All Questions
Tagged with fa.functional-analysis fa.functional-analysis or
9,772 questions
21
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2
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Is there an L^p tauberian theorem?
From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
- 13k
2
votes
3
answers
3k
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What function has fourier series the harmonic series?
I know that this is on the boundaries of what's allowed, but hopefully someone'll answer before it gets closed!
What (periodic) function has Fourier series the harmonic series? I really want the ...
- 26.8k
3
votes
3
answers
2k
views
Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
- 131
2
votes
2
answers
1k
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Decoupling lemma for the Lambda(p) problem
I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
- 13k
3
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3
answers
1k
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Minimizing a functional
I have wondered the problem in http://www.helsinki.fi/~hmkokko/Stuff/Esdale/index.html for over year without success. If we try to minimize the functional equation
T(\theta ) = \int_0^L\frac {dx}{v_0\...
4
votes
3
answers
451
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uniformity for Banach algebras - a three space property?
Let $A$ be a commutative, unital Banach algebra and $I \subset A$ an ideal such that $I$ with the relative norm is a uniform Banach algebra and $A / I$ with the quotient norm is uniform as well.
Does ...
- 1,338
26
votes
2
answers
2k
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When is a locally convex topological vector space normal or paracompact?
All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
- 26.8k
23
votes
4
answers
5k
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Are proper linear subspaces of Banach spaces always meager?
Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-...
- 353
10
votes
5
answers
1k
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What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
- 2,745
11
votes
2
answers
932
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A group action of the Heisenberg group with special symmetries
Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
9
votes
5
answers
870
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Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
- 629
9
votes
1
answer
611
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opposite Banach space
I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the opposite of V is the complex vector space with the same elements, the same operations except switch scalar ...
- 41.1k
12
votes
3
answers
530
views
Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 4,462
21
votes
2
answers
2k
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In a Banach algebra, do ab and ba have almost the same exponential spectrum?
Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
- 2,154
3
votes
1
answer
2k
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Hilbert Space as direct sum of subspaces with cyclic vectors
Ok,so this should be easy, however I havent taken functional analysis for a while. But given a compact self-adjoint operator on a hilbert space H(over the complex numbers), we define v to be a cyclic ...
- 31
9
votes
1
answer
395
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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
3
votes
1
answer
914
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Range of a Certain Linear Operator
Consider the following hermitian form on the sobolev space H^1(I), of an interval I:
g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I.
Riesz representation ...
- 31
40
votes
5
answers
10k
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Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
21
votes
5
answers
4k
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Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 629
8
votes
3
answers
698
views
L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...
- 101
2
votes
1
answer
493
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Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
12
votes
4
answers
877
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Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 1,059