All Questions
12,823 questions
2
votes
1
answer
846
views
Integral involving exponential of fractional power
Can anything be said about the Fourier integral
$\int_{-\infty}^{\infty} \exp\left[ika - (\gamma + ik)^{2/3}\right]dk$
where $a > 0$ and $\gamma > 0$?
Can it be related to some special ...
- 185
37
votes
1
answer
1k
views
A question of Erdős on equidistribution
In his book Metric Number Theory, Glyn Harman mentions the following problem he attributes to Erdős:
Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that
$$\lim_{N\...
- 2,151
3
votes
2
answers
766
views
Borel vs measure for all Borel measures
Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
- 18.7k
5
votes
2
answers
641
views
Percolation Model and Complex Probabilities
Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$.
I would like to know, if can we ...
- 2,044
5
votes
4
answers
1k
views
Is any continuous linear operator from a dual Banach space to a separable Hilbert space the strong-operator limit of a net of adjoint operators of less or equal norm ?
Let $E$ be an arbitrary Banach space and let $T:E^{*}\rightarrow\ell^{2}$
be a linear continuous operator. Is it true that $T$ must be the
$so$-limit (i.e., limit w.r.t. the strong operator topology) ...
- 4,060
4
votes
5
answers
2k
views
Analytic Functions over Fields other than Real or Complex Numbers
Let K denote either the field of real numbers or the complex fields. An analytic function over $K^n$ is a function that can be represented locally by a convergent power series in n variables with ...
user2529
2
votes
1
answer
2k
views
Uniqueness of the logarithm function
In my Analysis class, we started to prove a theorem that said:
Let a > 1. So there is a unique increasing function $f:(0,\infty)\to\mathbb{R}$ so that:
$f(a) = 1$
$f(xy) = f(x) + f(y)\quad\forall ...
- 183
4
votes
0
answers
487
views
Convolutions and Toeplitz Operators
Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...
- 2,044
7
votes
0
answers
4k
views
Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]
Possible Duplicate:
What’s an example of a space that needs the Hahn-Banach Theorem?
It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, say). ...
- 25.6k
4
votes
1
answer
2k
views
Proof of Green's theorem in Apostol book [closed]
Hello,
I got a question about one point in the Green's theorem proof that appears in Apostol's Mathematical Analysis, first edition, Ed. Addison Wesley. More exactly, in the theorem 10-42, ...
- 41
2
votes
2
answers
354
views
A bound on linear functionals over cotype 2 spaces
This is a modification of the somewhat naive question that I asked below.
Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
- 2,151
2
votes
1
answer
373
views
What is the partial derivative in this expression?
The question is similar to this one Implicit derivative?
Let $x_1, x_2, x_3$ be three points in $\mathbb{R}^3$, $A=(a_{ij})$ is a $3\times 3$ matrix with $a_{ii}=0$ and $a_{ij}=\frac{1}{|x_i-x_j|}$ ...
- 1,858
2
votes
2
answers
505
views
Decreasing coefficients?
For any integer $n\ge 3$, let $P(x)=\sum\limits_{i(=2k)\ge 0}^{n}\binom{n}{2k}(1-x)^k$, $Q(x)=\sum\limits_{i(=2k+1)\ge 0}^{n}\binom{n}{2k+1}(1-x)^{k+1}$. Define $\frac{P(x)}{Q(x)}\equiv\sum\limits_{i=...
- 1,858
1
vote
2
answers
734
views
Polynomial series
Consider the following polynomial series:
$S(x) = \sum_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$
Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this ...
- 13
6
votes
2
answers
5k
views
Can I relate the L1 norm of a function to its Fourier expansion?
I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
- 185
5
votes
1
answer
562
views
Do upper-semicontinuous polyhedral multifunctions have Lipschitz continuous selections?
We are interested in the following question (definitions and references are given below):
Main Question: Given an upper-semicontinuous polyhedral multifunction $F:R^n \rightarrow R^m$, is there ...
- 115
16
votes
2
answers
4k
views
Usefulness of Frechet versus Gateaux differentiability or something in between.
If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux ...
- 943
1
vote
1
answer
311
views
References for weak ellipticity
There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as ...
- 3,383
9
votes
7
answers
25k
views
a^b = b^a when a is not equal to b.
I am attempting to find a closed form solution or a nice series for $x^{x+1}=(x+1)^{x}$. First of all I looked at $a^b=b^a$. Fixing a, this means finding out when $a^{1/a} = x^{1/x}$. $f(x)=x^{1/x}$ ...
- 466
28
votes
7
answers
13k
views
Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
- 18.7k
7
votes
2
answers
714
views
Infinite product experimental mathematics question.
A while ago I threw the following at a numerical evaluator (in the present case I'm using wolfram alpha)
$\prod_{v=2}^{\infty} \sqrt[v(v-1)]{v} \approx 3.5174872559023696493997936\ldots$
Recently, ...
- 975
11
votes
1
answer
2k
views
Algebraic properties of the algebra of continuous functions on a manifold.
Does the algebra of continuous
functions from a compact manifold to
$\mathbb{C}$ satisfy any specific
algebraic property?
I'm not sure what kind of algebraic property I expect, but I feel that ...
- 855
12
votes
3
answers
2k
views
To what extent is convexity a local property?
A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
- 1,846
4
votes
1
answer
675
views
Name for topology making group action continuous
Fix a set $X$ with right $G$-action. Give $X$ a topology $\tau$ and make $G$ a topological group. (These topologies need not make the action continuous).
We can define another topology $\tau'$ on $...
- 2,895
8
votes
3
answers
1k
views
Locally complete space is topologically equivalent to a complete space
Can someone please tell me where I can find a citeable reference for the following result:
Call the metric space $(X, d)$ "locally complete" if for every $x \in X$ there a neighbourhood of $x$ which ...
- 2,895
17
votes
3
answers
2k
views
Recursions which define polynomials
There are many examples (Somos sequences, special polynomials related to rational solutions of the Painleve equations) when a recurrence relation, which a priori produces a sequence of rational ...
- 13.5k
1
vote
3
answers
379
views
monotonicity from 4 term-recursion.
In determining the monotonicity of coefficients in a series expansion (which appeared in one of my study), I come across the following problem.
Let $p\ge 2$ be an integer, and $$6p^3(i+3)d_{i+3}=6p^...
- 1,858
6
votes
1
answer
2k
views
Entire solutions of polynomial ODE's
Consider a differential equation of the form
$$f^{(n)}(z)=P(z,f(z),f'(z),...,f^{(n-1)}(z))$$
where $P$ is a polynomial in $n+1$ variables, with the initial condition (for definitness)
$$f^{(k)}(0)=0,\...
- 520
4
votes
3
answers
3k
views
Examples of Banach spaces and their duals
There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued ...
- 8,512
11
votes
0
answers
657
views
For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?
The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
- 637
12
votes
1
answer
1k
views
How to best distribute points on two concentric circles?
An N-subset $\{x_1,\dots,x_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le N}|x_k-x_j|\...
1
vote
1
answer
713
views
Linear elliptic partial differential equation with analytic coefficients
Consider the second order linear elliptic differential equation
$$Lu=(\sum_{i=1}^d{\partial^2\over\partial\theta_i^2}+{\partial b\over\partial \theta_i}{\partial\over\partial\theta_i})u=exp(i\theta_1)...
- 11
6
votes
5
answers
1k
views
smooth Gelfand-duality
Assume $M$ is a compact smooth manifold (without boundary). What can we say about the spectrum of the $\mathbb{R}$-algebra $A=C^{\infty}(M)$? The elements of $M$ give rise to rational points of $A$, ...
- 63.1k
1
vote
3
answers
5k
views
rules for operator commutativity?
Hi, my apologies for a rather non-specific question. I wonder if there is a general set of conditions under which operators are commutative in functional analysis. Most that I've found is that "...
- 222
38
votes
5
answers
21k
views
Criteria to determine whether a real-coefficient polynomial has real root?
Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not?
I ...
- 667
65
votes
9
answers
12k
views
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
- 651
1
vote
0
answers
660
views
Fractional Fourier transform [closed]
Let $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ be the Fourier transform. Is there any reasonable definition of fractional Fourier transform (i.e. operator $A$ such that $A^{\alpha}=T$ for $\...
- 3,323
10
votes
2
answers
5k
views
Approximate a probability distribution by moment matching
Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
- 1,839
42
votes
6
answers
14k
views
A question regarding a claim of V. I. Arnold
In his Huygens and Barrow, Newton and Hooke, Arnold mentions a notorious teaser that, in his opinion, "modern" mathematicians are not capable of solving quickly. Then, he adds that the exception that ...
- 8,842
14
votes
6
answers
3k
views
What's a natural candidate for an analytic function that interpolates the tower function?
I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
- 4,466
12
votes
3
answers
646
views
Radii and centers in Banach spaces
Suppose I have a Banach space $V$ and a set $A \subseteq V$ such that for all $\epsilon > 0$ there exists $v$ such that $A \subseteq \overline{B}(v, r + \epsilon)$. Does there exist $c$ such that $...
- 1,331
5
votes
0
answers
558
views
continuous selection of a multivalued function?
The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
- 1,839
2
votes
0
answers
197
views
Generating cones having no surjections [in operator spaces]
Is this little toy known ?
Let $E$ be some Banach space, and let $K$ be the closed unit ball
of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$
be the natural ...
- 4,060
32
votes
19
answers
23k
views
Good books on theory of distributions
Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
1
vote
3
answers
2k
views
A question on weak derivative - Sobolev spaces
Let $\Omega$ be an open set in $R^n$, and $f \in L^1_{loc}(\Omega)$, such that for each multiindex $\alpha\in N^n$, $|\alpha| = l$ f has weak derivative $D^\alpha f$ in $L^p(\Omega)$, with $1\leq p\...
- 783
81
votes
4
answers
8k
views
Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?
Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...
- 7,329
1
vote
1
answer
359
views
Convergence of operators to the identity on Banach spaces
Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. That is, $U_r \subseteq U_{r'}$ if $r \le r'$ and $U_\infty = \...
- 8,512
3
votes
1
answer
2k
views
Approximating a multiple sum with an integral
Hi,
I want to approximate a multiple sum of the form
$$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$
where each $x_i$ is an integer between $0$ and $n$,
by an integral
$$\int_{x_1+x_2+\...
- 491
5
votes
1
answer
288
views
Is there a notion of "Morse index" for geodesics in a manifold with indefinite metric that is well-behaved under cutting and gluing?
More generally, I'm interested in the situation of Lagrangian mechanics. And actually my question is local, so you can work on $\mathbb R^n$ if you like. I will begin with some background on ...
- 54.6k
10
votes
3
answers
4k
views
Estimate for tail of power series of exponential function?
I would like to have an estimate for the infinite series
$$
\sum_{k=B}^\infty \frac{A^k}{k!},
$$
where $A$ is a large positive quantity and $B$ is just a little bit bigger than $A$, namely, $B = A + C ...
- 12.8k