All Questions
12,777 questions
1
vote
1
answer
463
views
Topology and convergence for conformal maps of disk
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, centre $0$. Write $\mathcal{S}$ for the holomorphic injective maps $\{ f : \mathbb{D} \to \mathbb{D} | f(z) = e^{-\lambda} z + O(z^2) \}$ i.e. ...
- 2,895
21
votes
8
answers
11k
views
Nice applications of the spectral theorem?
Most books and courses on linear algebra or functional analysis present at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many ...
- 4,874
18
votes
1
answer
2k
views
Borel Lemma for vector-valued functions
The classical Borel Lemma states that for an arbitrary sequence $(v_n)_{n \in \mathbb{N}_0}$ of complex numbers there is a smooth function $f\colon \mathbb{R} \longrightarrow \mathbb{C}$ with Taylor ...
- 8,098
3
votes
6
answers
1k
views
Reference for complex analysis jargon
I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
logarithmic capacity
transfinite diameter
Green's function of a compact ...
- 741
4
votes
1
answer
632
views
Polylogarithm inequality
Recall the polylogarithm $Li_s(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}.$
For $|z|<1,$ is $\Re \left( \frac{Li_1(z)}{Li_2(z)} - \frac{2}{3}\frac{Li_2(z)}{Li_3(z)} \right)>0?$
The numerics suggest ...
- 1,306
3
votes
1
answer
801
views
Restriction of a linear functional equation to surface of a sphere
Let $f_i : R \rightarrow R$ and $g_j: R \rightarrow R$ be unknown functions, for $i = 1, \cdots, N$ and $j = 1, \cdots, K$. Let $A$ be a $K \times N$ matrix whose columns are unit-length vectors ${\...
- 93
7
votes
2
answers
1k
views
Weighted Poincaré inequality
Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...
- 2,133
24
votes
3
answers
2k
views
The third axiom in the definition of (infinite-dimensional) vector bundles: why?
Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
- 243
0
votes
1
answer
384
views
spectral measure
how to calculate spectral measure for a given normal operator for example right shift operator?
- 1
0
votes
1
answer
12k
views
HOW TO Generate Equation of a Curve Given (x,y) pairs - algorithm? [closed]
Hi,
How can I generate the equation of a curve that matches all arbitrarily given (x,y) pairs? I would like a polynomial of nth degree, where n does not matter, as long as the curve passes thru all ...
- 11
0
votes
1
answer
255
views
Average compared with discrete average for some $\lbrace -1,1 \rbrace$ polynomials
Let $k>0$ be a positive integer. Set $n=4k.$ Let $R(t)$ be a polynomial of degree $n-1$
with coefficients in $\lbrace -1,1 \rbrace$.
Consider the discrete average
$$
D(n,R) = \frac{\sum_{j=0}^{n-...
- 1,641
1
vote
2
answers
1k
views
Riemann Integral of Banach space valued functions
Does anyone know of a good undergraduate or graduate text that gives a brief rundown of the Riemann integral on Banach space valued functions?
3
votes
0
answers
178
views
One-parameter groups acting on dual Banach spaces
Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...
- 18.7k
3
votes
1
answer
995
views
Plurisubharmonic exhaustion functions without critical points at infinity
A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.
For example, Stein manifolds are weakly pseudoconvex (in this ...
- 7,902
3
votes
1
answer
896
views
separability of a certain space of continuous functions, II
This is a follow-up of a question of mine with a similar title. I am interested in Morse homology (on Hilbert manifolds), more specifically with "generic" perturbations of the metric tensor (under the ...
- 2,935
0
votes
1
answer
2k
views
separability of a certain space of continuous functions
Let $O$ be an open subset of the separable Hilbert space $H.$ Let $E$ be a separable Banach space. Is it true that $C^0_b(O;E),$ the space of bounded continuous maps $O\rightarrow E$, endowed with ...
- 2,935
3
votes
0
answers
223
views
Extension of positive operators and Bauer-Namioka
When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
- 31
25
votes
1
answer
3k
views
Does there exist a measurable function which is not a.e. "strongly" measurable?
More specifically, letting $I=[0,1]$, do there exist $f,E$ with $E$ a (necessarily nonseparable) Banach space and $f$ a bounded Lebesgue measurable function $I\to E$ such that $f$ is not equal almost ...
- 3,584
5
votes
2
answers
909
views
Is there an infinite−dimensional Banach subspace in C^∞([0,1]) ?
More specifically, with $I=[0,1]$ let $E=(X,\mathcal T\ )=C^\infty(I)$, where $X$ is the underlying (say real) vector space and $\mathcal T\ $ is the (standard projective limit) topology of uniform ...
- 3,584
4
votes
2
answers
3k
views
Proper holomorphic map from unit disk to punctured unit disk
It is easy to see that there cannot be a proper holomorphic map from the punctured unit disk to the unit disk in the complex plane. What about the other direction; does there exist a proper ...
- 1,159
0
votes
0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
- 1,222
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
6
votes
1
answer
536
views
The identity $\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})$
As in the famous Euler product identity, the primes occur on
only one side of the following:
$\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})\ .$
My basic question: Does this ...
- 17.6k
13
votes
1
answer
3k
views
metric on the space of real analytic functions
Hello,
this question may be simple but I couldn't find a reference.
Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of ...
- 223
1
vote
2
answers
980
views
About Schauder Basis [closed]
Suppose M is a compact Lie Group, is there a Schauder basis for L^1(M)?
- 37
4
votes
2
answers
2k
views
The Frechet derivative and Lagrange multipliers on Banach spaces
I am interested in questions of the following form: minimize $H(f)$ given $G(f) = 0$ where $H$ and $G$ are operators of type $X \to R$ where $X = R \to R$. An example is:
Minimize $$H(f) = \int_{-1}^...
- 493
5
votes
3
answers
794
views
Is there a "Riemann mapping theorem" for a circle in C^2 ?
The Riemann mapping theorem says that if you have a simple closed curve in $\mathbb{C}$, then there is an essentially unique way to map a holomorphic disc to the interior. Is there any reasonable ...
- 8,723
8
votes
1
answer
2k
views
Is there a manifold structure on a space of conformal maps?
I would be very grateful for any information or pointers for the following:
1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
- 81
18
votes
1
answer
1k
views
Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
4
votes
3
answers
695
views
magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal
What can you say about a function defined on a square region of the complex plane, if the integral of the function along any horizontal, vertical or diagonal of the square is equal ? - an analytic ...
- 41
2
votes
2
answers
991
views
An extension of the Hardy-Littlewood-Polya inequality?
Let $x,y$ be vectors in $\mathbb{R}^n$ and let's use the notation $\hat x$ for the vector $x$ with its components sorted in increasing order.
The Hardy-Littlewood-Polya inequality states that
$$ x\...
- 6,541
3
votes
1
answer
806
views
On the Nyman-Beurling equivalent form for RH
Now, I am new to functional analysis. So please dont be harsh.
I was going through some papers by Balazard & Saias, Baez-Duarte, etc. that discussed and delved deep into details of approximating ...
- 731
11
votes
1
answer
8k
views
Double Orthogonal Complement
Let $V$ be a complex inner product space. If $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in W$....
- 1,199
1
vote
1
answer
353
views
Separability of the space of bounded continuous maps
Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric ...
- 2,935
4
votes
2
answers
452
views
Is every bounded representation of Z unitarisable when all sets are measurable?
For the purpose of this question, a group is amenable iff there exists a Følner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...
- 3,897
2
votes
0
answers
304
views
Question in complex analysis arising from large $N$ gauge theory
This is a question in complex analysis that comes up in the treatment of large
$N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...
- 5,546
18
votes
2
answers
1k
views
Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform
The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator
$$
\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.
$$
It satisfies $\...
- 43.2k
33
votes
2
answers
6k
views
Which almost complex manifolds admit a complex structure?
I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau'...
- 4,343
3
votes
0
answers
1k
views
weak regularity conditions for regions to assure boundary of measure zero
Let $\Omega \subset \mathbb{R}^d$ be a region ( bounded, simply connected, open set ). What are some regularity conditions to assure the boundary $\partial\Omega$ is a set of (lebesgue-)measure zero? ...
- 131
1
vote
2
answers
422
views
Finding representatives of PSL_2(Z) orbits
Given $\tau$ in the upper half plane, what is a good, systematic way to find a representative in the usual fundamental domain for the $PSL_2(Z)$-orbit of $\tau$? For example, let $\tau=\frac{2}{3} + \...
- 1,329
2
votes
0
answers
695
views
Pole data of meromorphic matrix function
Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable.
Recall that such a $T$ is said to have a right pole of order $r$ ...
- 1,214
3
votes
0
answers
216
views
Picard Fuchs and Lefschetz trace
In Clemen's book "A Scrapbook of Complex Curve Theory", he discusses in Chapter 2 how the infinite sum giving the period of the Legendre curve matches (mod p) the sum giving the number of points over ...
- 661
22
votes
3
answers
7k
views
Subspace of $L^2$ that lies in $L^\infty$
Let $E$ be a closed subspace of $L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true that $E$ is finite dimensional?
PS. This is actually a question from the real analysis qualifier. I came ...
19
votes
1
answer
5k
views
Intuition for the Hardy space $H^1$ on $R^n$
the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities.
In particular, a ...
- 5,327
8
votes
3
answers
841
views
Holomorphic function with a.e. vanishing radial boundary limits
Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...
- 742
5
votes
1
answer
577
views
Does generator of continuous time random walk map heat kernel from L^2 to L^2?
Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi_{xy}$, where, given $e=\{x,y\}\in E$, we write $\...
- 269
8
votes
1
answer
1k
views
Applications of the Theorem of Gelfand-Naimark
Hi,
I am interested in the correspondence of algebraic results about C(X) (the space of continuous functions $X\to {\mathbb C}$(complex numbers) or $X\to {\mathbb R}$(real numbers) and topological ...
- 891
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
2
votes
1
answer
4k
views
Relation between complex analysis and harmonic function theory [closed]
There are some theorems in harmonic function theory that resemble results in complex analysis, like:
Holomorphic functions and complex functions are analytic;
Cauchy's integral formula in complex ...
2
votes
0
answers
200
views
Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21