All Questions
12,935 questions
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62
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To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
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78
views
Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?
In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality:
$$\...
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72
views
Fourier coefficient of close functions
Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...
- 1
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303
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Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
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183
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Comparison of orthogonal complements in L^2 and H^1 spaces
Let $\Omega$ be a bounded domain with smooth boundary, and let $L^2(\Omega)$ and $H^1(\Omega)$ the usual $L^2$ and $H^1$ function spaces on $\Omega$, respectively. We call $\phi \in H^1(\Omega)$, and $...
- 11
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86
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Solution to the differential equation $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle =-\frac{\partial g(X)}{\partial x_{1}} $?
If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $...
- 912
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103
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How should I understand the completeness relation of the form $\sum_{n} \phi_n(x) \overline{\phi_n}(y)=\delta(x-y)$?
Let $A$ be an unbounded self-adjoint operator on $L^2(\mathbb{R})$ and we are assuming the $L^2$ functions to be complex-valued.
We further assume (e.g. compactness of resolvent) that there exists an ...
- 3,477
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190
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Reference request for the dual space of the Bochner Space $L^1(\Omega ; X) $
Let $\Omega\subset\mathbb{R}^N$ be open and let $X$ be a Banach Space. Let $L^1(\Omega ; X)$ denote the space of all strongly measurable (sometimes also referred to as Bochner Measurable) functions, $...
- 1
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98
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
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66
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Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463
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76
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Linear dependence of the derivatives of a vector valued function
Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function
$$
g:\mathbb{R}^5\rightarrow\mathbb{R}^5
$$
given by
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 8,998
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79
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Reverse Hölder for possibly singular matrix weights
I'm working on some nonlinear partial differential equations and I have been led to the following puzzle. Let $W(x)$ be a symmetric positive semidefinite-valued function of $x \in \mathbb{R}^d$ (a &...
- 981
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238
views
On the spectrum of a compact pertubation of a skew-adjoint operator
Let $A\colon \text{dom}(A)\subset H \to H$ ($H$ is a Hilbert space) be a skew-adjoint (i.e. $A^{*}=-A$), closed and densely defined operator. Then the essential spectrum is the set of spectral values $...
- 173
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150
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Prove or disprove the compactness of an operator
Consider $X=L^{2}(0,\pi, \mathbb{R})$.
Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator.
We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
- 39
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36
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Regarding significance of spectral variation under algebraic operations
I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...
- 27
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answers
54
views
Perturbation of first eigenvalue
Let $\Omega$ be a bounded domain in $R^N$ and suppose $\gamma$ is some smooth bounded function in $\Omega$. Let $ -\Delta \phi_0 = \lambda_0 \phi_0 $ in $ \Omega$ with $ \phi_0=0$ on $ \partial \...
- 1,385
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92
views
Positive definite matrix and Hörmander theory
Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$
Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set
$$
\varphi_{\...
- 101
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answers
59
views
Identification of vector valued function
Do you know of a good reference for a proof of the fact that $L^2(0,T,L^2(\Omega))$ and to $L^2([0,T]\times \Omega)$ can be identified?
- 235
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90
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Cousin of Fourier transform for rescaling and translating functions
Is there a cousin of the Fourier transform which obeys the following property: if I rescale and translate a function, then its transformed form, at each frequency, only depends on the original ...
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171
views
Explanation of a step in a preprinted work
I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct.
I do not ...
- 159
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126
views
On the convergence of operators and their spectra
We consider a sequence of operators $\{L_n\}_{n=1}^\infty$. Each operator $L_n$ is a densely defined (possibly unbounded) closed linear operator on a real Hilbert space $H_n.$ The domain of $L_n$ is ...
- 721
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225
views
Evaluate the projection operator norm with respect to maximum norm
Consider the normed space $\left(\mathbb{R}^{3},\|\cdot\|\right)$ where $\|\cdot\|=|\cdot|_{\infty}$ is the usual maximum norm. Consider the 2 -dimensional vector subspace $\left \{ (x,y,z):x+y+z=0 \...
- 111
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129
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Dirac operator on 4-dimensional rectangle with the periodic boundary conditions is self-adjoint? What is its spectrum?
Let us think of the Euclidean Dirac operator $iD^k \gamma_k$ on the rectangle $[-1,1]^4$ with the periodic boundary conditions.
The covariant derivative $iD^k$ carries a gauge potential term and we ...
- 3,477
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answers
83
views
Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 159
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71
views
Sufficient condition to be increasing, following a vector field
Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
- 449
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241
views
Can a non-reflexive space embed into a reflexive space?
My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
- 11
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57
views
Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 173
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122
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Sharp, salient and opposite cones
I have been reading about star shaped sets and support cones from this article.
Can anyone please help me with examples the difference between a sharp and dull cone.
How come a salient cone has a ...
- 27
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203
views
Regarding definition of convex cone and apex
I have been reading about star shaped sets and support cones from this article. I am wondering about the definition of the cone as to why is it defined this way? Why is it $C-a$? I am familiar with ...
- 27
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0
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63
views
increasing property of the heat equation on the interval
$Q_T^l=\{(x,t); 0<x<l, 0<t\leq T\}$,
$u_l\in C^{2,1}(Q_T^l)\cap C(\bar Q_T^l)$ is the solution of $\frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l$...
- 119
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votes
1
answer
116
views
Using Hölder's Inequality to prove the following equation [closed]
Here $B_r$ presents the open ball of radius r in $R^3$. So I hope to know how to prove the following inequality.
$$
\int_{B_r} |u|\le Cr^{\frac{9}{5}}\left(\int_{B_r} |u|^2\right)^{\frac{1}{5}}\left(\...
- 65
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0
answers
94
views
Is the space of affine continuous functions a Baire space
Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
- 675
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1
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78
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The conditions used to prove upper semicontinuous of generalized directional derivative (in Clarke sense)
Let $X$ be a reflexive Banach space, $z, x, v \in X$. $\{z_i\}, \{x_i\}$ and $\{v_i\}$ are arbitrary sequences converging to $z, x$ and $v$, respectively.
I would like to know under which conditions ...
- 15
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0
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143
views
Minimax problem : uniqueness of a solution
Let $n\geq2$. Is it true that the minimax problem:
$$
\min_{p\in\mathcal{P}}\max_{H\in\mathcal H}p^tH\bar{p},
$$
where
$\mathcal H\subset\mathcal{M}(n)$ is a strictly convex bounded subset of ...
- 4,034
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votes
0
answers
113
views
Existence of distance-preserving mappings for general norm in vector space
We say a mapping $f:\mathbb R^n\to \mathbb R^n$ be 1-Lipschitz with respect to a norm $\|\cdot\|$ if $\|f(x)-f(z)\|\le\|x-z\|$ holds for all $x,z\in\mathbb R^n$. Such a mapping are sometimes called a ...
- 601
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52
views
How to know if two special functions are related by an elementary function?
Suppose I have two special functions $f_1$ and $f_2$. Is there an algorithm which can tell me whether there exists elementary $g$ such that $f_1 = g\circ f_2$? Furthermore, is there any possibility to ...
- 333
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98
views
Viscous stress equation in Newtonian fluid
In this Wikipedia entry, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is
$$
\tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\...
- 9
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0
answers
540
views
The definition of essential spectrum for general closed operators
I've asked this problem in MSE several days ago, see here. But there is no reply up until now. Maybe I wrote things too complicated there and so I'll write a very clean problem here. For background ...
- 1
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0
answers
43
views
Minimal condition on set for an optimisation problem
We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{E}(\...
- 181
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0
answers
136
views
Fractional Laplacian of smooth cut off functions
Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$
...
- 537
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210
views
Reed-Simon Vol. IV: Question regarding convergence of eigenvalues
I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
- 91
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0
answers
21
views
Estimatives for elliptic systems involving the laplacian
Considering the problem
\begin{equation}
\left\{
\begin{array}[c]{11}
\Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\
\Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\
\end{...
- 101
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votes
0
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75
views
Extracting the point mass measure of some type of positive measures
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals.
Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
- 4,058
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votes
0
answers
80
views
Cyclicity of composition operators
Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ ...
- 5,405
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votes
0
answers
116
views
Is there a proper term for a "continuum-convex" set?
Let $X$ be a Banach space, and for any compactly supported Borel probability measure $\mathbb{P}$ on $X$, define the mean $\mu_\mathbb{P}$ by $\mu_\mathbb{P}=\int_X x \, \mathbb{P}(dx)$.
I want to say ...
- 708
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votes
0
answers
40
views
Time regularity of traces
I have a question about the time regularity of the traces in one dimension.
Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(...
- 11
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votes
0
answers
129
views
Sturm-Liouville result
Suppose $n \ge 2$ an integer and consider finding the first eigenvalue of
$$ -\partial_\theta \left( \omega(\theta) \psi'(\theta) \right) = \mu_1 \omega(\theta) \psi(\theta)$$ for $ 0<\theta<\...
- 1,385
0
votes
0
answers
148
views
A Grönwall-type inequality for $u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s$?
Quick question: Are we able to show a Gronwall-type inequality assuming that $$u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s,$$ where $\alpha$ is nondecreasing (or constant) and $\beta$ is ...
- 167
0
votes
0
answers
60
views
How to prove this estimate involving the Stein Derivative?
Recall the Stein Derivative,
$$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$
I want to show that,
$$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \...
- 537
0
votes
0
answers
70
views
Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99