All Questions
9,781 questions
1
vote
0
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100
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Conditions on a measure to satisfy certain relation on moments.
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^...
- 173
0
votes
0
answers
65
views
Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
- 103
1
vote
0
answers
103
views
Generalized bilinear estimates
Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have
$$
\|(u^2)_x\| _{X^{s,b'-1}} \leq c \|u\|_{X^{s,b}} \|u\|_{X^{-...
- 425
2
votes
0
answers
262
views
A specific projection and compactness on the Bargmann-Fock space
Let $F_2$ be the Bargmann Fock space defined as the space of entire functions $f$ on $\mathbb{C}$ such that \begin{align*} \int_{\mathbb{C}} |f(z)|^2 e^{- |z|^2} dA(z) \end{align*} ($dA$ is just ...
0
votes
0
answers
137
views
$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$
Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
- 113
1
vote
1
answer
224
views
Can symmetrizing a contraction increase the speed of convergence?
Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
- 199
1
vote
0
answers
104
views
On generalization of Wigner semi circle
I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...
0
votes
0
answers
298
views
High dimensional beta integral (question following the previous post)
Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,649
3
votes
0
answers
217
views
Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user14319
2
votes
0
answers
290
views
Consequence of Modified Young's inequality
Let $f\in L^1(\mathbb R^n)$. Define operator $T_f(g)=|f|\ast g$ for functions $g$ on $\mathbb R^n$. The set of measurable functions $f$ on $\mathbb R^n$, such that $T_f$ is bounded from $L^p(\mathbb R^...
- 415
1
vote
0
answers
400
views
Uniqueness of differential adjoint operator
I wonder if someone can help me on what is, probably, a simple question but is baffling me at the moment!
In standard texts on functional analysis, something like the following is written
Let $L\...
- 11
1
vote
1
answer
219
views
fourier transform of cumulative function
Hi
I've encountered a test that uses the cumulative value of a finite time series to deterime the data set's stationarity.
I would like to know the characteristics of this test in frequency space,...
- 11
4
votes
0
answers
140
views
When is $A^*A$ invertible for Banach space?
Let's consider a linear functional $A$ from smooth objects to smooth ones. It is first order operator in the sense that it extends to be a map from $W^{k+1,p}$ to $W^{k,p}$. Assume that we have $L^2$ ...
- 527
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 351
2
votes
0
answers
366
views
Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
- 888
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
0
votes
1
answer
319
views
Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 789
1
vote
0
answers
57
views
Looking for CDFs that I can integrate a particular transformation of
I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate
$$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
- 11
13
votes
0
answers
564
views
Symmetric (extended) Haagerup tensor product
Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...
- 18.7k
0
votes
2
answers
146
views
representation of compact supported distribution
Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
- 9
1
vote
1
answer
154
views
Question about coercivity of a functional
Hi!
Let $(M,g)$ be a compact Riemannian manifold without boundary of dimension $2m$. Let
$$T:W^{2,2}(M)\rightarrow L^{2}(T^{*}M\otimes TM)$$
be a second order, linear, differential operator (...
- 1,727
1
vote
0
answers
202
views
Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
- 11
0
votes
1
answer
142
views
A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 17
0
votes
0
answers
35
views
Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T
Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...
- 345
0
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0
answers
164
views
Can we separate Toeplitz matrices for negative and positive eigenvalues?
Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
- 9
1
vote
0
answers
660
views
Fractional Fourier transform [closed]
Let $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ be the Fourier transform. Is there any reasonable definition of fractional Fourier transform (i.e. operator $A$ such that $A^{\alpha}=T$ for $\...
- 3,323
4
votes
0
answers
90
views
$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology
This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
3
votes
0
answers
356
views
Stability of convex sets w.r.t. integration over [0,1]
In the preprint on pages 19−20, first using Hahn−Banach, one proves
Lemma 38. For any closed convex set $U$ in a real Hausdorff locally convex space $E$ and for any Riemann integrable $\gamma:[0,1]\...
- 3,584
2
votes
0
answers
93
views
Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces
Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
- 21
11
votes
0
answers
657
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For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?
The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
- 637
5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 922
1
vote
0
answers
119
views
Boundedness of Riesz transforms.
The Riesz $R_i$ transform on $\mathbb{R}^n$ is defined by
$$ R_if(x)= \int_{\mathbb{R}^n} \frac{t_i-x_i}{\vert x-t \vert^{n+1}}f(t) dt$$
for a Schwartz function $f$ on $\mathbb{R}^n$. Can you please ...
- 583
1
vote
1
answer
359
views
Convergence of operators to the identity on Banach spaces
Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. That is, $U_r \subseteq U_{r'}$ if $r \le r'$ and $U_\infty = \...
- 8,512
5
votes
0
answers
200
views
Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
- 8,512
0
votes
0
answers
93
views
Infinite limit in all points
Do there exist a Banach space (possibly nonseparable) $X$ and a mapping $F: X\to X$ such that
$$
\lim_{x\to a} \|F(x)\| = +\infty \quad \forall a\in X\quad?
$$
0
votes
1
answer
222
views
Bounding near the boundary for a Sobolev function.
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
- 3,217
4
votes
0
answers
94
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Algebraic conditions of separability
Let $X$ be a real vector space (without any norm), and $Y$ be a convex subset of $X$, $0\notin Y$. The goal is to find a hyperplane $L$ passing through 0 such that $Y$ lies in a closed halfspace ...
- 109k
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
9
votes
1
answer
395
views
Is there a coalgebraic characterisation of the hyperfinite II_1 factor?
Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
- 5,094
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
1
answer
111
views
Log-nonexpansive functions: terminology and references
During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.
(Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
- 28.6k
3
votes
0
answers
188
views
Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
- 265
2
votes
0
answers
53
views
Topology of fibers of operators under C^{\infty} convergence
A smooth family of maps $f_t : L^2 (\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is given, with $t \in (0,1]$. Suppose that when $t \rightarrow 0$ the family restricted to balls of given radius converges ...
1
vote
1
answer
138
views
Estimating norms of derivatives
Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\...
- 1,701
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 143
6
votes
0
answers
639
views
Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 4,060
0
votes
1
answer
234
views
A property of "Schwartz" quadratic forms
Consider $K(x, y)$, $f(x)$ Schwartz functions and $g(y)$ a tempered distribution. Suppose $$K(x, y) = K(y, x)$$ Define
$$h(t) = \int f(x - t) K(x, y) g(y - t) dx dy$$
It appears to me $h(t)$ is a ...
- 1,368
2
votes
0
answers
176
views
A limit involving a regularizing kernel
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...
- 2,222
0
votes
0
answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
1
vote
1
answer
254
views
Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
user7807