All Questions
9,959 questions
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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\...
- 161
0
votes
0
answers
104
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Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
2
votes
0
answers
262
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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 ...
2
votes
1
answer
362
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Quasinilpotent example [duplicate]
Possible Duplicate:
Quasinilpotent operator
Do you know any example of a quasinilpotent operator such that every its power is non-compact?
Of course direct sum of nilpotent operators(or Volterra ...
CommunityBot
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1
answer
437
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Möbius Transform of a Continuous Possibility Function
In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.
There is ...
- 1,343
2
votes
2
answers
551
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L^2 basis of class functions on a compact Lie group that are point-wise small
Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...
- 23k
5
votes
2
answers
412
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Wiener Tauberian Theorem for nonunimodular group
Is there a nonunimodular group for which Wiener's Tauberian theorem is true?
Is a locally compact topological group whose volume grows polynomially with radius always unimodular?
- 1,152
2
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1
answer
680
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spectra of sums in (Banach) algebras
A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".
Answer there led me to the following question.
If for ...
CommunityBot
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26
votes
4
answers
5k
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Can $L^{2}$ be represented as a space of functions (not equivalence classes)?
Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
- 4,597
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0
answers
138
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Notion of simplicity of a function(al)
Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).
Specifically, intuitively one could ...
- 59
1
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1
answer
559
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Sum of a Gaussian and an independent second moment constrained random variable
I am studying the (asymptotic) behavior of the p.d.f of the random variable $Y = X + Z$, where $X$ is an r.v. with any distribution function $F(x)$ such that $\int_{-\infty}^{\infty} x^2 \mathrm{d}F(x)...
CommunityBot
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4
votes
1
answer
287
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Second conjugate operators to operators on $c_0$
I posted my question at MS but unfortunately it is still without a response, so let me ask it here.
We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,...
- 31.5k
6
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0
answers
8k
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Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
1
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1
answer
931
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Finding a counter example for Minkowski's integral inequality for $p=\infty$ [closed]
Dear All,
As we know that this following Miskowski's integral inequality is true for $1\leq p<\infty$
$
[\int_{S_1}|\int_{S_2}F(x,y)d\mu_1(x)|^pd\mu_2(y)dy]^{1/p} \leq \int_{S_2}[\int_{S_1}|F(x,y)...
- 41.1k
1
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1
answer
233
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Structure of Measurable Subsets of the Unit Square
If A is a (Lebesgue-)measurable subset of the unit square that has positive measure, does there exist a subset B contained in A that has a product structure (is the product of two subsets of the real ...
- 54.2k
1
vote
1
answer
219
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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,...
CommunityBot
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3
votes
2
answers
403
views
Cesaro bounded Operator which is not power bounded
Good evening!
Let X be a banachspace and T a bounded linear operator on X.
The cesaro avearges of T are defined as:
$A_n:=\frac{1}{n} \sum\limits_{j=0}^{n-1}T^j $
We call T cesaro bounded if: $\...
- 31.5k
7
votes
1
answer
2k
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On the Paley-Wiener theorem
Is there any even Schwartz function whose restriction to $[0,\infty)$ is monotone and whose Fourier transform is compactly supported? In other words, is there any entire analytic function satisfying ...
- 2,753
3
votes
2
answers
428
views
Approximating smooth functions with polynomials subject to constraints.
Suppose that we are given a smooth function $h:\mathbb{R}^n \to \mathbb{R}$ which satisfies $h \circ F= h \circ G$ for two polynomial functions $F,G:\mathbb{R}^m \to \mathbb{R}^n$ (i.e. each component ...
- 1,540
4
votes
1
answer
2k
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Existence of weak limits
Background: Three months ago, I asked this question, which is a bit related to the following (if the answer to it was Yes, then answer to this one would be Yes too, but since that was a No, it still ...
CommunityBot
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0
votes
1
answer
396
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Characterization of Measureable Sets [closed]
Every countable union of rectangles in R2 is a Lebesgue measurable set. Is the converse true, too?
Specifically, I wonder whether the following statement is true:
Let A be a set in the unit square ...
- 25.8k
7
votes
1
answer
682
views
$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$
Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...
- 31.5k
17
votes
4
answers
2k
views
Banach-Mazur applied to a Hilbert space
The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm.
...
- 61.9k
2
votes
1
answer
2k
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Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
CommunityBot
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4
votes
1
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775
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Algebraically simple Banach algebras
There are plenty of semi-simple Banach algebras - this broad class includes C*-algebras and algebras of bounded operators on a given Banach space. On the other hand, it seems unlikely to me that there ...
- 11.1k
2
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2
answers
411
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Functional Minimization: When is this heuristic rigorous?
I'm trying to solve a functional minimization problem of the following form:
$$\arg\min_{f:\mathbb{R}\rightarrow [0,1]} h(f)$$
where $h$ is some expression in terms of several integrals over $f$.
I ...
CommunityBot
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14
votes
0
answers
2k
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Schwartz kernel theorem for A-linear operators
Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
- 17.1k
4
votes
0
answers
500
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Laplace Transform: Are there theorems similar to the Bernstein Theorem?
Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b)
Are ...
- 93
2
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0
answers
137
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Invariant linear manifolds for multiplication by the independent variable in L^2 (R)
In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-...
- 31
2
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1
answer
230
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Completing The Space Sections in a Vectorbundle
Hi there.
Assume $(M,g)$ is a Riemanian manifold and $E\to M$ is a
vector bundle with a bundle metric $\langle\cdot,\cdot\rangle$. We then have the pre-Hilbert space $H_0:=\Gamma_c^\infty(E)$ of ...
- 8,559
5
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1
answer
461
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Is there a standard notation for a "shift space" in functional analysis?
I'm writing up some notes on the nLab about things like embedding spaces and infinite spheres and similar things (can't link to them yet as I haven't put them up yet). One aspect that crops up time ...
- 31.5k
10
votes
1
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784
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When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
- 2,203
3
votes
2
answers
461
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Complemented subspaces isomorphic to $c_0$ in $\mathcal{B}(E)$ [closed]
It is well known that neither
1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$
nor
2) $c_0$ is a quotient of $\mathcal{B}(H)$
for a Hilbert space $H$. Can we replace $H$ above ...
CommunityBot
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1
vote
2
answers
1k
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Conformal transformations and harmonic analysis on the sphere
Consider the $n$-dimensional sphere $S^n$. I'm especially interested in the $n=4$ case. The Hilbert space $L^2(S^n)$ can be decomposed into a direct sum of eigenspaces of the Laplacian, which are ...
CommunityBot
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13
votes
1
answer
4k
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Modulus of Continuity
I originally posted this question on math.stackexchange (https://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2), but it's been a few days and I haven't received any correct ...
CommunityBot
- 1
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 ...
- 35.5k
7
votes
2
answers
1k
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Yang Mills gradient/heat flow on 4-torus
The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta ...
- 362
1
vote
1
answer
244
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Oscillatory integral decay & sublevel set growth
I am trying to understand how estimates on sublevel integrals imply estimates on oscillatory integrals. Specifically in this article by M. Greenblatt it says on page 7:
By well-known methods ...
- 96.4k
9
votes
1
answer
893
views
Perturbations of an operator that disconnect the spectrum
The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under ...
CommunityBot
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1
vote
0
answers
159
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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
10
votes
1
answer
680
views
A problem concerning $L^2([0,1]\times[0,1])$
Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let $...
- 61.9k
3
votes
2
answers
3k
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Uniqueness of power series
Is there two sequences of real numbers $a_i$ and $b_i\neq 8$, not depending on $x$, such that $x^8=\sum_{k=1}^{\infty}a_kx^{b_k}$ for all $x$?
If $\displaystyle\sum_{k=1}^{\infty}a_kx^{b_k}=\sum_{k=1}...
- 61.9k
4
votes
3
answers
2k
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looking for a book on banach manifolds
Hi,
I am looking for a book on Banach manifolds. Can somebody recommend me something.
Thanks in advance.
leo
- 12.5k
0
votes
2
answers
1k
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Jordan form of compact operator [closed]
Let $X$ be Banach space over a field $\mathbb{C}$.
Consider the Banach space $L_c$ of compact operators in $X$. Let $A^0\in L_c$ be fixed and
$\lambda^0\neq 0$ his eigenvalue with algebraic ...
- 95
0
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1
answer
130
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Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 39.9k
18
votes
1
answer
564
views
Is the space of Hankel operators complemented in B(H)?
Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$.
Let ...
- 25.8k
15
votes
3
answers
8k
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What is an isomorphism of Banach spaces?
The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
CommunityBot
- 1
4
votes
2
answers
1k
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How to use DFT to solve this minimization problem?
This is a problem when I'm reading a paper.
Equation:
$min\{\sum_p(S_p-I_p)^2+\beta((\partial_xS_p-h_p)^2+(\partial_yS_p-v_p)^2) \} $
where $S,I,h,v$ are all $M*N$ matrices and p stands for every ...
- 86
3
votes
1
answer
394
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Topological weak mixing and $\omega$-linearly-independent sequences generated by composition operators
A research problem on which I am currently working requires a construction in topological dynamics of the following type:
Let $T \colon X \to X$ be a continuous transformation of a compact metric ...
- 4,624
8
votes
0
answers
196
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Parametrizing derivations from the algebra of smooth functions on a manifold to its dual
$\newcommand{\Der}{\operatorname{Der}}$
$\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
- 25.8k