All Questions
12,823 questions
11
votes
3
answers
3k
views
A two-variable Fourier series and a strange integral
I have recently had occasion to investigate the Fourier series of the function $f(x,y)=\log({2+\cos 2\pi x} +\cos{2\pi y})$. Accordingly, define
$I(m,n)=\int_{0,0}^{1,1}f(x,y)\cos{2\pi mx}\cos{2\pi ...
- 13.1k
3
votes
3
answers
571
views
Truncated product of $\zeta(1)$?
This is my first question. It appeared while solving a research problem in cryptography. I am computer science student, so I apologize for lack of mathematical rigor in this question. Thanks for any ...
- 41
6
votes
7
answers
8k
views
Existence of an extreme point of a compact convex set
The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points.
It seems this implies that a compact convex ...
- 169
3
votes
1
answer
1k
views
characterization of continuous functionals in weak-star topology
Reading Wojtaszczyk's Banach spaces for analysts, I'm trying to understand his proof that the space of all continuous linear functionals on $(X^\star,\sigma(X^\star, X))$ is $X$.
To show the $ \...
- 922
5
votes
1
answer
807
views
Self-adjoint extension of locally defined differential operators
The following is well known. Given a symmetric differential operator, like $\partial_x^2$, defined on smooth functions of compact support on $\mathbb{R}$, $C_0^\infty(\mathbb{R})$, one can count the ...
- 21.5k
15
votes
3
answers
5k
views
Zariski open sets are dense in analytic topology
How does one show that if $U \subseteq \mathbb{C}^n$ is nonempty and Zariski open then $U$ is also dense in the analytic topology on $\mathbb{C}^n$?
- 685
5
votes
1
answer
403
views
Nonlinear Nuclear Operators ?
Is there a "right" definition of the nuclear
operator in the nonlinear framework ? Of course, such an operator
must be compact, while a linear operator should be "nonlinearly"
nuclear iff it is ...
- 4,060
6
votes
5
answers
2k
views
Asymptotic series for roots of polynomials
Let $f(z) = z + z^2 + z^3$. Then for large $n$, $f(z) = n$ has a real solution near $n^{1/3}$, which we call $r(n)$. This appears to have an asymptotic series in descending powers of $n^{1/3}$, ...
- 14k
11
votes
2
answers
2k
views
Hypoellipticity of square root of laplacian
It is a well known result (sometimes called the Weyl lemma) that the laplacian in $\mathbb{R}^n$ is hypoelliptic, i.e. if $f$ is a distribution s.t. $\triangle(f)$ is smooth in an open set, than $f$ ...
- 2,479
4
votes
2
answers
340
views
Embeddings of Weighted Banach Spaces
Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
- 2,044
1
vote
2
answers
605
views
How many ways can we characterize gamma function?
First let's state a well-known characterization of gamma function.
If f is a positive function on positive real numbers such that:
(1).f(x+1)=xf(x);
(2).f(1)=1;
(3).logf is convex,
then f(x) is gamma ...
3
votes
1
answer
363
views
"exchange" of real analyticity and integration
Sorry for the impreciseness of the title. It is merely meant for an analogy.
Exchange of limiting operations and integrations are basically derived from Lebesgue's dominated convergence theorem. For ...
- 1,839
28
votes
2
answers
3k
views
Simulating Turing machines with {O,P}DEs.
Qiaochu Yuan in his answer to this question recalls a blog post (specifically, comment 16 therein) by Terry Tao:
For instance, one cannot hope to find an algorithm to determine the existence of ...
- 47.3k
3
votes
3
answers
584
views
Polynomials and L^p(R)
As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
- 11.8k
5
votes
3
answers
759
views
How to estimate the growth of a recurrence sequence
If we have a linear recurrence sequence where each term depends on all previous terms, say
$a_n = \sum_{k=0}^{n-1} \binom{n}{k} a_k, \quad a_0 = 1$
is there any way to estimate the growth of a_n in ...
- 1,839
4
votes
2
answers
1k
views
What's the space of smooth functions in L^2(R)?
Maybe this question is not appropriate here.
Let R be real numbers, and L^2(R) the square integrable functions, now what's the space of smooth functions in L^2(R)?
Edit:Sorry for the ambiguity. Let'...
- 2,709
3
votes
1
answer
473
views
Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?
It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable ...
- 371
0
votes
1
answer
288
views
The Quantum Operations On The Bipartite Systems
Given two distinct and noninteracting quantum mechanical
systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces
$\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space
of the ...
- 1
6
votes
2
answers
3k
views
Dense inclusions of Banach spaces and their duals
This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
- 8,512
24
votes
1
answer
2k
views
How many ways are there to globalize Harish Chandra modules?
Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...
- 4,909
7
votes
2
answers
1k
views
Relation between full elliptic integrals of the first and third kind
I am working on a calculation involving the Ronkin function of a hyperplane in 3-space.
I get a horrible matrix with full elliptic integrals as entries. A priori I know that the matrix is symmetrical ...
- 71
7
votes
1
answer
446
views
at which rational points does the Hypergeometric function take rational values
A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric ...
- 605
8
votes
3
answers
606
views
Compact Hausdorff and C^*-algebra "objects" in a category.
This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
- 26.8k
-1
votes
1
answer
1k
views
cauchy product for general case [closed]
How to multiply this series:
$$(\sum_{t=-\infty}^{\infty}a_{t})(\sum_{k=-\infty}^{\infty}b_{k})$$
- 39
0
votes
2
answers
259
views
Existence of an "anti-additive" (or "never linear") map?
(I've edited this question)
I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent $...
- 365
3
votes
3
answers
2k
views
"Interesting" properties of sets of natural numbers
On Wikipedia, there is a list of properties of sets of reals, which are in some sense "interesting": just have a look.
I could not find a comparable list of properties of sets of natural numbers (...
5
votes
3
answers
2k
views
When can a function be recovered from a distribution?
What properties does a distribution (in the generalized function sense) has to have in order to be a function. That is, when is $T(\varphi) = \int f \varphi$ for some $f$?
- 259
19
votes
5
answers
18k
views
Visualization of Riemann–Stieltjes Integrals
The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...
- 5,935
13
votes
0
answers
816
views
How hard is it to make a differential operator Hermitian?
Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...
- 54.6k
4
votes
1
answer
822
views
What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?
I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$?
(For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...
- 41
0
votes
1
answer
198
views
An integral arising in statistics(2)
The integral I am interested in is:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
$$\pi/2-\...
- 267
7
votes
2
answers
684
views
Yet more on distortion
I would like to elaborate a little bit on my previous question which can be found
here.
Firstly, let me recall that a separable Banach space $(X, \| \cdot \|)$ is said to be
arbitrarily distortable ...
- 576
1
vote
1
answer
2k
views
spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]
Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(...
- 11
8
votes
3
answers
2k
views
Definition of a von Neumann algebra
Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in ...
- 37.8k
7
votes
4
answers
946
views
On operator ranges in Hilbert & Banach spaces
Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ...
- 8,512
2
votes
3
answers
713
views
Is there a "Bezout's theorem" for analytic curves?
Let $\varphi_1(u,v)$ and $\varphi_2(u,v)$ be two entire or meromorphic functions in the two complex variables $u$ and $v$. If they are both polynomials, then Bezout's Theorem says that the set of ...
- 371
11
votes
1
answer
813
views
Approximation to divergent integral
Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...
- 111
12
votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 3,323
5
votes
2
answers
862
views
Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
- 93
101
votes
1
answer
8k
views
Dropping three bodies
Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...
- 8,336
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
5
votes
0
answers
537
views
Conditional probabilities in Banach spaces
This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
- 8,512
6
votes
3
answers
1k
views
Real-analytic manifolds in real-analytic sets
Let $U\subset \mathbb{R}^n$ be open, and let $f:U\to\mathbb{R}$ be real-analytic. We consider the zero set $Z:=f^{-1}(\{0\})$.
For a paper I am writing, I am looking for the best reference to the ...
- 6,548
1
vote
4
answers
411
views
Sum and interpolation of hurwitz zeta functions
$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$
Can I apply Euler-Maclauren formula to this sum?
where $a\in(0,0.5)$, p is a natural number, and $x$ is a ...
- 267
4
votes
3
answers
2k
views
Algebraic Dual / Continuous Dual
Let $E$ be an infinite dimensional Banach space, let $E^{\ast}$ denote
its continuous (i.e., Banach space) dual, and let $E'$ be its algebraic
dual. Clearly, $E^{\ast}$ is a proper vector subspace of $...
- 4,060
0
votes
1
answer
412
views
An integral arising in statistics
The integral I need:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc ...
- 267
6
votes
3
answers
423
views
Infinite electrical networks and possible connections with LERW
I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
Given a lattice L, we turn it into a ...
- 85.6k
11
votes
3
answers
1k
views
Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
1
vote
1
answer
1k
views
How can I calculate the characteristic function of these distributions? [previously: difficult integral]
How to compute this integral in general case?
$$t(x)=\int_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$
Mathematica can compute it when q is known. For example,for q=1 this integral is
$$\exp(-{\...
- 267
32
votes
11
answers
23k
views
A book for problems in Functional Analysis
I want to know if there's any book that categorizes problems by subjects of Functional Analysis.
I'm studying Functional Analysis now a days and I really need to solve some problems in order to ...