All Questions
3,629 questions with no upvoted or accepted answers
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131
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Uniqueness of solution of Volterra Integral Equation with deviating argument
In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$:
\begin{equation}...
- 259
1
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94
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Interpolation theory: equivalence of norms
Consider the interpolation space $Z=(X,Y)_{\theta,p}$. In the case $Y\subseteq X$ do we have that, for all $a>0$ the following norm:
$$N_a:x\mapsto\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \...
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0
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58
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Extension of a result about measurable, additive functionals
Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
- 869
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110
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On functions obtained from Gaussian Quadrature integration
Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...
- 1,209
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122
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Mackey topology characterising property
Let $V$ be a topological $k$-vector space.
Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.
...
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389
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The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
- 263
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74
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Elliptic pde L^p theory via adjoint theory
Let $ T:X \rightarrow Y$ denote some linear operator and suppose we know its one to one (here $X$ and $Y$ are Banach spaces). I believe their is results that say $Ker(T^*)= (R(T))^\perp$ (where ...
- 1,385
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147
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Property of Fixed Point Function
Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...
- 1,127
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110
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Using semigroup theory for nonautonomous semilinear equations
We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...
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36
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A variation on Sylvester equation
Let $X$ be a finite measure space and $D,M$ be bounded linear operators on a $(L^1(X;\mathbb C))^2$. $D$ is a diagonal operator matrix whose entries are multiplication operators by the invertible ...
- 3,388
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92
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Extension of a derivation
Let I be a closed left ideal of a Banach algebra A and let D:I\to I* be a derivation. Does D extend to a derivation from A to A*?
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71
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Kernel of Radon transform in $\mathbb{R}^3$
Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$:
$$(Rf)(H):=\int_{l\subset ...
- 21.8k
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73
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relative compact on nonlinear term
On the paper: Decay of Solutions to Nonlinear Schrodinger
Equations.
Let $u$ be a solution of the equation
$$Hu+|u|^2u=0,$$
where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
- 1,915
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50
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Representability of smooth invertible Lipschitz functions by a finite composition of near-identity functions
Theorem 1 of this paper shows that
For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \...
- 6,853
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122
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Completed Tensorproduct
I am trying to understand the completed tensorproduct. This can be defined as follows:
Given a topological ring $R$ and two linearly topologized rings $A$ and $B$ with fundamental systems of open ...
- 309
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49
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On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
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53
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Limit contration rates and expansion rate solenoid map
Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
- 1,043
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196
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Compact embedding result
Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
- 395
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863
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Why is $H^{1/2}$ a Hilbert space?
Let $n\in\mathbb{N}$ and $\Omega \subseteq \mathbb{R}^n$ sufficiently smooth. Then we have the Hilbert space $H^1(\Omega)$ and the trace operator $\operatorname{tr}: H^1(\Omega) \to L^2(\partial \...
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259
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Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?
The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
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50
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Necessary conditions for a function to be represented as symmetric tensor product
Let $f(\cdot,\cdot)$ be any function of 2 arguments. Suppose also that the following equation holds $$f(y,x)+f(x,y)=g(x) h(y) +g(y)h(x) \quad \forall x, y$$ for arbitrary $h(\cdot)$ and $g(\cdot)$. ...
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169
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Conditional Expectation for von Neumann algebra
Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...
- 233
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95
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An example of a sequence of finite projections
Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...
- 4,058
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922
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A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
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80
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Unclear inequality of L2 norms (Poisson equation for modeling flow)
I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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94
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Show that the norm's bound is an exponent
Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...
- 343
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74
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Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
user127450
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75
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Proof of Lemma 7.1 Bonsall and Duncan
In the proof of Lemma 1 in section 7 (A functional calculus for single Banach algebra element) of the book Complete normed algebras by Bonsall and Duncan, the last line says
$$\phi\left(\frac{1}{2\pi ...
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299
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How can I obtain this inequality (from Evan's PDEs)
I am self studying PDEs from Evans' "Partial Differential Equations" textbook. Currently, I am going through Theorem 1 from Section 5.7 (Rellich-Kondrachov compactness theorem) and am having ...
- 185
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93
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Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
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211
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Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
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387
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Fast growing unbounded functions in the Sobolev space $H^1(\Omega)$
I am looking for unbounded functions that grow rapidly fast near the origin, but are in the Sobolev space $H^1{(\Omega)}$, where $\Omega$ is a unit square centered at the origin.
I already know about ...
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80
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Is $Iso(V)$ a deformation retract of $GL(V)$ when $V$ is a finite dimensional linear normed space
Assume that $V$ is a finite dimensional real or complex normed linear space. Let $Iso(V)\subset GL(V)\subset L(V)$ be the space of linear isometric endomorphisms, invertible endomorphism and ...
- 356
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63
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Are the sets whose convex hull surface admits multiple representations a shy set of sets?
Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...
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324
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Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
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403
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Dual of $C(X)$ with the compact open topology
Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compact-open topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong ...
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Fourier Lapalacian over periodic end
This is a technical question on Taubes' paper: Gauge theory over periodic end. on Page 378.
Recall that:
Let $Y$ be a closed manifold, with $b_1=1$, and $\tilde Y$ be the $\mathbb Z$-covering of $...
- 1,915
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117
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trace inequality for Dirichlet Neumann operator
Does there exists a Sobolev trace inequality of the form
$$ \|U(x, 0)\|_{L^{q}((a, b))} \leq C\sqrt{q}\| \nabla U \|_{L^{2} (\mathcal C)} ; \forall U\in H^{1}_{0, L} (\mathcal C)$$ and for any $q>...
- 402
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159
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Embedding standard function spaces into superstructure
I have a question concerning the precise handling the usual function spaces like $L^2$ in the context of the superstructure. In their paper
Benci, Vieri; Luperi Baglini, Lorenzo. Generalized ...
- 16.6k
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192
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A nested sequence of closed subspaces of $\ell^2$
Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to u$.
Is it possible extract a ...
- 433
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45
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Shifting Sobolev norms in a hyperbolic estimate
Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
- 4,143
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217
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Status of an open problem in isometric aspect of Banach space theory
The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$
Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach ...
- 623
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122
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Discrete Wavelet Transform and Gaussian decay
I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
- 11
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0
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86
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An equality of inner products of holomorphic curves
The following is the main result in the paper by Vinnikov, Putinar, Alpay: A Hilbert space approach to bounded analytic extension in the ball, 2003, Communications on Pure and Applied Analysis. The ...
- 91
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61
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Realizing $N$-body Hamiltonian operator from $2$-body operator
Let $N\in\mathbb{N}$, and consider the formal $N$-body Schrodinger operator
$$\sum_{j=1}^{N}-\partial_{x_{j}}^{2}+2c\sum_{1\leq j_{1}<j_{2}\leq N}\delta(X_{j_{1}}-X_{j_{2}}), \tag{1}$$
where $c\in\...
- 873
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291
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Exponential decay of a convolution
Let $z=(x,y) \in \mathbb{R}^N \times (0,+\infty)$, and let
$$
P_m(z)=y^{2s} |z|^{-\frac{N+2s}{2}} K_{\frac{N+2s}{2}}(m|z|),
$$
where $N \geq 3$ is an integer, $0<s<1$ and $K_{\frac{N+2s}{2}}$ ...
- 459
1
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0
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126
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identity involving spectral functions
Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true :
$$ A^* f(AA^*) = f(A^* A) A^*$$
- 519
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0
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261
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If $H$ is a Hilbert space, is the projective tensor product $E\:\hat\otimes_\pi\:H$ isometrically isomorphic to $E\:\hat\otimes_\pi\:H'$?
Let
$E$ be a $\mathbb R$-Banach space
$H$ be a $\mathbb R$-Hilbert space
$E\:\hat\otimes_\pi\:H$ denote the completion of the tensor product of $E$ and $H$ with respect to the projective norm
By ...
- 167
1
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0
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237
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On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
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0
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146
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Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
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