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

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17 views

Is the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ in $L^\infty(\Omega)$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of $$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$ $$\partial_\nu u = \alpha \quad \text{on ...
5
votes
0answers
41 views

Weak*-closure of finite rank operators on dual space

Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is ...
0
votes
0answers
12 views

The effective strategy for choosing epsilon_init of random initialization in neural networks [on hold]

gays. i meet some problem in a description about choosing epsilon_init for random initialization in neural networks. enter image description here i don't know why the epsilon_init is related to the ...
-2
votes
0answers
32 views

Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in L2: ‎‎‎‎‎$‎$‎‎‎‎d‎‎{R}(‎A‎,B)=‎\sqrt{‎{‎‎‎‎\paralle‎l ‎P‎{R(A)}‎‎- ‎‎‎‎‎‎ ...
2
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1answer
50 views

If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?

Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact. Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$. Is then $u \in L^\infty(0,T;X_1)$? To apply ...
-2
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0answers
46 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first? [migrated]

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
0
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0answers
54 views

smoothness of boundary under Riemann mapping

Suppose there is a smooth Jordan curve separating the complex plane. For complicity, assume the curve is given by a graph $(x, \phi(x))$, where $\phi(x)$ is smooth, bounded, and derivatives are ...
5
votes
1answer
229 views

Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of ...
3
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0answers
32 views

Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
-1
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0answers
118 views

The eigenvalue of operator $-\Delta$

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We know the eigenfunction of Laplacian operator $-\Delta$ is an orthonormal basis of $L^2$. Let $\{\omega_n\}$ denote the ...
0
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1answer
69 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
4
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0answers
57 views

Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$. Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that ...
0
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0answers
59 views

Continuous inclusions in Hilbert-Sobolev space $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ [on hold]

I have to prove that $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ with $s \in \mathbb{R}$, $k \in \mathbb{N}$ and $s-k > n/2$, where $\mathcal{E}^k(\mathbb{R}^n):=\lbrace u: ...
2
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0answers
133 views

Can we do better than zero padding of FFT?

My background is in signal processing, and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...
3
votes
1answer
111 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
2
votes
1answer
105 views

Duality argument to get $L^\infty-L^2$ inequality

In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that $$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$ where the norms are $L^p$ norms. He states ...
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votes
0answers
37 views

weakly p- summable sequence

Let $ (x_{n}) $ be a weakly $ p$-summable sequence in $ X $ and $ ( x^{\ast}_{n})$ a weakly null sequence in $ X^{\ast} $. Let $ i_{n} : Y_{n}\rightarrow X$ be the natural injection and $ p_{n} : ...
5
votes
2answers
134 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
1
vote
1answer
94 views

$L^1$ convergence to equilibrium of solutions of heat equation

Let $u$ and $v$ be the weak solutions of $$u_t - \Delta u = f$$ $$u(0)=u_0$$ and $$-\Delta v = f$$ $$|\Omega|^{-1}\int_\Omega v =0$$ on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...
10
votes
1answer
179 views

Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first examples in the standard functional analysis course, when one learns about $\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
3
votes
1answer
88 views

Is the module action $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map: $$\gamma: M\times M^*\to M^*: (a,f)\to af$$ where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
0
votes
1answer
106 views

Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition. $$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$ Of course if $f\in H^s(\mathbb{R}^3)$ ...
0
votes
0answers
71 views

Matrix inequality between a traceless matrix and identity

Given a traceless matrix $C\in M_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$.
3
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0answers
73 views

Invertibility of group Laplacian in $\ell^1$

Let $G$ be a discrete group and let $S$ be a generating set for $G$; assume that $S$ is symmetric (i.e., $g\in S$ iff $g^{-1}\in S$). Let $L=L_S=\frac{1}{|S|}(\sum_{g\in S} g-1)$ be an element of the ...
2
votes
1answer
132 views

To determine if a 2 variable symmetric function is addition formula of one variable function or not?

Since $$f(x+y)=f(y+x)$$, So an addition formula must be symmetric. If we define $$f(x+y)=U(f(x),f(y))$$ If we define $f(x)=p$ and $f(y)=q$ $$f(x+y)=U(p,q)$$ and because of $f(x+y)=f(y+x)$, ...
1
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0answers
47 views

Can a semigroup be defined on a Banach algebra? [closed]

I simply need to know that how a semigroup of operators (say $\{T(t)\}_{t\geq 0}$) is defined on any Banach algebra (say $X$)? For $(f,g\in X)$ now the so called product is also there i.e. $f.g\in X$. ...
1
vote
1answer
97 views

The space of loops as a Banach space [closed]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...
1
vote
1answer
83 views

p-summable sequence

Let Y be a closed linear subspace of X and suppose that Y does not have copy of l1 .Does each weakly p-summable sequence in X/Y has a subsequence that's the image of a weakly p-summable sequence in X ...
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0answers
27 views

Total measure and Riesz theorem [migrated]

As I specified in the other question I asked, analysis is not my field, so I'm sorry if my question is trivial. Riesz representation theorem states the following: Given $X$ a locally compact ...
1
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0answers
62 views

Weak convergence in $L^2(0,T;X)$

In the book Navier Stokes Equations by Constantin and Foias, the folloiwng argument is made: Let $u_m$ converges weakly to $u$ in $L^2(0,T;V)$ where $$ V=\overline{\{f\in ...
1
vote
0answers
63 views

Does bounded and closed equal compact for sets of Borel probability measures?

Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...
2
votes
1answer
125 views

reference request: simple facts about vector-valued $L^p$ spaces [closed]

I learned basic results (regarding weak convergence) about Banach-space valued functions of a single real variable when learning PDE. (See e.g. Appendix E in Evans's Partial Differential Equations) I ...
0
votes
0answers
48 views

Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
2
votes
1answer
87 views

The inverse of Laplacian operator for different orders

I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you! Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open ...
8
votes
3answers
286 views

Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters

Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...
1
vote
1answer
106 views

When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
2
votes
0answers
65 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
4
votes
2answers
181 views

Heat kernel asymptotics for small distances

I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies $$p_t(x, y) = \frac{1}{(4\pi ...
1
vote
0answers
73 views

Can Gradient be controlled by Curl and Divergence in Morrey spaces

In $L^p(\mathbb{R}^3)$, it holds for $1< p< \infty$ and $\mu\in C^\infty_0(\mathbb{R}^3)$, $$\|\nabla\mu\|_p\leq C \left( \|\operatorname{div} \mu\|_p + \|\nabla\times\mu\|_p \right).$$ So, how ...
2
votes
1answer
210 views

Decoupling in mixed norm spaces

Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|_{L^p}$ into an $L^2$ sum of $\|f_\theta\|_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where ...
5
votes
2answers
184 views

Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?

Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces. So it is ...
0
votes
0answers
22 views

Is there a “natural” way to interpolate between a set of bound state wave functions?

Consider for example the Coulomb potential, $-Z/r$, for which there exist a set of bound states with energy $\epsilon_n := {-Z^2 \over 2 n^2}$ (in Hartree). If I want the "wavefunctions" for some ...
1
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0answers
52 views

Kirillov orbit Method for Complex nilpotent groups

Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...
2
votes
1answer
298 views

Pointwise convergence of Fourier series, Fefferman's article

This is the first time I ask a question here, so sorry if I make any mistake in the way I ask it. I'm studing Fefferman's article Pointwise Convergence of Fourier Series, and I have two questions: ...
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0answers
48 views

The closed support of νν is a set of infinite 1-dimensional Hausdorff measure

if $\nu$ is non-zero measure,and $\sum\limits_{|n|\neq 0}\frac{|\hat{\nu(n)}|^2}{|n|}<\infty$,$\hat\nu(n)$ is the Fourier transform of the measure $\nu$. why the closed support of $\nu$ is a set ...
6
votes
1answer
345 views

Who gave the generalized Stone-Weierstrass Theorem?

Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on ...
0
votes
0answers
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
votes
0answers
60 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
votes
1answer
108 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
votes
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 ...