# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

5,346 questions

**0**

votes

**0**answers

41 views

### Transformation of Fourier Transform

Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...

**1**

vote

**1**answer

82 views

### Growth rate of bounded Lipschitz functions on compact finite-dimensional space

Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-...

**1**

vote

**0**answers

26 views

### Potential for a Monotone Operator

[Cross-posted from math.stackexchange]
I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...

**0**

votes

**0**answers

28 views

### Optimal transport between Gaussian mixtures and their centers

I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...

**4**

votes

**1**answer

126 views

### Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...

**0**

votes

**0**answers

70 views

### Hilbert-Schmidt Operators

I am reading Brezis's book : Functional Analysis, Sobolev Spaces and Partial Differential Equations and trying to solve an exercise on Page 499. The question is as follows: Assume that $\Omega = (0, 1)...

**1**

vote

**0**answers

58 views

### Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...

**6**

votes

**1**answer

136 views

### Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...

**-1**

votes

**0**answers

74 views

### Weak closure of subsets of the unitary sphere of a Banach space

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define
$$
B_\varepsilon=\{x\in X:\|x-...

**5**

votes

**1**answer

80 views

### Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces

Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...

**3**

votes

**0**answers

124 views

### Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \left\{ r\ |\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\limits_{k=...

**4**

votes

**1**answer

81 views

### Semi-continuous fields of C*-algebras having dimension one on a dense set

Given a Hausdorff, locally compact space $X$, let us consider a semi-continuous field $\{A_x\}_{x\in X}$ of C*-algebras over $X$, such that $A_x$ is one-dimensional for every $x$ in a dense subset $D$ ...

**2**

votes

**0**answers

95 views

### Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...

**4**

votes

**0**answers

55 views

### Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?

I posted the following question on StackExchange a few months ago (https://math.stackexchange.com/questions/2898620/notions-of-beta-h%C3%B6lder-smoothness-when-beta-in-1-2-are-they-equivalent), but ...

**1**

vote

**0**answers

36 views

### Uniqueness of solution of Volterra Integral Equation with deviating argument

In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$:
\begin{equation}...

**2**

votes

**2**answers

115 views

### Continuous embedding of the Skorohod space D(0,1) into L^2(0,1)

Let $D(0,1)$ be the Skorohod space with the Skorohod topology, i.e. the space of real-valued càdlàg-functions on $[0,1]$ with topology induced by the metric
$$d(f,g) = \inf_{\varphi \in \Lambda} \left\...

**2**

votes

**0**answers

62 views

### k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

**0**

votes

**0**answers

49 views

### Convolution with an analytic semigroup

Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution
$$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$
...

**1**

vote

**0**answers

34 views

### Extension of a result about measurable, additive functionals

Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...

**4**

votes

**2**answers

339 views

### If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am ...

**1**

vote

**0**answers

80 views

### On functions obtained from Gaussian Quadrature integration

Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...

**0**

votes

**0**answers

45 views

### Existence of a function satisfying some integral conditions

I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...

**4**

votes

**1**answer

116 views

### Taylor spectrum of commuting operators

Taylor spectrum of commuting operators
Fom the following paper (M. Ch—o, H. Motoyoshi, B. Na¡cevska Nastovska: On the joint spectra of commuting tuples of operators and a conjugation) we have
Let ...

**2**

votes

**1**answer

108 views

### Relation between Optimal Transport Cost and Difference between Topological Invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...

**0**

votes

**0**answers

93 views

### Is this basis a Schauder basis?

Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that
$\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent,
$(...

**2**

votes

**1**answer

127 views

### “Compactness in Measure” in Function Spaces

In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". ...

**-1**

votes

**1**answer

135 views

### Riesz representation theorem for Hilbert-to-Hilbert mappings [closed]

Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz ...

**2**

votes

**0**answers

41 views

### Can we extract information from signature (rough path theory) to construct part of signal?

This question is related to rough path theory. Consider we have obtained signature obtained from a set discrete data points postulating linear from one data point to another. Such signature are used ...

**0**

votes

**0**answers

59 views

### Continuity of the positive part of a linear functional

Let $X$ be a vector space containing a convex cone $C$ which induces a (pre) order on $X$ by $x \geq y \Leftrightarrow x-y\in C$. Given a sequence of linear functionals $x^*_n:X\rightarrow\mathbb{R}$ ...

**4**

votes

**1**answer

85 views

### Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?

I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...

**2**

votes

**1**answer

92 views

### Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...

**-2**

votes

**0**answers

51 views

### Estimate on the difference between the measure of the sublevels of two functions in terms of their $L^1$ distance

Fix $R\gg 1$. How can I estimate the difference between the Lebesgue measures $$\mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): f(x)>0\} - \mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): g(x)>0\...

**3**

votes

**1**answer

212 views

### What are the almost periodic functions on the complex plane?

The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...

**3**

votes

**2**answers

98 views

### Product of concave functions and harmonic mean

I discovered something interesting, and I would like to know whether it is a known result or not. Say that a function $f: \Omega \subset \mathbb{R} \rightarrow \mathbb{R_+^*}$ is $\alpha$-concave if $...

**3**

votes

**0**answers

54 views

### Uniform homeomorphism of Frechet nuclear spaces

Given a Frechet space X which is uniformly homeomorphic with a nuclear (Frechet) space, does it follow that X is also nuclear?

**3**

votes

**3**answers

124 views

### Sum of subspaces is closed iff inclination is positive

It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination
$$\widehat{(M,N)} := \inf_{x\in M, \|x\|...

**1**

vote

**1**answer

51 views

### Reference on vector-valued convex conjugate

The following definition of convex conjugate is taken from Wiki:
Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$
Denote the dual pairing by
$$\langle \cdot ,...

**1**

vote

**1**answer

63 views

### Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously,
It should be at least twice differentiable.
It should have a ...

**2**

votes

**0**answers

91 views

### Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?

If $A=Mat_{n\times n}(\mathbb{C}) $, Is $\ell_2(A)$ a Hilbert $A$-module with Opial property?
Opial property: If ($w-\lim x_n=0 $) then $
(\liminf \lVert x_n\rVert<\liminf \lVert x_n-y \...

**2**

votes

**0**answers

63 views

### Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$

I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball:
If you pick n points uniformly at random from the surface of a d
dimensional sphere of ...

**-1**

votes

**0**answers

55 views

### A complicated integral / a complicated Laplace transform involving the error function

For some reason I am interested in solving a complicated integral, which is
$$\int_0^\infty (x+1)erf\left(\frac{-c_1+c_2x}{b}\right)e^{-c_3x^2-c_4x}dx,$$ where all $c_i$ are positive real numbers and $...

**1**

vote

**1**answer

123 views

### Convolution with Schwartz class function

Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution).
Consider distribution as follows:
$$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...

**3**

votes

**0**answers

64 views

### Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far:
1. There exists a density defined ...

**2**

votes

**0**answers

23 views

### Funk transform of density supported on an embedded curve

A Funk transform is a certain invertible linear transformation on the space of square-integrable functions on $S^2$. I think its domain can be extended to include densities supported on embedded ...

**3**

votes

**1**answer

109 views

### Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?

$A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra ...

**2**

votes

**1**answer

210 views

+50

### Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...

**1**

vote

**0**answers

92 views

### Mackey topology characterising property

Let $V$ be a topological $k$-vector space.
Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.
...

**0**

votes

**0**answers

78 views

### Can we have Levy area for N dimensional process?

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what ...

**2**

votes

**1**answer

123 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 ...

**3**

votes

**1**answer

55 views

### Connectness of $K$ and the existence of non-trivial $M$-summands on $C_0(K)$

For a locally compact Hausdorff space $K$, we denote by $C_0(K)$ the Banach space of continuous functions from $K$ to $\mathbb R$ which vanish at infinity equipped with the usual supremum norm.
For ...