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

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0
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1answer
46 views

How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper. Equation: $min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$, where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
0
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0answers
42 views

Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
0
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0answers
22 views

Interchange summation and differentiation [migrated]

I asked this question already on math.stackexchange, but did not receive any answers see here Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that ...
6
votes
0answers
169 views

A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...
2
votes
0answers
69 views

Multivariate Maximal Hilbert Transform

One way to define the maximal Hilbert transform of a function, $f$, is by $$\mathcal{H}[f](x):=\underset{\varepsilon>0}{\sup}\;\int_{|t|\geq\varepsilon}\dfrac{f(t)}{x-t}dt,\quad x\in\mathbb{R},$$ ...
7
votes
0answers
88 views

Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map $$ \alpha\colon X\hat{\otimes} X^* \to B(X)^*, $$ defined one simple tensors as $$ \alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...
5
votes
1answer
135 views

Analytic perturbation of eigenfunctions

Consider a domain $\Omega_0 \subset \mathbb{R}^n$, and deformations of $\Omega_0$, called $\Omega_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is ...
3
votes
1answer
63 views

Composition of spectral measures

Let $f: \mathbb{R}\rightarrow \mathbb{C}$ be a measurable function, $H$ some Hilbert space and $$ f_E := \int_{\mathbb{R}} f dE$$ for some spectral measure $E$. Now, my question is: When do we have ...
1
vote
1answer
61 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
2
votes
1answer
126 views

Is this structure a Banach bundle?

Let $X$ be a Banach space. Put $Y=\{ \phi\in X^{*}\mid\;\; \parallel \phi \parallel\leq 1\;\; \&\;\; \phi \neq 0\}$ which is a locally compact Hausdorf space with the weak star topology. ...
1
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0answers
89 views

A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology. Is there a continuous map ...
1
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0answers
43 views

Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain $$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$ $$u|_{\partial\Omega} = 0$$ where $a$ is real-valued and satisfies $C_1 \leq a(r) \leq C_2$ ...
0
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0answers
39 views

Searching for conditions?

I have this operator $$Au(t)=\int_0^1 G(t,s) f(s,u(s)) ds$$defined from $H^1_{0}$ to $H_0^1$ and satisfy the problem: $$\begin{cases} -(Au)''(t)=f(t,u(t)), t\in[0,1]\\Au(0)=Au(1)=0\end{cases}$$ Where ...
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0answers
56 views

Example of topological vector space [closed]

Can Someone provide me any link or research paper about the working of definition of topological vector space(open neighborhood definition) on R to become a vector space?
5
votes
1answer
82 views

Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that $$ ||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ? $$ where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
0
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0answers
71 views

Explicit formula for Bergman kernel on the unit ball

On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...
1
vote
1answer
127 views

Multiplication of generalized functions

I would like to know if there is any associative algebra $( A(\mathbb{R}), +, \cdot )$ such that: EDIT: Fan Zheng found an inconsistency in my requirements. Therefore I changed the function space in ...
2
votes
1answer
142 views

Asymptotic behaviour of eigenvalues

If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
2
votes
1answer
101 views

Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...
1
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0answers
62 views

Weak (Sobolev) derivative and the Frechet derivative (chain rule) [closed]

Let $A: H^s(\Omega) \to H^1(\Omega)$ be a bounded linear map. Let $u \in H^s(\Omega)$. Let $f:\mathbb{R} \to \mathbb{R}$ be nonlinear and Lipschitz such that $f(u) \in H^s(\Omega)$. Is it possible to ...
3
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2answers
175 views

Does this C*-algebra embed into a simple nuclear C*-algebra?

Let $\mathcal K$ denote the C*-algebra of compact operators, and fix an embedding $\phi_n:M_n(\mathbb C) \to \mathcal K$ for each $n\in \mathbb N$. Define the C*-algebra $A := \{(a_n)_{n=1}^\infty ...
1
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0answers
43 views

Fractional Poincare inequality on closed manifold

Let $u \in H^{\frac 12}(M)$ on a compact closed Riemannian manifold. Can someone refer me to a source where the inequality $$\lVert u - \bar u \rVert_{L^{2^*}} \leq C|u|_{H^{\frac 12}}$$ is proved, or ...
0
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0answers
16 views

Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
0
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0answers
50 views

Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map ...
2
votes
1answer
111 views

Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?

I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that $$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...
0
votes
0answers
39 views

Interpolation of Banach spaces: theta = 0, 1

Let $A_1 \subset A_0$ be Banach spaces with continuous embedding. Is $(A_0, A_1)_{i, \infty} = A_i$ for $i =0, 1$, with equivalent norms? Here, $(\cdot, \cdot)_{\theta,p}$ denotes the $K$-method of ...
1
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0answers
89 views

A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$

The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
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0answers
24 views

Topological tensor products of spaces of holomorphic functions of slow growth

Let $X$ be a Banach space, $M$ be a complex manifold, and $\Omega$ a relatively compact domain in $M$. We consider the space $\mathcal{A}^{-\infty}(\Omega, X)$ of $X$-valued holomorphic functions of ...
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0answers
46 views

Hilbert scales of covariance operators

Assume we have 2 covariance operators(positive definite trace class) $S$ and $T$ on Hilbert space $\mathcal H$ with corresponding eigenpairs $\{e_j,\lambda_j\}$ and $\{f_j,\lambda_j\}$. Assume that ...
0
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0answers
63 views

Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1. Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...
2
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0answers
106 views

Generalising the $L^2$-Norm to Compact Quantum Groups

For a compact quantum group (in the sense) of Woronowicz, we have a noncommutative $C^*$-algebra replacing ${\mathbb C}(G)$. This $C^*$-algebra is endowed with a linear functional generalizing the ...
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0answers
52 views

Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems? $$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...
0
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0answers
27 views

Causal (Volterra type) differential equation with local Lipschitz condition

Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ ...
2
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0answers
44 views

Hilbert c*-module over approximately finite c*-algebra

Is there a construction of a Hilbert c*-module over an approximately finite c*-algebra using Hilbert c*-modules over finite algebras? How do we get a Hilbert c*-module over an inductive limit of such ...
1
vote
1answer
99 views

Identifying the weak limit of a gradient (Bochner spaces)

Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f_n \to f$, $f_n(0)=f(0)=0$ and $f_n$ Lipschitz with Lipschitz constant ...
1
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0answers
73 views

How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?

We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$ By Helson-Kahane-Katznelson-Rudin Theorem, it follows that, "Let $F$ be a function on $\mathbb C$ and if ...
2
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0answers
81 views

Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two ...
1
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0answers
69 views

Normal points of an operator and discrete eigenvalues

Let $\mathcal{H}$ and $\mathcal{L}({\mathcal{H}})$ denote a separable Hilbert space and the set of bounded linear operators on it respectively. As a graduate student entering the field of ...
1
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0answers
88 views

Limit of a simple function including a zero of the Riemann Zeta function

Lets consider : $$F(x)= \sum_{n\in\mathbb{N}} n^{-s_0} e^{2i\pi nx}$$ This function is well defined for $x>0$ (Abel summation formula proves it) and I would like to show that if $s_0$ is a zero ...
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0answers
61 views

Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question: Given ...
1
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1answer
120 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
3
votes
2answers
110 views

Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & ...
3
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0answers
48 views

Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} ...
4
votes
1answer
171 views

Is the space of Radon measures a Prohorov space?

Consider the spaces $C_c(\mathbb{R})$ of compactly supported continuous functions equipped with the inductive limit topology and the Banach space $C_0(\mathbb{R}) = \overline{C_c(\mathbb{R})}^{\, ...
5
votes
1answer
165 views

Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully). Suppose $g$ is holomorphic on ...
0
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0answers
78 views

Second variation of the domain functionals [closed]

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
1
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0answers
187 views

On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$ with $s_0$ a zero of the Riemann Zeta function in the critical strip. This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
1
vote
0answers
93 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
2
votes
3answers
461 views

Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...
6
votes
1answer
187 views

Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality $$ |\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...