Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
1
vote
0
answers
149
views
(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\...
- 11
1
vote
0
answers
305
views
Adjoint operator in sobolev space
Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta u\|...
- 51
1
vote
0
answers
104
views
On generalization of Wigner semi circle
I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...
1
vote
0
answers
275
views
Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
- 11
1
vote
0
answers
232
views
From positive definite function to Følner sequence ----- a question on amenability and nuclearity
We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
- 11
1
vote
0
answers
341
views
spaces of smooth functions with bounds on partial derivatives
EDIT: As there were no takers at all... I have added below a possible approach I came up with...
I would like to ask the following elementary but tricky question about the density of spaces of smooth ...
- 1,338
1
vote
0
answers
291
views
(DFM) vs (DFS) spaces, Banach scales
This question is related to one I posted before: https://mathoverflow.net/questions/117492/dfm-spaces-and-the-c-infty-topology
According to Dineen (in "Complex Analysis in Locally Convex Spaces",p.15)...
- 11
1
vote
0
answers
618
views
A question on a variant of Hardy's inequality.
I would like to know a proof of a variant of Hardy's inequality below. Could anyone introduce me a reference or give me a proof? Thank you very much for your assistance.
Set $0<r<\frac{2(n-s)...
- 11
1
vote
0
answers
294
views
Galerkin method for existence for PDE with nonsymmetric bilinear form
Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in L^2(0,T;...
- 113
1
vote
0
answers
153
views
Functional Analysis Generalizations: indeterminated inner product and functions over manifolds
There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
- 11
1
vote
0
answers
784
views
How to find the tensor product of modules that we don't know a basis for them?
Hi
I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[X] \otimes_{F} F[Y] \...
- 11
1
vote
0
answers
61
views
Multi-completely monotone functions
Consider a $C^{\infty}$ nonnegative function $f(x,y,z)$, $x,y,z>0$ and let $\lambda f(\lambda x, \lambda y,\lambda z) \equiv f(z,y,z)$ for any $\lambda > 0$ (positive homogenity). Define
$$
g_{...
- 1,329
1
vote
0
answers
119
views
Boundedness of Riesz transforms.
The Riesz $R_i$ transform on $\mathbb{R}^n$ is defined by
$$ R_if(x)= \int_{\mathbb{R}^n} \frac{t_i-x_i}{\vert x-t \vert^{n+1}}f(t) dt$$
for a Schwartz function $f$ on $\mathbb{R}^n$. Can you please ...
- 583
1
vote
0
answers
52
views
Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\...
- 71
1
vote
0
answers
237
views
bivariate polynomial
Hello,
Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex.
If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where $|...
- 9
1
vote
0
answers
316
views
"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
- 29
1
vote
0
answers
205
views
Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
1
vote
0
answers
266
views
Nuclear Space problem
I need to show that if X is compact,then C(X) is nuclear.Also is the condition X is metrisable
necessary. I am at present attending a conference "Recent Aadvances in Operator Theory". This
problem ...
- 2,106
1
vote
0
answers
289
views
Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
- 61
1
vote
0
answers
346
views
HyperKaehler manifolds are Ricci-flat
Hi,
I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = g(...
- 93
1
vote
0
answers
187
views
Injective modules over Fourier algebra
Is there any article on injective modules over Fourier Algebras?
Do we have anything about injectivity of $A(G)$ as a $A(G)$-bimodule?
- 71
1
vote
0
answers
125
views
base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
- 2,649
1
vote
0
answers
125
views
Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
1
vote
0
answers
465
views
Reference book for a stronger version of Goldstine theorem
Let me quote the proof of Goldstine theorem from wikipedia ( http://en.wikipedia.org/wiki/Goldstine_theorem ):
Given an $x^{**} \in B_{X^{**}}$, a tuple $(\phi_1, \dots, \phi_n)$ of linearly ...
- 311
1
vote
0
answers
119
views
Particular types of basis on a normed vector space of finite dimension
Is it true that on every normed vector space $V$ of dimension $n$ there exists a basis of norm $1$ vectors $v_i$, such that $\|\sum_{i=1}^n\epsilon_iv_i\|\geq 1$ for all possible combinations of $\...
- 31
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 351
1
vote
0
answers
153
views
The existence of the solution of the perturbed KdV Equation(semi-group operator)
Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...
- 61
1
vote
0
answers
271
views
When Pelczynski's property (V*) forces (V) in the dual space?
Recently, I have been asked if are there any sufficient conditions for a Banach space with Pelczynski property (V*) to have dual with property (V). Since it was a long standing open problem, I guess ...
1
vote
0
answers
1k
views
Can you prove the monotonicity of the function (or find a counter example)?
Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative ...
- 11
1
vote
0
answers
136
views
Boundedness of Integral
Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point.
Define the integral
$$
Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta
$$
and ...
- 233
1
vote
0
answers
180
views
iterated traces for sobolev functions
It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose ...
- 2,041
1
vote
0
answers
178
views
Inequalities between self-adjoint operators
Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$...
- 329
1
vote
0
answers
135
views
Inequality involving BV norm and a regularizing kernel
In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ...
- 2,222
1
vote
0
answers
202
views
Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
- 11
1
vote
0
answers
114
views
Mappings preserving convex compactness
Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...
- 11
1
vote
0
answers
393
views
Unambiguous "weak" vector valued $L^{+\infty}$ spaces?
For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
- 3,584
1
vote
0
answers
149
views
Banach spaces with simple best approximate solutions
Let $\langle V,||.||\rangle$ be a Banach space such that:
$\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$
$\;\;$ that ...
user5810
1
vote
0
answers
2k
views
Derivative total variation function
Hello everyone,
I would like some help on proving the following statement:
Let $f\in\mathrm{BV}[a,b]$, i.e. $f$ is of bounded variation and let $T(x)$ be the total variation of $f$ on $[a,x]$ for $x\...
- 11
1
vote
0
answers
318
views
Fourier series/transform of an amplitude-limited sinusoid
I am trying to estimate the amplitude of an original unlimited sine wave from a measurement of the power spectral density (PSD) of an amplitude-limited version. I expect that I may be able to do so ...
- 11
1
vote
0
answers
159
views
variational problem under convexity constraints
I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert ...
- 1,839
1
vote
0
answers
215
views
Classification of Self similar sets
I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are ...
- 11
1
vote
0
answers
378
views
Adjoint operators in LCS
Before my main question let me start with the following notions.
Let $X$ and $Y$ be locally convex spaces and let $T \colon X \rightarrow Y$ be a linear mapping. The adjoint of $T$ is an operator
$T^...
- 145
1
vote
0
answers
693
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
- 1,649
1
vote
0
answers
369
views
Infinite internal direct sums of subspaces
Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a ...
1
vote
0
answers
869
views
Limit of two hypergeometric functions (2F1)
Hi,
Does anyone know whether there is a known function/distribution that corresponds to the limit:
$\lim_{\epsilon\rightarrow0^+} \mathfrak{Re}\left[f(x+i\epsilon) - f(x-i\epsilon)\right]$
when $f(...
- 131
1
vote
0
answers
477
views
A norm ratio inequality
Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.
Then
$$
\frac{||z||_3}{||z||_2} \le
K_n
||z/y||_\infty
\frac{||y||_3}{||y||_2}
$$
where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ ...
- 6,541
1
vote
0
answers
400
views
Uniqueness of differential adjoint operator
I wonder if someone can help me on what is, probably, a simple question but is baffling me at the moment!
In standard texts on functional analysis, something like the following is written
Let $L\...
- 11
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
1
vote
0
answers
237
views
Variation of a function
There are probably some of you guys who already know some of the terms that I am going to use so in order to be not so boring I will put the definition to the end.
Let $f$ be a piecewise expanding ...
- 11
1
vote
0
answers
2k
views
Extension Operators for Sobolev spaces
Let $\Omega\subset\mathbb{R}^d$ be a bounded domain with Lipschitz smooth boundary and $\delta>0$ sufficiently small so that
$
\Omega_\delta = ${ $x\in\Omega : dist(x,\partial\Omega)>\delta $ }$...
- 3,217