All Questions
930 questions
3
votes
0
answers
306
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
3
votes
1
answer
518
views
Connection between the Fourier transform of f and |f|
If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
\|\widehat{f}\|_{L^{p'}}\...
- 425
3
votes
1
answer
1k
views
Duality map in strictly convex Banach spaces
Say $(B,\|\cdot\|)$ is a finite dimensional, strictly convex Banach space. Is it true that the map $\phi:B^*\rightarrow B$ which takes a linear functional $f$ with $\|f\|=1$ into the unique unit norm ...
- 39
3
votes
2
answers
1k
views
Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
- 3,738
3
votes
1
answer
365
views
Final topology of surjective linear map on Banach space
Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...
- 315
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
3
votes
1
answer
233
views
Extending linear isometries from subspaces of $\ell_p^n$
Take $p\in (1,\infty)\setminus \{2\}$. Let $X$ be a subspace of $\ell_p^n$ and let $U\colon X\to \ell_p^m$ ($m\geqslant n$) be a linear isometry. Is it possible to extend $U$ to a (non-surjective) ...
- 331
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
3
votes
0
answers
160
views
Elements of vector-valued $L^1$-spaces
Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$...
- 799
3
votes
1
answer
233
views
Show identity for a norm on Fréchet differentiable functions on a Banach space
Let $E$ be a $\mathbb R$-Banach space, $v:E\to(0,\infty)$ be continuous with $$\inf_{x\in E}v(x)>0\tag1,$$ $r\in(0,1]$ and$^1$ $$\rho(x,y):=\inf_{\substack{c\:\in\:C^1([0,\:1],\:E)\\ c(0)=x\\ c(1)=...
- 167
3
votes
1
answer
173
views
Weak Lebesgue spaces and an estimate for BV functions
Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that
$$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$
holds for a.e. $...
- 839
3
votes
1
answer
479
views
Extension of positive functionals II
This is a follow-up to
Extension of positive functionals.
Assume that $X=R^n$ with the canonical order (I indicate with $K$ the positive cone, $x \in K$ iff $x_i \ge 0$ for all $i$) and let $L:M \to R$...
- 6,035
2
votes
2
answers
252
views
When is the Minkowski sum of weighted compact sets $w_1 B_1 + w_2 B_2 + \ldots$ (with $w \in L^1$) closed?
Let $B_1,B_2,\ldots,$ be compact subsets of $\mathbb R^d$ and $w_1,w_2,\ldots$ be nonnegative numbers summing to $1$. Consider the set
$$
A := w_1 B_1 + w_2 B_2 + \ldots = \left\{\sum_{n=1}^\infty w_n ...
- 6,853
2
votes
0
answers
382
views
Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)
Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.
I am ...
- 53
2
votes
1
answer
453
views
Weak convergence of probability measures on weak versus strong dual
The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
2
votes
1
answer
168
views
Regarding representation of an outer function
Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
- 111
2
votes
1
answer
169
views
How to choose minimisers in a continuous way
Let $\langle X, X' \rangle$ be a dual pair equipped with the weak and weak* topologies.
Let $C$ be a weak* compact subset of $X'$ with nonempty interior. For each $x \in X$, let $M(x)$ be the set of ...
- 869
2
votes
2
answers
2k
views
Separable quotients of non-separable Banach spaces?
I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of $l^{\infty}=l^{\infty}(\mathbb{N})$ with the property that $l^{\infty}/M$ is separable. I ...
- 281
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
2
answers
223
views
Relating function value to $L^2$ norm in Holder space
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 410
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
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
$$\frac{\partial u(t,x)}{...
2
votes
1
answer
228
views
Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set
$$\tag{1}
\int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
- 809
2
votes
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
2
votes
0
answers
107
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
2
votes
1
answer
236
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 835
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 835
2
votes
1
answer
142
views
Estimating a solution to an Euler-type ODE
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number.
Let $u(r)$ be a function on $[1,\infty)$ ...
- 969
2
votes
1
answer
702
views
Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces
I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.
In the notation of the paper, let
$H,H_1$ be separable $\...
- 167
2
votes
1
answer
315
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum
How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
- 2,239
2
votes
2
answers
254
views
Is there any work on distributional vector fields?
I know that people think about weak solutions to PDEs by turning a differential equation into an integral equation. Have people studied any analogue to this where we start with an integral equation, ...
- 182
2
votes
1
answer
320
views
Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
- 807
2
votes
1
answer
155
views
Tempered distributions at non-coinciding points and density of Schwartz functions
In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind.
Let us consider the Schwartz space $\mathcal{...
- 3,477
2
votes
1
answer
265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
238
views
Hilbert-irreducible Banach space
A Banach space $X$ is called Hilbert-irreducible if it satisfies the following condition:
If a subspace $Y\subset X$ satisfies the parallelogram equality, then $Y$ is necessarilly a one ...
- 356
2
votes
2
answers
634
views
Continuous upper envelope of upper semicontinuous function
Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...
- 1,399
2
votes
0
answers
184
views
Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$
Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...
- 1,372
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
1
answer
713
views
Reference request: numerical analysis of PDEs and integro partial differential equations
I'm very new to the field of numerical analysis of PDE and integro partial differential equations.
My advisor (who is not a specialist in this area) highly recommended to read
Randall J. LeVeque'...
user102027
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
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
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
- 657
2
votes
1
answer
255
views
On the infimal convolution of two norms on $\mathbb R^n$
$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation*}
K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1),
\end{equation*}
\begin{equation*}
M:=M_{n,...
- 128k
2
votes
1
answer
105
views
Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define
$$
\begin{split}
K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\
M_n(a,t) &:= ...
- 6,853
2
votes
1
answer
525
views
Sequential Continuity in dual spaces of separable Banach Spaces
Is the following true?
Let $X$ and $Y$ be separable Banach spaces and consider their dual spaces $X^*$ and $Y^*$ equipped with weak* topology. Suppose that a linear map $T:X^*\to Y^*$ is sequentially ...
- 165
2
votes
0
answers
115
views
Application of Bishop-Phelps Theorem
Consider a real Banach space $X$ and its continuous dual $X^*$. The Bishop-Phelps Theorem states that the set
$$A^*=\{x^*\in X^* \mid x^* \text{ attains its supremum on } \text{ the unit ball } \...
- 193
2
votes
1
answer
471
views
Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?
Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...
- 175
2
votes
1
answer
166
views
Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
- 131
2
votes
1
answer
296
views
An abstract characterisation of weak* topologies
Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if ...
- 153