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Envelopes of functions with respect to some convex cone $\mathcal{F}$

Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
J.R.'s user avatar
  • 291
0 votes
0 answers
79 views

Definition of Moore-Penrose inverse for unbounded self-adjoint operators?

I know there is a concept of Moore-Penrose or pseudoinverse of a matrix. I would like to know if one can define it for densely defined unbounded self-adjoint operators on Hilbert spaces as well. ...
InMathweTrust's user avatar
3 votes
1 answer
79 views

Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$

Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
P. P. Tuong's user avatar
2 votes
1 answer
474 views

Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?

Let $$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$ be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
Balaji Mallikarjun S's user avatar
6 votes
3 answers
282 views

Extreme points of the dual unit ball of a Banach algebra

Let $A$ be a unital Banach algebra. Let $f\in A^*$, $\Vert f\Vert=1$ satisfy that there exists a maximal left ideal $L\subset A$ such that $L\subseteq\ker{f}$. Question: Is $f$ an extreme point of ...
Onur Oktay's user avatar
  • 2,605
2 votes
1 answer
121 views

Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$

For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite, $$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$ Using the triangle inequality $|\eta-\xi|\le |\eta|...
Dispersion's user avatar
2 votes
0 answers
94 views

Representations of the commutation relations and renormalization

I had a hard time deciding whether this question is more appropriate for physicsSE or MO, so I have cross-listed it for the time being. The physicsSE post can be found here. In the lecture Is ...
CBBAM's user avatar
  • 721
0 votes
1 answer
117 views

Validity of approximation method for von Mangoldt function

I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
Brendan Thorne's user avatar
0 votes
0 answers
60 views

Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$

Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one. However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
107 views

Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property

I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
user273331's user avatar
1 vote
0 answers
86 views

Gamma convergence via density argument: Looking for references

I am looking for a reference or result dealing with Gamma via density argument. Let me elaborate more my wish. I am actually trying to establish the Gamma convergence (precisely only the liminf) of a ...
Guy Fsone's user avatar
  • 1,101
1 vote
1 answer
122 views

distance in the matrix algebra w.r.t. the nuclear norm

Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
Krzysztof's user avatar
  • 375
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
3 votes
1 answer
327 views

Derivative norm estimates

Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$. QUESTION. Does this norm estimate hold? ...
T. Amdeberhan's user avatar
7 votes
1 answer
281 views

Norm in the minimal tensor product of C*-algebras

Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
Hannes Thiel's user avatar
  • 3,497
7 votes
0 answers
250 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
3 votes
1 answer
176 views

Are measurable maps with countably separated image in a Banach space always strongly measurable?

Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
Packo's user avatar
  • 285
3 votes
1 answer
187 views

Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
Cauchy's Sequence's user avatar
15 votes
1 answer
602 views

Topological spaces in which countable intersections of dense open sets have dense interior

In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense. Now consider the following strengthening of the Baire ...
Gro-Tsen's user avatar
  • 32.5k
8 votes
0 answers
115 views

optimal regularity for the Neumann heat equation on Lipschitz domains

$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
leo monsaingeon's user avatar
4 votes
1 answer
252 views

Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra

Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put $$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
Andromeda's user avatar
  • 175
2 votes
1 answer
134 views

Density of smooth functions in weighted Sobolev space

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
Bastien's user avatar
  • 23
3 votes
2 answers
137 views

Non-complete space verifying uniform boundedness

Recently, I have seen the so-called uniform boundedness theorem, which says: Let $(X,∥⋅∥)$ be a Banach space and $(Y,∥⋅∥)$ be a normed linear space. Let $A⊂B(X,Y)$ be a pointwise bounded family of ...
Tomas smith Smith's user avatar
4 votes
3 answers
315 views

Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$

Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
Bogdan's user avatar
  • 1,759
5 votes
1 answer
263 views

Counter example for Hadamard Differentiability

I am having a hard time while trying to fully understand Hadamard differentiability. I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
Matthis's user avatar
  • 53
5 votes
2 answers
364 views

Euler-Lagrange equations for minimizer of energy with indicator function

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
BBB's user avatar
  • 93
4 votes
1 answer
196 views

(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?

In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like Uniform boundedness of the partition function in the limit where lattice ...
Isaac's user avatar
  • 3,477
4 votes
0 answers
148 views

Some questions on Hardy's spaces

In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
A beginner mathmatician's user avatar
3 votes
1 answer
68 views

Infinite direct sum decomposition of the heat semigroup on $\mathbb R$

This question is based on a very similar question posted by me yesterday. A very nice solution was provided by Aleksei Kulikov. Here I modify my question slightly. Let $Q_t$ be the heat semigroup on $...
Ribhu's user avatar
  • 407
7 votes
1 answer
243 views

Isoperimetric inequality, but $L_p$ norm

I would like to consider the isoperimetric problem of $L_p$ norm: Given a region in $\mathbb R^2$ such that the boundary is a curve $C(x,y)$, where $\int_{C}(|\mathrm dx|^p+|\mathrm dy|^p)^{1/p}$ is a ...
JetfiRex's user avatar
  • 843
2 votes
1 answer
141 views

(Sub)Optimality of random transport

Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
yfful's user avatar
  • 25
3 votes
1 answer
219 views

Moment problem, ergodicity and spectral gap on the space of tempered distributions

Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$. Just below formula [5] in p.122 of the Fröhlich paper, ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
248 views

Characterization of normed spaces based on violation of parallelogram law

For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied. I would like ...
Hikaru's user avatar
  • 213
0 votes
0 answers
97 views

Sufficient condition for weak convergence in Banach spaces

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
an_ordinary_mathematician's user avatar
1 vote
2 answers
221 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 ...
Ali Taghavi's user avatar
1 vote
1 answer
130 views

Existence of solutions to a series of integral equations

I am trying to solve the following integral equation analytically: $$ \sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T], $$ where $(f_n(t))_n$ is the unknown ...
Gustave's user avatar
  • 617
5 votes
2 answers
521 views

Functions whose product with every $L^1$ function is $L^1$

Let $\mu$ be a probability measure and $f$ a measurable function whose product with any integrable function is integrable: $$ \int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$ ...
KhashF's user avatar
  • 3,599
6 votes
1 answer
250 views

The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by $$Lu = \partial_i(a^{ij}...
MathsGoose's user avatar
5 votes
2 answers
432 views

Does closedness of the image of unit sphere imply the closed range of the operator

Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
Anupam's user avatar
  • 585
3 votes
1 answer
377 views

A more general product rule for weak derivatives?

Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
Bogdan's user avatar
  • 1,759
2 votes
0 answers
47 views

Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces

Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
kiliroy's user avatar
  • 56
0 votes
0 answers
46 views

What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?

This is related to a question I recently asked on math.SE. Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
glS's user avatar
  • 342
2 votes
1 answer
89 views

Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances

Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$: $$ d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|. $$ Let $\rho$ be the Levy-Prokhorov metric on the ...
ssss nnnn's user avatar
  • 177
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
Javier's user avatar
  • 69
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
121 views

On mollifiers acting between $L^2$ and Sobolev spaces

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
S.Z.'s user avatar
  • 557
15 votes
2 answers
889 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 ...
David Roberts's user avatar
  • 35.5k
2 votes
1 answer
79 views

There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?

Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
Ilovemath's user avatar
  • 677
3 votes
0 answers
118 views

Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?

Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
Stefan Schrott's user avatar
2 votes
1 answer
142 views

Bounded differentiation operator on compact intervals with $L^2$ norm

It is known that the differentiation operator $D$ is not bounded on $C^1([0,1])$ with $L^2$ norm (counterexample: $f(x)=x^n$). Now I am wondering whether there is an infinitely dimensional subspace ...
graham's user avatar
  • 153