All Questions
12,780 questions
4
votes
1
answer
521
views
Basic sequences in $\ell_p$
Let $p\in [1,\infty)\setminus\{2\}$. Suppose $(e_n)$ is a basic sequence in $\ell_p$ (or $L_p$) equivalent to the basis of $\ell_p$ ($L_p$). Is there a subsequence $(e_{n_k})$ such that $[e_{n_k}]$ is ...
- 181
6
votes
0
answers
98
views
Do the translates of integrable function approximate its radial part?
For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (...
- 415
12
votes
1
answer
838
views
A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
- 1,143
5
votes
1
answer
349
views
information on an Euler product
The following Euler product came up in some sieving applications:
$f(z, s) = \prod_{\mbox{primes}} \left(1-\frac{z}{p^s}\right).$
What is known about this function? (Analytic continuation? ...
- 96.4k
3
votes
2
answers
340
views
Perturbing upper-semi Fredholm operators
Let $T\colon X\to X$ be an upper-semi Fredholm operator acting on a $B$-space $X$ (the range of $T$ is closed and kernel is finite-dimensional) with complemented range. Suppose $S\colon X\to X$ is ...
- 181
18
votes
3
answers
2k
views
Perron, Fourier
Perron´s formula is in some sense just Fourier inversion, but I have never seen proven it that way in a textbook. I take this must be because the conditions for the Fourier inversion formula to hold ...
- 20.2k
2
votes
1
answer
2k
views
If any perfect set is uncountable in a metric space which is not complete?
We know that every ferfect set $E$ in a complete metric space $X$ is uncountable. My question is if there exists a metric space which is not complete, but every ferfect set in it is uncountable. The ...
- 21
1
vote
2
answers
139
views
Given f(t) = \sum_k C_k exp(2 pi i w_k t ) + noise. Need to estimate C_k and w_k .
Simpliefied setup.
Assume I am given some function f(t).
I know that it is constructed as $f(t) = \sum_{k=1...M} C_k exp(2 \pi~ i~ w_k t ) + noise(t)$.
where $noise(t)$ is some random set of numbers ...
- 24.9k
1
vote
1
answer
287
views
General compactness criterion in functional spaces
What follows is a total boundness criterion in the space $L^1(X)$, where $X$ is arbitrary space with probabilistic continuous measure (Lebesgue space). Of course, all such spaces $X$ and hence $L^1(X)$...
- 109k
8
votes
1
answer
776
views
distinct zero points for polynomial
I met an interesting phenomenon. Suppose $f(z)=\frac{1}{p(z)}$ where p(z) is a polynomial in $\mathbb{C}[z] $. If there exists a $ k \in \mathbb{N} $ and $ k>1 $ such that after you take $k$-th ...
- 283
2
votes
1
answer
214
views
union of Stone-Cech remainders
Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
- 607
2
votes
1
answer
499
views
Hölder estimates for the Complex Monge-Ampere equation
If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{...
- 3,383
3
votes
0
answers
354
views
Hurwitz Spaces and Rauch Variational Formulas
I have read in some papers about Rauch-type variational formulas on Hurwitz spaces, and I would like to know what exactly is the theory behind them.
A Hurwitz Space $H_g^d$ is the space of coverings ...
3
votes
1
answer
393
views
A Sobolev-type inequality with weights
In the study of a particular PDE I found myself wanting to prove the following inequality:
$( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r |f''|^2]...
- 143
8
votes
0
answers
964
views
Etymology of the O-notation for algebras of holomorphic functions
The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
- 1,111
20
votes
1
answer
995
views
Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
- 16.2k
2
votes
2
answers
408
views
Elliptic function with constant real part on the unit square diagonals?
Consider the following even meromorphic doubly periodic function with poles at the gaussian integer lattice.
$H(z) = \prod_{n \in \mathbb{Z}} {1 \over{ 1 - {1 \over{\cosh\left(2\pi\left(z-n\right)\...
- 181
3
votes
0
answers
409
views
Continuous function sort
If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
- 529
3
votes
2
answers
771
views
Special values of a doubly periodic meromorphic function
Consider the following function: $G(z) = \prod_{n \in \mathbb{Z}} {1 \over{\tanh^2\left(\pi\left(z-n\right)\right)}}$.
By constuction, it has poles at $z=m+in$ with $m,n \in \mathbb{Z}^2$.
...
- 181
4
votes
1
answer
195
views
Can $-1/a_2$ belong to the range of a schlicht function $z+a_2z^2+\cdots$? Or is $-1/a_2$ necessarily an omitted value?
Is there an example of a schlicht function $f(z)=z+a_2z^2+a_3z^3+\cdots$, which is analytic and injective on the open unit disk $\mathbb{D}$, such that $-1/a_2$ belongs to the range $f(\mathbb{D})$? ...
- 115
1
vote
1
answer
735
views
complex convex functions
I call a function f defined and valued on a domain A in the plane convex if it maps convex areas to convex areas. Some obvious example of convex functions. If f is also a bijection, what can we say ...
- 55
13
votes
2
answers
776
views
Properties of orthogonality-preserving c.p. maps between $C^*$-algebras
Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map.
(Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then $\phi(a)\phi(...
- 1,786
6
votes
1
answer
522
views
Need there be infinitely many Gaussian primes along lines that contain at least one?
Greetings from EuroCG 2012, from which I post via iPod, so apologies for lack of problem motivation, background research and mathematical formatting.
Question:Suppose L is a horizontal or vertical ...
- 873
2
votes
1
answer
373
views
Is it true that $c_0(X)^* = \ell_1(X^*)$ ?
I'm trying to prove this that but I can't . Any help/reference ?
- 151
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
5
votes
2
answers
566
views
Help with a mellin-type integral
greetings . i've been trying to do this integral for many days now, with no clue on how to attack it . the integral is a mellin inverse of some kind, and appears in analytic number theory .
$$I(x)=\...
- 181
5
votes
1
answer
664
views
Are piecewise linear curves dense among Hölder curves?
Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and
$\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$
is finite.
There are at least two ...
- 4,304
1
vote
1
answer
686
views
analytic continuation of a Laplace transform from a countably infinite set of points?
Let $f(\lambda)=\int_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely ...
- 351
7
votes
2
answers
921
views
Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
- 101
2
votes
0
answers
564
views
Young inequality in weighted spaces
Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...
- 1,205
24
votes
3
answers
4k
views
Self-dual normed spaces which are not Hilbert spaces
Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-...
- 2,239
2
votes
0
answers
508
views
About a Christoffel-Darboux-type sum
I've been using the Christoffel-Darboux identity for the Hermite polynomials,
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$
for some ...
- 2,358
2
votes
2
answers
2k
views
A point in the weak closure but not in the weak sequential closure
I'm trying to find a proof of this counterexample by von Neumann:
Let $x_{mn}\in \ell^2$ be defined by
$$x_{mn}(m)=n \quad,\quad x_{mn}(n)=m \quad,\quad x_{mn}(k)=0 \hbox{ otherwise,} $$
and let $S=\...
- 131
4
votes
1
answer
899
views
Algebraic relationships between elliptic functions
If $f$ and $g$ are two elliptic functions with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of two variables with constant coefficients.
...
-2
votes
1
answer
3k
views
Multiplying two Fourier series gives one Fourier series, but what are the new coefficients? [closed]
If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as
$\sum_n A_n \sin(n \pi x)=\sum_m B_m \sin(m \pi x)\sum_p C_p \sin(p \pi x)$
is there any easier way to compute $A_n$ from $B_m,...
- 149
5
votes
2
answers
805
views
Bedford-Taylor theory
The Dirichlet problem for the Complex Monge-Ampere equation on a bounded pseudoconvex domain in $\mathbb{C}^n$ was studied in Bedford-Taylor's seminal paper wherein they defined $(dd^{c} u)^n$ for ...
- 3,383
8
votes
2
answers
1k
views
Approximation by polynomials
Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x_0, ..., x_m$ be different numbers from $[a,b]$.
Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x_i)=f^{...
- 277
4
votes
1
answer
266
views
Exotic uniform algebras
The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region ...
- 43
5
votes
3
answers
821
views
are the smooth vectors of a Frechet space dense?
Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is ...
- 342
7
votes
2
answers
484
views
Extension of weakly compact operators from $\ell_1$ into $c_0$
Is every weakly compact operator from $\ell_1$ into $c_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
0
votes
0
answers
191
views
Asymtotic Complexity Analysis using logarithms and binomial coefficients
On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} \frac{1}{n}\log_{2}\left(\dbinom{k_{1}}{p_{1}}\...
- 101
7
votes
0
answers
266
views
Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
- 879
15
votes
3
answers
1k
views
Extreme points of unit ball in tensor product of spaces
Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively.
Let $e(B_1), e(B_2)$ be corresponding extreme points sets.
Consider the unit ball $B$ in tensor product $...
- 580
36
votes
2
answers
3k
views
Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
- 22.8k
23
votes
8
answers
8k
views
Grothendieck on topological vector spaces
In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on topological vector spaces (TVS), apparently, he told Bernard Malgrange ...
7
votes
3
answers
4k
views
Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 1,399
5
votes
1
answer
3k
views
Inner product of linear bounded operators between Hilbert spaces
Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\...
- 5,327
0
votes
1
answer
221
views
Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
9
votes
2
answers
1k
views
polynomials with minimal $L_\infty$ norm on multiple disjoint intervals
It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
- 223
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