All Questions
12,823 questions
3
votes
7
answers
692
views
Splines, harmonic analysis, singular integrals.
Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)
One of the references I'm ...
- 661
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
7
votes
1
answer
577
views
Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?
Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}_1$ and let $\mathbb{F}_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}_n$ on $\mathcal{R}$.
Does the von ...
- 1,222
11
votes
4
answers
10k
views
Approximation with continuous functions
Is it true that for every function $\mathbb{R} \to \mathbb{R}$ there exists a sequence of continuous functions $f_n(x): \mathbb{R} \to \mathbb{R}$ such that for any $x \in \mathbb{R}$ $f_n(x)$ ...
- 2,821
0
votes
1
answer
607
views
sum of fractional parts
Any hints how to compute this sum
$$\sum_{i=1}^{N-1}\left[i\frac{K}{N}\right]^{p}?$$
where K < N , $\left[\cdot\right]$ denotes fractional part,
$p\in N$
- 267
18
votes
1
answer
5k
views
Unbounded linear operator defined on $l^2$
Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(...
- 2,821
1
vote
0
answers
3k
views
Notation for space of Lipschitz continuous functions
The Lipschitz norm of a function over a domain $D \subseteq \mathbb R^n$ is easy to define: $$\|f\|_{\mathrm{Lip}} = \sup_{x,y \in D} \frac{|f(x) - f(y)|}{|x-y|}.$$ Is there a standard notation for ...
- 8,512
1
vote
1
answer
250
views
Has coarse continuity been known already?
In need for something equivalent to the continuity-definition of real functions I use the following definition of "coarse-continuity" for sequences. Has it been known already? Has it even got a name? ...
- 13
1
vote
1
answer
782
views
Inverse of (1+x) ln(1+x) - x
Define
g(x) = (1+x) ln(1+x) - x.
One can check that g is strictly monotonically increasing for x>=0 by checking its derivative is ln(1+x). So g is invertible and its inverse is also strictly ...
- 13
6
votes
4
answers
363
views
A sequential optimizing task
Find distinct positive real numbers $x_1$ , $x_2$ , ... of least supremum such that, for each positive integer $n$, any two of 0, $x_1$ , $x_2$ ,..., $x_n$ differ by $1/n$ or more.
Note that the ...
- 2,437
1
vote
2
answers
395
views
Are there always nontrivial real solutions to $A_1 x^5 + B_1 y^5 + C_1 z^5 = 0$ and $A_2 x + B_2 y + C_2 z = 0$?
Firstly, are there always nontrivial real solutions to the sysytem
of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$,
for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, ...
- 65
8
votes
9
answers
5k
views
Ways to prove an inequality
It seems that there are three basic ways to prove an inequality eg $x>0$.
Show that x is a sum of squares.
Use an entropy argument. (Entropy always increases)
Convexity.
Are there other means?
...
user2529
3
votes
1
answer
1k
views
Adjoint/transpose of wavelet transform
I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...
- 170
9
votes
2
answers
877
views
Sum f(p) over all primes convergent with sum f(n) over all natural numbers divergent?
The sum $\sum_{n=1}^{\infty} 1/n^{s}$ is convergent for all real $s>1$ and diverges for all real $s \le 1$. The same holds for the sum $\sum_{p \ prime} 1/p^{s}$. Thus, for the functions $f(n)= 1/n^...
- 4,014
10
votes
2
answers
960
views
Stone-Weierstrass for cones
A version of the Stone-Weierstrass Theorem asserts: If A is a linear subspace of C(K), the set of continuous functions on a compact space, and if A is a subalgebra that contains the constant functions ...
- 101
4
votes
1
answer
228
views
When can closedness of the range of an operator be checked on a positive cone?
Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements ...
user2412
2
votes
2
answers
874
views
Dimension of the space of harmonic functions on the unit ball
Is the dimension of the space of $H^2(B)$ harmonic functions on unit ball $B\subset\mathbb{R}^d$ countably or uncountably infinite?
- 21
5
votes
2
answers
3k
views
Uniform convergence of difference quotient
Let $\phi\in C^\infty_c(\mathbb R)$ be a smooth function with compact support.
For $h>0$ define the difference quotient $\phi_h\in C^\infty_c(\mathbb R)$ by $\phi_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$...
- 3,184
4
votes
3
answers
3k
views
Distributional derivative of non continuously differentiable functions
Hello,
let $f$ be a continuously differentiable function on $R^n$. Then its classical derivative and its distributional derivative coincide.
It is known (cf. Rudin, Functional Analysis, Sect. 6.13) ...
- 5,327
6
votes
1
answer
581
views
A puzzling question on real interpolation
Suppose an operator $T$ is bounded on $L^2$ and also bounded from $L^{1}$ to $L^{1}$-weak. Then by Marcinkewicz interpolation one gets that $T$ is bounded on every $L^{p}$ for p between 1 and 2. ...
- 9,007
16
votes
2
answers
1k
views
Evaluation of a combinatorial sum (that comes from random matrices)
I'm looking for an elementary combinatorial/generating function/etc proof of the following result:
For nonnegative integers $r$,
$$\frac{1}{r!} = \sum_{p_0+p_1+\cdots = r} \frac{1}{(p_0!)^2(p_1!)^2\...
- 2,151
3
votes
2
answers
1k
views
Spectral decomposition for an arbitrary linear combination of position and momentum operators
Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i $\frac{...
- 195
3
votes
1
answer
2k
views
What is the pure intuition for topological continuity and topology? [closed]
I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity.
The ...
- 191
3
votes
6
answers
8k
views
Functional Analysis and its relation to mechanics
Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there ...
- 33
2
votes
1
answer
931
views
Root Finding for Raytracing (Ray and Meta-Ball Intersection)
The motivation behind this is to find the points of intersection between a ray and a level set of a potential function $g$, built in terms of a basic potential function $f$ (the building is explained ...
- 3,165
10
votes
2
answers
3k
views
roots of sum of two polynomials
I believe that there is no common theory for finding roots of polynomial sum. In my case I have
$$P_{n}(x)+AQ_{n}(x)$$.
I am wondering how roots of this sum depend on $A$?
- 267
4
votes
4
answers
385
views
Is anything known about $w^*(x)=\sup_y w(x+y)/w(y)$ for measurable functions w on $R^n$
In my recent studies (fourier multipliers on weighted Lp spaces) I have to deal with this kind of transformation: if w is a measurable function on $R^n$, define
$w^*(x)=\sup_y \frac{w(x+y)}{w(y)}$.
...
- 783
3
votes
3
answers
2k
views
motivation for compactness [duplicate]
Possible Duplicate:
How to understand the concept of compact space
Hello,
I am learning some analysis on my own and
what is the motivation to consider compactness?
eg. I do not understand why ...
- 317
15
votes
1
answer
13k
views
Fourier transforms of compactly supported functions
One manifestation of the uncertainty principle is the fact that a compactly supported function $f$ cannot have a Fourier transform which vanishes on an open set. As stated, this phenomenon applies ...
- 2,243
1
vote
4
answers
620
views
Do there exist nonconstant functions such that...
Do there exist nonconstant real valued functions $f$ and $g$ such that the expression:
$$f(x) -v/g(x)$$
is maximized at $x = v$ for all positive real $v$?
- 13
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
23
votes
3
answers
6k
views
Density of smooth functions under "Hölder metric"
This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
- 505
14
votes
9
answers
9k
views
Greatest power of two dividing an integer
Does anyone know of a closed form for the function on $\mathbb{N}$ which returns the greatest power of two which divides a given integer?
To be more precise, any positive integer $n\in\mathbb{N}$ can ...
- 1,035
41
votes
2
answers
4k
views
Must the set of lines through the origin on which a nonconstant entire function is bounded be finite?
If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \...
- 4,014
8
votes
4
answers
4k
views
Approximation by exponential polynomials
Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of order $n$.
Define $E_n$ to be the collection of all exponential ...
- 1,795
7
votes
1
answer
845
views
Treating Differential Operators as Numbers
In Penrose's book (The Road to Reality, chapter 21) he gives an example of Oliver Heaviside's observation that you can treat differential operators like numbers:
The differential equation $(1+D^2)y = ...
- 1,412
21
votes
1
answer
2k
views
Trigonometry related to Rogers–Ramanujan identities
For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric numbers:...
- 13.5k
1
vote
2
answers
644
views
Projection of a gradient and the gradient of a projection
Hi,
I am trying to figure out if there are any functions, and then for which, where one can say that the gradient of the projection is the same as the projection of the gradient.
In this case a ...
- 13
8
votes
3
answers
3k
views
Convergence of orthogonal polynomial expansions
"Everyone" knows that for a general $f\in L^2[0,1]$, the Fourier series of $f$ converges to $f$ in the $L^2$ norm but not necessarily in most other senses one might be interested in; but if $f$ is ...
- 11.4k
25
votes
4
answers
3k
views
Can a conditionally convergent series of vectors be rearranged to give any limit?
Warmup (you've probably seen this before)
Suppose $\sum_{n\ge 1} a_n$ is a conditionally convergent series of real numbers, then by rearranging the terms, you can make "the same series" converge to ...
16
votes
0
answers
910
views
Polynomials with presumably positive coefficients
After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...
- 13.5k
1
vote
0
answers
309
views
Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
26
votes
3
answers
11k
views
L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
- 833
-1
votes
1
answer
366
views
A classical analysis problem
Define $$x_{k+1}(t)=\frac{3x^4_k(t)+6(1-t)x_k^2(t)-(1-t)^2}{8x_k^3(t)},$$
with $x_0(t)=1$. It is not difficult to see $x_k(t)$ converges to $\sqrt{1-t}$, whose (Maclaurin expansion) has negative ...
- 223
2
votes
3
answers
813
views
Closed form of divergent infinite product?
Okay, we know that
$$ \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\cdot\pi^2}\Big) $$ .
Is there some known (trigonometric(?)) function that is equal to the following infinite ...
- 4,829
11
votes
0
answers
1k
views
Is the Fourier-Transform a bounded operator on Lorentz spaces L(2,q)?
It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^n)$ continuously into $L^\infty(\mathbb{R}^n)$ and $L^2(\mathbb{R}^n)$ continuously into $L^2(\mathbb{R}^n)$.
Then, by ...
- 111
3
votes
1
answer
280
views
An analogue of an old proposition
For the absolute value $|C|=(C^*C)^\frac{1}{2}$ and the
Hilbert-Schmidt norm
$\parallel C\parallel_{HS}=(trC^*C)^\frac{1}{2}$ of the operator $C$. The
following inequality is shown by Araki et al in ...
- 223
4
votes
0
answers
283
views
Easy to find roots
Is there a smooth function $f:\mathbb{R} \to \mathbb{R}_{\geq 0}$ such that:
1) $\lim_{x \to \infty} = \lim_{x \to -\infty} = 0$
2) $\forall x > 0$, $f'(x) < 0$
3) $\forall x < 0$, $f'(x) &...
- 3,165
6
votes
1
answer
2k
views
Approximation by analytic functions
Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...
- 3,343
5
votes
1
answer
429
views
A plausible positivity
After getting stuck with the
previous positivity
(it probably sounds too complex),
I would like to give a version of the problem which is of most interest to me.
Consider a sequence of real numbers
$...
- 13.5k