views
Is the canonical map $\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$ injective?
If $A,B$ are $\mathbb R$-Banach spaces, let $A\:\hat\otimes_\pi\:B$ denote the completion of the algebraic tensor product of $A$ and $B$ with respect to the projective norm. Let $X,Y,E,F$ be $\mathbb ...
- 167
1
vote
0
answers
59
views
What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
- 5,979
1
vote
2
answers
604
views
Generalized limits
Cross-posted from Math SE.
The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:
...
- 2,203
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852