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

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42 views

absolutely continuous of two probability measures

Suppose $X_t$ satisfies $$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...
1
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1answer
64 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
1
vote
1answer
97 views

How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...
-2
votes
0answers
15 views

Approximate point spectrum is complement of set of points of regular type [migrated]

I have a question concerning the approximate point spectrum of a closed linear operator. I need to show that the approximate point spectrum is the complement of the set of points of regular type, ...
-3
votes
0answers
32 views

Averages of bounded function [migrated]

For a continuous $f\colon\mathbb{C}\longrightarrow[0,1]$, what could be said about $\mathcal{F}=\{f_r\colon r>0\}$ where $$\forall\,r>0,\,\forall\,z\!\in\!\mathbb{C},\quad ...
0
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0answers
52 views

Bounded-open topology vs norm on $L\left(X,Y\right)$

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform ...
-2
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0answers
37 views

periodic function satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf? [on hold]

Can a periodic function f(x) satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf?
4
votes
3answers
300 views

reflexive banach space

I want to ask this non-expert question: What does it mean geometrically for a Banach space to be reflexive? Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
3
votes
2answers
97 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
1
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0answers
102 views

Dual space of $l^p(\mathbb{Z},X)$

Let $X$ be a Banach space, $p \in [1,\infty)$ and $l^p(\mathbb{Z},X)$ the usual sequence space taking values in $X$. Is it always true that $(l^p(\mathbb{Z},X))^* = l^q(\mathbb{Z},X^*)$ and ...
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1answer
72 views

Can functional invariants of dynamical systems be used in data science (or parameter identification)? [on hold]

Given a functional of the form: $$ F[x]=\int^{t}_{0} \mathcal{L}(x^{(n)},x^{(n-1)},...,x,\tau)\,\text{d}\tau+g(x(t)) $$ Where $x$ is in $\mathbb{R}^{m}$ and is in ...
-1
votes
1answer
110 views

Analysis of Sobolev spaces [on hold]

I just wanted to know wthether the following is OK or not. Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
0
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0answers
168 views

Theorems about matrices with entries from $0,1,-1$? [on hold]

Consider matrices which are of the form $\left [ \begin{matrix} 0 && A\\ A^T && 0 \\ ...
2
votes
0answers
89 views

Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
2
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0answers
71 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...
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votes
0answers
86 views

How to use Holder's Inequality to prove two Banach spaces are equal $(\mathbb R^n,\|\cdot\|_3)*= (\mathbb R^n,\|\cdot\|1.5)$ [closed]

How can one prove that Banach spaces $(\mathbb R^n,\|\cdot\|_3)*= (\mathbb R^n,\|\cdot\|1.5)$
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0answers
60 views

Da Prato's notion of Symmetric Operator

For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...
2
votes
0answers
94 views

Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on $\mathbb{R}^{+}$): $$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$ We call ...
3
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1answer
52 views

Relatively compact sets in Ky Fan metric space

Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E ...
3
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0answers
60 views

Is Wiener's Tauberian theorem true in Wiener space?

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$. Is the following true? ...
7
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0answers
151 views

Lipschitz-free spaces of $\mathbb R^n$

We define $$ \text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and } \sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty. \} $$ It is well-known ...
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0answers
51 views

Maximization of Funcional [closed]

Let $p$ be a nondecreasing function defined on $[\frac{1}{4}, \frac{1}{2}]$ that takes values on $[0,1]$, that is $p:[\frac{1}{4}, \frac{1}{2}]\rightarrow [0,1]$. Find a maximum of a functional ...
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0answers
89 views

Topological properties of space of Radon measures

Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...
1
vote
1answer
83 views

About the upper bound on the roots of the matching polynomial

Heilman and Lieb had proven that if a graph had $d$ as its maximum vertex degree then the roots of the matching polynomial are bounded from above by $2\sqrt{d-1}$. Is there a modern exposition of ...
1
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1answer
85 views

Orthogonal compact operators on an infinite dimensional Hilbert space [closed]

How do I show that when $H$ is an infinite-dimensional Hilbert space we can find two compact positive operators $u,v$ with infinite dimensional image and $u \perp v$? This statement can be found at ...
2
votes
2answers
86 views

Infinite direct sum of $l_2^{(n_k)}$ contains a complemented isometric copy of $l_2$

How do I show that for any increasing sequence $(n_k) \subseteq \mathbb{N}$, the space $\left( \oplus _{k=1} ^\infty l_2 ^{(n_k)} \right) _\infty$ contains a complemented isometric copy of $l_2$?
1
vote
1answer
104 views

Is a Fréchet Montel space distinguished?

Based on a couple of references, it seems that the answer is yes, see for example Boneta-Dierolf, 1992 and Bierstedt-Bonet, 1989. However, from a comment to the answer of this MO question, I infer ...
5
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0answers
83 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
0
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0answers
84 views

Banach space dual to $L^\infty(I,H^1(M))$

What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus. Any references that show ...
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0answers
26 views

Lipschitz continuity of a composition operator

Let $P\colon U\subset \mathbb{R}^K\rightarrow \mathbb{R}^K$ be $C^\infty$, $\Omega \subset \mathbb{R}^d$ open and bounded, $m \in \mathbb{N}$, $p\in [1,\infty]$ such that $m>\frac{d}{p}$, $f \in ...
4
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1answer
76 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
0
votes
1answer
41 views

Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...
4
votes
1answer
71 views

Monotonicity of a ratio of conditional expectation operator

Let a pair of random variables $(X, Y)$ over a finite product space $\mathcal{X}\times \mathcal{Y}$ be given. The conditional expectation operator is defined as $$(T_Yf) (y):=\mathbb{E}[f(X)|Y=y],$$ ...
0
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0answers
74 views

Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is $$\text{Tr}(P(A)) = ...
1
vote
1answer
264 views

Euler-Lagrange Equation and “Eigen Value ”

I posted this question on Math.SE, but I could not get any help. The eigenvalue $\lambda(t)$ is characterised as the minimum of the Rayleigh quotient (where $t$ is a scalar variable) ...
0
votes
1answer
68 views

Motivating the Bessel translation operator

In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ...
1
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0answers
142 views

Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...
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0answers
51 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
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0answers
70 views

an elementary proof without using K-convexity constant?

Suppose that $A$ is an $n\times d$ matrix $(n\gg d)$ with orthonormal columns, and $g\sim N(0,I_d)$. I wish to show that $$ \mathbb{E}\sup_{x:\|Ax\|_1 = 1} \langle g,x\rangle \lesssim \sqrt{\log ...
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1answer
91 views

Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?

We formally write the solution of nonlinear Schrödinger equation (NLS) as follows: $$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$ where $U(t)= e^{it\Delta} $(free ...
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0answers
22 views

Completion of a vector space inside a given Banach space [migrated]

Good evening, Here is my question: let $X$ be a normed vector space and $Y$ be a Banach space such that $X$ is continuously embedded into $Y$ (this will be denoted by $X\hookrightarrow Y$ in the ...
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0answers
25 views

asymptotically nonexpansive mappings [migrated]

Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ such that ...
0
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0answers
50 views

global bi harmonic functions in a riemannian manifold

Any help will be appreciated thanks! Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain ...
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0answers
52 views

Relation between locally convex calculus and Kriegl & Michor's “convenient setting”

I have a very general question regarding the book "Kriegl, Michor: The Convenient Setting of Global Analysis.": Is the differential calculus of locally convex spaces (see here, for instance) ...
4
votes
2answers
254 views

Question about projections in von Neumann algebras

Let $M$ be a von Neumann algebra, and let $\mathcal{P}$ be the set of nontrivial (not equal to $0$ or $e$) projections of $M$. Define $p,q \in \mathcal{P}$ to be equivalent if there exist projections ...
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0answers
57 views

What does integrability of a strictly monotonic function imply about the tails of that function?

In particular, if $f:\mathbb{R}_{+}\rightarrow[0,1]$ is a strictly monotonic decreasing function and $f$ is integrable then does it necessarily hold that $f^{-1}(1/t)=o(t)$?
5
votes
0answers
100 views

Approximation in the tensor square of a weakly exact von Neumann algebra

Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
0
votes
1answer
150 views

Sobolev's lemma on manifolds

Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying ...
2
votes
0answers
95 views

About the small set expansion conjecture

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - ...
0
votes
0answers
23 views

Uniform Boundedness of the Szasz-Durrmeyer Operators on Variable $L^{p(\cdot)}$ Spaces

I am stuck with the uniform boundedness problem associated with the Szasz-Durrmeyer Operators on $L^{p(\cdot)}$ spaces. The Szasz-Durrmeyer operator can be defined as $$M_{n}(f;x)=\sum ...