2,445 reputation
730
bio website math.uni-trier.de/~wengenroth
location Universität Trier
age 47
visits member for 2 years, 10 months
seen 1 hour ago
I am Professor for Mathematics at the Unversität Trier (Germany)

1d
comment decomposition of tempered distributions by entire analytic functions
I guess you mean by $g^\vee$ the inverse Fourier transform. Since Fourier transformation is an isomorphism $\mathscr S'\to\mathscr S'$ (with the weak*-topology) you need to show that $\sum\limits_{k=1}^K \phi_k \hat f$ converges to $\hat f$ and hence $\langle \sum\limits_{k=1}^K \phi_k \hat f,g \rangle \to \langle \hat f ,g\rangle$ for all $g\in\mathscr S$ (where$\langle f,g \rangle$ denotes the duality between $\mathscr S'$ and $\mathscr S$). For this it would be enough to show $\sum\limits_{k=1}^K \phi_kg \to g$ in $\mathscr S$ and this will depend on bounds for the derivatives of $\phi$.
1d
comment Continuity of a Functional
The question is not clear. What is the domain of $T$? Do you mean by $F^{-1}(s)$ the multiplicative inverse $1/F(s)$?
2d
awarded  Nice Answer
Nov
20
comment Between compact and locally uniform: What is the name of this convergence?
Very informative answer. A reference would be nice.
Nov
19
comment Universal maps between topological spaces
@Oblomov Surjectivity is clearly necessary but universality is much stronger (the identity on $\mathbb R$ is surjective but not universal with $f(x)=x+1$).
Nov
16
comment When is a `1-form' with continuous coefficients exact?
Very long second...
Nov
11
comment Counterexample for closed graph theorem in unmetrizable case
For an infinite-dimensional Hilbert space $X$ consider $Y=X$ endowed with the finest locally convex topology and the identical map. On the other hand, if the domain $X$ is ultrabornologial the closed graph theorem holds for so-called webbed spaces introduced by de Wilde. This is a very large class of locally convex spaces which contains all Banach spaces and is stable w.r.t. closed subspaces, separated quotients, and countable products and direct sums. See, e.g., Introduction to Functional Analysis of Meise and Vogt.
Nov
5
comment Rank of a sequence of covariance matrices
In general, $ZX_i$ are only in $L^1(\Omega,P)$ so that the covariance is not necessarily defined. $Z\in L^\infty$ would be okay.
Oct
31
comment “Partition” of a smooth function in $\mathbb R^2$
@GHfromMO Formally, you are right. The statement means: For all $f\in C^\infty(\mathbb R^2)$ with $f(0,0)=0$ there are $g_1,g_2 \in C^\infty(\mathbb R^2)$ such that $f(x,y)=g_1(xy,x)+g_2(xy,y)$ for all $(x,y)\in\mathbb R^2$.
Oct
30
comment Large Deviations: Exponential decay in normed spaces
There is a Banach-space version of Azuma's inequality due to Assaf Naor, theorem 1.5 in [web.math.princeton.edu/~naor/homepage%20files/AR-notes.ps]. There you need however special properties of the norm (modulus of smoothness of power type 2).
Oct
28
comment Examples of topologies compatible with a given dual pair
The answer to your question depends very much on what you mean by concretely. If the strong topology $\tau$ on $\mathscr E'$ (Schwartz' notation for $C^\infty(\mathbb R)'$) is concrete enough you can restrict it to $\mathscr D= C^\infty_c(\mathbb R)$ to obtain a compatible topology for the dual pair.
Oct
28
comment Examples of topologies compatible with a given dual pair
You mean $\langle\varphi,f\rangle = \int \varphi(x)f(x)dx$, right?
Oct
19
comment How to formulate approximation from above?
It would be most natural to assume the continiuty of $X_{j+1} \hookrightarrow X_j$ so that you have a projective limit of Banach spaces. Then $X_\infty=\bigcap_j X_j$ is a Frechet space.
Oct
18
comment Is every Montel locally convex vector space compactly generated?
The Mackey-Ulam theorem is discussed in Bonet's and Perez-Carreras' book "barrelled locally convex spaces". They write that if a product $\mathbb R^I$ fails to be bornological then the cardinality $d$ of $I$ is "strongly inaccessible". In particular, $a<d$ $\Rightarrow$ $2^a < d$ which is certainly wrong for $c=card(\mathbb R) = 2^{\aleph_0}$.
Oct
17
comment Is every Montel locally convex vector space compactly generated?
I don't have any references at hand, but I think that products of "moderate cardinality" are bornological, in particular, I believe that $\mathbb R^{\mathbb R}$ is bornological even without assuming the continuum hypothesis.
Oct
16
answered Is every Montel locally convex vector space compactly generated?
Oct
16
comment Is every Montel locally convex vector space compactly generated?
A counterexample to the Banach-Dieudonne theorem for non-metrizable spaces was first given by Komura [link.springer.com/article/10.1007%2FBF01361183]
Oct
16
accepted Consistent price index
Oct
15
comment Is every Montel locally convex vector space compactly generated?
Kriegl-Michor really mean by $kX$ the finest topology (not necessarily locally convex) making all inclusions of compact subsets continuous. The proof uses the the Banach-Dieudonne theorem for which metrizability is quite essential.
Oct
14
comment Consistent price index
Excellent. Although it looks slightly simpler than Bjorn Kjos-Hannssen's solution it is essentially the same. Unfortunately, I can's share the bounty and Bjorn was a little bit faster.