Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
5,340 questions
2
votes
0answers
86 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
0answers
47 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
0answers
28 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
2answers
108 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
$$d(f,g) = \inf_{\varphi \in \Lambda} \left\...
2
votes
0answers
60 views
k-rank numerical range of an operator
Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
0
votes
0answers
32 views
get the function that passes through n k-tuples of values with rounded value [on hold]
well basically I need to get the function that passes through n k-tuples (in my particular case k is 4 and n is as big as needed) but the results of the function are rounded, I've already seen a way ...
-1
votes
0answers
57 views
A property about a MNC
Let $E, \left \|\cdot \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the family of all relatively compact sets, and $Ker \...
0
votes
0answers
46 views
Convolution with an analytic semigroup
Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution
$$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$
...
1
vote
0answers
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
2answers
322 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 ...
0
votes
0answers
69 views
+50
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
0answers
42 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
1answer
111 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
1answer
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
0answers
90 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
1answer
125 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
1answer
132 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
0answers
37 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
0answers
57 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
1answer
82 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
1answer
89 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
0answers
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 $$\mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): f(x)>0\} - \mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): g(x)>0\...
3
votes
1answer
211 views
What are the almost periodic functions on the complex plane?
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
3
votes
2answers
98 views
Product of concave functions and harmonic mean
I discovered something interesting, and I would like to know whether it is a known result or not. Say that a function $f: \Omega \subset \mathbb{R} \rightarrow \mathbb{R_+^*}$ is $\alpha$-concave if $...
3
votes
0answers
50 views
Uniform homeomorphism of Frechet nuclear spaces
Given a Frechet space X which is uniformly homeomorphic with a nuclear (Frechet) space, does it follow that X is also nuclear?
3
votes
3answers
123 views
Sum of subspaces is closed iff inclination is positive
It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination
$$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
1
vote
1answer
49 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
$$\langle \cdot ,...
1
vote
1answer
63 views
Looking for a very particular kind of non-convex functions
I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously,
It should be at least twice differentiable.
It should have a ...
2
votes
0answers
90 views
Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?
If $A=Mat_{n\times n}(\mathbb{C}) $, Is $\ell_2(A)$ a Hilbert $A$-module with Opial property?
Opial property: If ($w-\lim x_n=0 $) then $
(\liminf \lVert x_n\rVert<\liminf \lVert x_n-y \...
2
votes
0answers
61 views
Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$
I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball:
If you pick n points uniformly at random from the surface of a d
dimensional sphere of ...
-1
votes
0answers
55 views
A complicated integral / a complicated Laplace transform involving the error function
For some reason I am interested in solving a complicated integral, which is
$$\int_0^\infty (x+1)erf\left(\frac{-c_1+c_2x}{b}\right)e^{-c_3x^2-c_4x}dx,$$ where all $c_i$ are positive real numbers and $...
1
vote
1answer
122 views
Convolution with Schwartz class function
Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution).
Consider distribution as follows:
$$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
3
votes
0answers
63 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
0answers
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
1answer
107 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
1answer
151 views
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
0answers
91 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
0answers
75 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
1answer
101 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
1answer
53 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 ...
0
votes
0answers
67 views
Eigenfunctions of square root of Laplacian in an arbitrary Riemannian manifold?
Please forgive me for my inability to pose a mathematics question properly.
In one dimension the eigenfunctions of Laplacian (simply double derivative) are also eigenfunctions of its square root (...
8
votes
1answer
176 views
Maps which are both completely positive and positive
Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
0
votes
0answers
32 views
Predictable Process Question (Da Prato & Zabcyzk 2014)
I've been looking at the 2014 edition of Da Prato & Zabcyzk and the sections on predictable processes. In particular, in their Proposition 3.7 (ii), they assert that if $\Phi$ is adapted and ...
4
votes
1answer
132 views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
9
votes
1answer
264 views
Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
0
votes
0answers
79 views
Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates
Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.
...
3
votes
1answer
193 views
Fundamental group and group measure space construction
Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
3
votes
1answer
215 views
Prescribing a gradient direction
Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...
4
votes
1answer
99 views
Power series in functions other than monomials
I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...
3
votes
1answer
68 views
Openness of invertibility in Fréchet spaces for families parameterized by compact spaces
Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T_k \equiv T(k, ...
