All Questions
12,779 questions
1
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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 ...
- 99
3
votes
1
answer
248
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relation between SOT-convergence of T and T'
I am trying to prove or to break the following statement (I assume that the statment is correct):
Assumptions: Let $H$ be a Hilbert-space (or more generally a reflexive space) and $T\in \mathcal{L}(H)...
- 65
3
votes
2
answers
403
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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: $\...
- 65
71
votes
16
answers
21k
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Is there a nice application of category theory to functional/complex/harmonic analysis?
[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in most ...
2
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0
answers
678
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Self-similarity of Riemann's "non-differentiable" function
I hope it doesn't seem inappropriate for me to raise on MO an unanswered question from MSE, indeed a question actually posed there by someone other than myself.
I want to ask the following:
1) ...
- 799
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0
answers
484
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Square and cube?
Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...
- 113
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3
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1k
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Is there a Plancherel Theorem for Gowers norms?
In the process of counting arithmetic sequences in sets, the Gowers norms
$$ ||f||\_{U^s[N]}^{2^s} = \frac{1}{N^s} \sum_{\vec{h} ,\\, n } \Delta_{h_1}\dots\Delta_{h_s}f(n) $$
where the sum is $ \...
- 22.8k
4
votes
1
answer
254
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M-bases for $C(K)$-spaces, $K$ -scattered
Recall that a biorthogonal system $\{(x_i, x^*_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x_i\}_{i\in I}$ is linearly dense in $E$ and $\{x_i^*\}_{i\in I}$ separates points. Let me ...
- 11.3k
0
votes
0
answers
213
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unbounded plurisubharmonic function
Reading Demailly's Algebraic and Complex geometry in particular chapter 3 about positive currents, tha author defines the unbounded locus $L(u)$ of a plurisubharmonic function $u$ to be the set of ...
- 75
1
vote
1
answer
289
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S-transformation of generalized Eisenstein series
I'm currently using generalized Eisenstein series to construct weight 2 modular forms under $\Gamma_1(N)$. They are defined as
$E_2^{\psi,\phi}(\tau) = \delta(\psi) L(-1,\phi) + 2\sum_{n=1}^{\infty} \...
- 63
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 ...
- 329
3
votes
2
answers
428
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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 ...
- 15.5k
4
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1
answer
626
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Reasonable crossnorm on Banach algebra tensor product constructed from isometric non-degenerate representations
The following is an abstraction of a more specific problem I've been grappling with, so I might give some extra unnecessary information. (The `bounded approximate
left identities' assumption is ...
4
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2
answers
447
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maximum of two plurisubharmonic
Let $u,v$ be two plurisubharmonic functions in a domain $\Omega\subset \mathbb{C}^n.$ Then
$w=\max$ {$u,v$} is plurisubharmonic. The support of $dd^c w$ is unclear in {$z: u(z)=v(z)$}
How to ...
- 750
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votes
3
answers
3k
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Reference request for translating from Top to C*-alg
Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
- 18.7k
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 ...
- 99
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 ...
- 113
2
votes
1
answer
262
views
Little Picard for (open) complex manifolds?
"Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The ...
- 3,806
17
votes
4
answers
2k
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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.
...
- 6,924
10
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5
answers
2k
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Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?
In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...
- 1,105
1
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0
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336
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Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) \left(\frac{b}{c}\right)^s\,\...
- 447
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4
answers
3k
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Representing a product of matrix exponentials as the exponential of a sum
In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
- 28.6k
4
votes
1
answer
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 ...
- 113
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4
answers
3k
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Riemann mapping theorem and smoothness on the boundary
Let $U\subset \mathbb C$ be open, bounded, simply connected, with $C^\infty$ boundary. Apply the Riemann mapping theorem to get a bilolomorphic isomorphism
$$
f:U\to \mathbb D
$$
between $U$ and the ...
- 43.2k
5
votes
1
answer
464
views
Viète's generalized infinite product yielding other converging values?
I took Viète's infinite product for $\frac{2}{\pi}$:
$\displaystyle \dfrac{2}{\pi} = \dfrac{\sqrt2}{2} . \dfrac{\sqrt{2+\sqrt2}}{2} . \dfrac{\sqrt{2+\sqrt{2+ \sqrt2}}}{2} \dots$
and made it generic:
...
- 4,169
2
votes
2
answers
650
views
Hurwitz's automorphisms theorem for infinite genus Riemann surfaces
Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...
- 1,169
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
4
votes
1
answer
1k
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A boundary-preserving map on the unit disk
We are given a (closed) ball $D^n$ and a (continuous) map $f: D^n \to D^n$, that is identity on the boundary of $D^n$.
Let $C$ be a subset of $D^n$, and let $f^{-1}(C)$ be the inverse image of $C$ ...
- 41
4
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2
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473
views
Galois cover via C star algebras
Hello to all, here's my question, I hope it's not too trivial. I haven't found reference for it so far.
We know that abelian C star algebras are the same as locally compact spaces.
Now what is the ...
- 399
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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-...
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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 ...
- 23
5
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1
answer
461
views
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 ...
- 26.8k
10
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1
answer
784
views
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 ...
- 976
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1
answer
1k
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Projection exists ⇒ Uniformly convex?
I know that: Let X be a uniformly convex Banach space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\...
- 43
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0
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457
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Quantum sheaves
Are the following definitions known?
Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:
(a) {0} and H lie in Sigma
(...
- 1,368
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 ...
- 1,368
1
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0
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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
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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 ...
- 93
1
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0
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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
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869
views
Complement of a subspace which is a cartesian product
Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ .
Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H =
U \oplus (V\times W)$ ?
See also Perturbations of an ...
- 2,753
2
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0
answers
320
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Hom of Nuclear spaces
Let V be a Nuclear space (not necessary Frechet, but complete). Let V^* be the dual space considered with the weak topology (i.e. pointwise convergence). Is it true that $V^*$ is also nuclear?
Is it ...
- 2,649
2
votes
1
answer
137
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Discriminant on boundary of semi-algebraic surface
Let
$P(t)$ be a polynomial in $t$ of degree $n$,
with some contiguous coefficients (not the first or last) being $x_1,\dots,x_k$
and the rest of the coefficients are fixed.
(E.g. $p(t)=1+2t+x_1t^2+...
- 15.8k
3
votes
0
answers
242
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Cauchy integral theorem and natural boundaries
Suppose one has function $f(z)$ analytic in the unit disk. Suppose closed loop $L$ lies in the disk except for one point $P$ on the boundary. Then the Cauchy integral theorem generally does not ...
- 17.6k
4
votes
4
answers
2k
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Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials
I am looking for an othogonal basis of functions over the interval $[0,1]$ with weight function $(1-x^2)^{\alpha-1/2}$. Gegenbauer polynomials are frustratingly close to what I need, but they are ...
- 203
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 ...
- 29.2k
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}...
- 39
0
votes
2
answers
1k
views
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
5
votes
0
answers
354
views
Weight-2 modular forms under $\Gamma(N)$
I'm looking for explicit bases of weight 2 modular forms under $\Gamma(N)$, for small N (<16 would be enough). (Ideally in terms of Theta- or Eta-Functions)
It seems to me that this should ...
- 63
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 $...
- 1,804
2
votes
1
answer
325
views
A Fact Of Quasiconformal Map
We just consider puntured unit disk $\triangle^{*}$ in $\mathbb{C}$. $f$ is a bounded quasiconformal map on $\triangle^{*}$. Why $f$ can extend to the origin,becoming quasiconformal map on the whole ...
