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

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47 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
2
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0answers
61 views

Dirichlet-to-Neumann Map is selfadjoint

Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$. For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...
2
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1answer
109 views

Representation of the elements of $c_0^\perp$ as integrals over ultrafilters

Let $$ X=\big\{\varphi\in\ell_\infty^{\,*}(\mathbb N) : \varphi(\{a_n\})=0\,\,\text{whenever $a_n\to 0$}\big\}. $$ If $\varphi_{\mathscr F}(\{a_n\})$ is the limit of $\{a_n\}$ with respect to the ...
5
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1answer
111 views

Orthonormal bases on Reproducing Kernel Hilbert Spaces

Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional ...
3
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193 views

Inverse problem for negative moments

Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length ...
1
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64 views

The real method of interpolation and operator ideals,

Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding) ...
7
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2answers
376 views

States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$ Now, of course there is also in classical physics and quantum ...
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220 views

Interpolation between $H^1$ and $H^1\cap L^1$

Suppose that $T:H^1(\mathbb{R}^3)\rightarrow\mathbb{R}$ is a linear bounded operator, with operator norm $M_2$. In particular, given $1\leq p\leq2$, there exist optimal constants $M_p\leq M_2$ such ...
3
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2answers
135 views

Weak convergence implies norm convergence for trace class operators?

It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. here. Because of the typical analogies of the Schatten ideals $C_p \subset B(H)$ (where $H$ is a Hilbert ...
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70 views

inverse Fouier transform from partial Fourier transform

It is known that a function $u(\mathbf{x})\in C_0^{\infty}(\Omega)$ can be reconstructed from its Fourier transform using inverse Fourier transform. $$u(\mathbf{x}) = \mathcal{F}^{-1} (\widehat{u}) = ...
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1answer
136 views

Dense subspaces of $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that $$\int_0^T\Vert ...
11
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1answer
373 views

An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
9
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162 views

Complemented subspaces in the dual of James' space $J$

James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...
2
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1answer
103 views

Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$. If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$, $$ ...
1
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1answer
137 views

Hermitian Projections on $C[0,1]$

If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by ...
5
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113 views

Integral-like concepts

I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect. The background is that I have proven a ...
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2answers
127 views

Antiproximanal subspace of $L_1[0,1]$

Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$. I read somewhere that $Y$ ...
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0answers
52 views

Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$ we have \begin{equation} ...
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0answers
58 views

Partial trace with spectral measure

I'm a physicist who needs mathematical advice: Let $A= \sum_{i=1}^{\infty} a_i P_{\phi_i}$ be a self-adjoint operator with projectors $P_{\phi_i}$ on the orthonormal eigenbasis $(\phi_i).$ Let $$B= ...
4
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1answer
121 views

Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...
3
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92 views

Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
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106 views

Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...
30
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2answers
400 views

If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
2
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0answers
106 views

$L^\infty-L^1$ smoothing effect for the heat equation

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ where $u_0$ is bounded initial data. Here ...
3
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0answers
87 views

Are Sobolev spaces on non-compact manifolds separable?

If $M$ is a Riemannian manifold that is not compact, is it true that the Sobolev spaces on $M$, $W^{k,p}(M)$, still be separable (for $p < \infty$)?
2
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1answer
96 views

Relation between two different definitions for relative sequential compactness

Building upon this question in Math.SE, I think the following might be rather of interest for MO. In the literature on measure theory, probability and functional analysis the definition of a subset ...
7
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1answer
446 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here: Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
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45 views

Coersivity of a bilinear form [closed]

I need to proof the coersivity of the following bilinear form. a,b and c are scalars, u is the velocity vector field and p is the pressure. Any help is much appreciated! $$ B(\textbf{u},\textbf{v}) = ...
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46 views

Doubts regarding pre-compactness of bounded sequence of measure valued functions

Given: $\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ . ...
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0answers
66 views

Gaussian heat kernel bounds on Riemannian manifolds

I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$ t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}}, $$ on a ...
2
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2answers
230 views

The space $L^p(\partial\Omega)$ in cited references

The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
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0answers
69 views

Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
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89 views

Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...
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67 views

$\sum_{n\in\mathbb{Z}^2}d\mu(x-2\pi n)=0\Rightarrow$ the summands are pairwise mutually singular

Let $\mu$ a finite measure supported by $\Gamma$ (smooth curve in $\mathbb{R}^2$) and absolutely continuos with respect to the arc length measure on $\Gamma$. Please why if ...
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0answers
29 views

Constant periodic Sobolev embedding

Dear mathoverflowers, I would like to have a reference regarding the optimal constant in the Sobolev embedding $$ \|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s}, $$ ($H^s$ denotes the standard L^2 ...
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2answers
322 views

Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$? [closed]

$$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{k!}$$ We can rewrite the equation as $$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{ \Gamma(k+1)} \tag{1}$$ because $x!=\Gamma(x+1)$ where $x$ is ...
6
votes
1answer
138 views

Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation: $$ -\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi, $$ where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
0
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1answer
164 views

$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
3
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1answer
144 views

Hermite-Kakeya Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited: Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...
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0answers
88 views

Functional composition of Hadamard product

Let $\Bbb K$ be a ring. Are there universal functions $$f,h:\Bbb K^{n\times n}\times\Bbb K^{n\times n}\times\Bbb K^{n\times n}\times\Bbb K^{n\times n}\rightarrow\Bbb K^{n\times n}$$ $$g:\Bbb ...
3
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1answer
172 views

difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$

This is cross-posted on MSE: http://math.stackexchange.com/q/1596565/9464 In the Partial Differential Equations by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1_0(\Omega)$ ...
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0answers
131 views

Sigma algebra generated by SOT versus of sigma algebra generated by WOT

Let $H$ be a non-separable Hilbert space. Let us denote $B_s$ ($B_w$), by the sigma algebra generated by the strong operator topology (weak operator topology) on $B(H)$. Question: Is $B_s$ the same ...
3
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1answer
174 views

Is the Fourier-Stieltjes algebra of a locally compact group semi-simple?

Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?
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103 views

Strong convergence

From this paper: ...
3
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154 views

Uniqueness of the reduced free product of unital completely positive maps

For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = ...
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1answer
172 views

The total variation of a complex measure

Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C_0(\Omega)^*$ , dual of the commutative C*-algebra $C_0(\Omega)$, is just the space of ...
2
votes
1answer
165 views

Polar decomposition

Let $x$ be a trace class operator on a Hilbert space $H$. Then $x$ induces unique normal functional on $B(H)$, which we denote it by $f_x$. Let us consider the polar decomposition $x=u|x|$ and ...
2
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0answers
61 views

Is the Wave Function a “Smooth” Function of the Potential?

Consider the Schroedinger equation in a spherically symmetric system. In the unit system under which energy is measured in Hartree and length in Bohr radius $a_0$, the schroedinger equation can be ...
4
votes
1answer
180 views

Is Lax-Milgram true without the separability assumption?

I read the Lax-Milgram Theorem in the Navier-Stokes Equations by Temam: Let $X$ be a separable Hilbert space (norm $\|\cdot\|_X$) and let $$ a:X\times X\to\Bbb{R} $$ be a bilinear continuous ...
2
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0answers
66 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...