All Questions
5,628 questions
5
votes
1
answer
550
views
Weakest assumption for pointwise convergence of Fourier series
This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose $f(x)=\sum_{...
- 31.5k
0
votes
1
answer
341
views
Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
0
votes
1
answer
155
views
Ratio of eventually close sequences
Let $a_n$,$b_n$ with $b_n>0$ be two bounded sequences which are eventually close to, respectively, two other sequences $\bar a_n$,$\bar b_n$ with $\bar b_n>0$, that is, for every $\epsilon >0$...
- 12.6k
9
votes
1
answer
1k
views
Why is this generality in Vitali's Lemma useful?
In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?
Vitali's Lemma:
...
CommunityBot
- 1
3
votes
1
answer
975
views
Generalized Cesàro means of a bounded sequence
While studying the convergence of a certain iterative algorithm, I have come across the following generalization of the Cesàro mean: given a sequence $\{a_k\}$ and an integer $m\geq 0$, define
$c_k^{(...
CommunityBot
- 1
0
votes
1
answer
337
views
Integral inequality
Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
- 882
3
votes
2
answers
1k
views
Function with all but mixed second partial derivatives twice differentiable?
Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
- 555
3
votes
1
answer
258
views
Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
- 2,270
3
votes
1
answer
2k
views
A question about a formal power series manipulation
I want to find a function $f(x,y)$ which can satisfy the following equation,
$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum _{n=1} ^\infty \frac{f(x^n,y^n)}{...
- 3,541
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
2
votes
2
answers
2k
views
Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?
Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
CommunityBot
- 1
2
votes
1
answer
190
views
Completeness for spaces of eventually bounded nets
Let $A$ be a directed set, and $\ell^\infty_A$ the (complex vector) space of all
eventually bounded nets $A\to \mathbb{C}$. We can define the limit superior seminorm on $\ell^\infty_A$:
$$
\vert\vert{...
- 42.8k
23
votes
4
answers
2k
views
Which is the correct ring of functions for a topological space?
There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...
CommunityBot
- 1
0
votes
0
answers
382
views
Lambert W-function
I asked this question MSE, but didn't get any answers. Maybe here someone can help.
I need to solve
$$
\theta \rho^{\theta}+r \theta>v
$$
where $\theta \in \mathbb{R}^{+}, -1 < r,v<1, \ 0&...
CommunityBot
- 1
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 586
6
votes
3
answers
1k
views
functional subrings
I should recall the notion of maximal subring of a commutative unitary ring $R$.
Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
CommunityBot
- 1
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
17
votes
12
answers
5k
views
Looking for an interesting problem/riddle involving triple integrals.
Does anyone know some good problem in real analysis, the solution of which involves triple integrals, and which is suitable for second semester Analysis students?
Thanks!
- 5,759
1
vote
0
answers
115
views
A question about smoothness
$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold :
$\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all ...
- 1,441
4
votes
0
answers
462
views
System of Equations Upper Bound
I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:
For $i=1,2,\...
CommunityBot
- 1
6
votes
1
answer
634
views
Arbitrary small positive lower semi continuous functions
This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...
CommunityBot
- 1
0
votes
1
answer
116
views
Root and sign of a complicated bivariate function
Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let
$$
\Phi(p,i) := \frac{1}{2^p+1}
+ \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right),
$$
where $\lg x$ is ...
CommunityBot
- 1
5
votes
1
answer
400
views
Estimating the volume of a semialgebraic set from above
Suppose $S$ is a subset of $\mathbb{R}^n$ of finite volume defined by a system of finitely many polynomial inequalities with integer coefficients. Can anyone describe an algorithm that, given such a ...
CommunityBot
- 1
1
vote
1
answer
199
views
On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis
Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...
- 91.8k
-8
votes
2
answers
1k
views
why do we need algorithms, and why is non-convex optimization difficult? [closed]
A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...
- 28.6k
1
vote
1
answer
393
views
On methods for dealing with recursively defined sequences
Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...
- 376
5
votes
1
answer
225
views
Extending Jordan loops
I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.
Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\...
- 44.8k
-1
votes
1
answer
4k
views
Lipschitz condition on the first derivative of a function? [closed]
If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?
- 2,135
0
votes
1
answer
372
views
Does this sequence converge to zero?
Description
Let $\{e_n\}$, $e_n\in \mathbb{R}^p$ be a sequence of vectors, $\{U_n\}$, $U_n\in\mathbb{C}^{p\times p}$ be a sequence of unitary matrices (that is $U_i^*=U_i^{-1}$, $^*$denonts conjugate ...
CommunityBot
- 1
0
votes
1
answer
298
views
Asymptotic behavior of convex functions
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly
positive. If $x_n$ is a sequence of points such that $f(x_n)\rightarrow 0$, show that (or
give a counterexample)...
- 91.8k
3
votes
3
answers
595
views
Every positive polynomial with rational coefficients is above a completely Q-factorized nonnegative polynomial ?
This question was originally asked in stackoverflow (https://math.stackexchange.com/questions/103941/every-positive-polynomial-is-above-a-completely-q-factorized-positive-polynomial) but as it has ...
CommunityBot
- 1
3
votes
2
answers
175
views
Decay rate of nonlocal differential operator?
Hi, Moers.
Let $m(\xi) \in S^0$, that is,
$$
|D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n.
$$
It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$.
...
- 425
2
votes
1
answer
403
views
The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 148k
7
votes
4
answers
3k
views
completeness axiom for the real numbers
Do any treatises on real analysis take the following as the basic completeness axiom for the reals?
"Let $A$ and $B$ be set of real numbers such that
(a) every real number is either in $A$ or in $B$;
...
- 2,249
0
votes
1
answer
138
views
question about the closed form of a function
Hi everyone! I have a question about how to find the closed form of a function defined by
$$\phi(\theta)=\inf_{x\geq 2}f(x;\theta)\equiv\inf_{x\geq 2}\frac{(x+2)^2}{\frac{1}{\theta}\left(\frac{x-1}{2}...
- 13k
26
votes
3
answers
7k
views
Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 2,409
9
votes
5
answers
2k
views
Homeomorphism of the rationals
In working with the classification of stable vector bundles on $\mathbb{P}^2$, I've found that I need to answer a fairly basic question from analysis/point set topology. Here it is.
Suppose $f:\...
- 11.5k
2
votes
1
answer
413
views
Technique: Compactness => (Finite -> Reals)
Context
I'm studying a classical results of Erdos and Lovasz, on colorings of the real line.
The theorem to be proved is as follows:
Let $m, k$ be two positive integers satisfying:
$$e(m(m-1)+1)k\...
CommunityBot
- 1
0
votes
0
answers
176
views
search for a function satisfying some conditions
Hi everyone, I would like to find a function
$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R_+}$$
satisfying the following conditions:
$$1-\frac{z\Psi'(z)}{\Psi(z)}+8s\Psi''(z)...
- 5,471
2
votes
1
answer
289
views
Can a simple curve intersect every subspace of dim 2 and avoid the origin?
Is there, e.g. in $\mathbb R^4$ a simple curve that does not contain the origin and intersects every subspace of dimension 2?
Sorry if the question is too easy, but I just cannot figure it out.
In ...
- 45k
43
votes
2
answers
4k
views
Square root of a positive $C^\infty$ function.
Suppose $f$ is a $C^\infty$ function from the reals to the reals that is never negative. Does it have a $C^\infty$ square root? Clearly the only problem points are those at which $f$ vanishes.
- 15.3k
5
votes
2
answers
774
views
Can we calculate the inner product of a semicontinous function with the Dirac delta function?
Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
- 3,322
3
votes
1
answer
464
views
smooth families of analytic functions
My question is essentially whether taking partial derivatives of a smooth family of analytic functions yields again a smooth family of analytic functions.
The precise question is the following:
Let $...
- 16.2k
1
vote
1
answer
3k
views
In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
CommunityBot
- 1
1
vote
1
answer
279
views
Conjecture that two nested convex curves have a point with the same slope
I'm trying to prove a conjecture and need some help.
Consider a continuous, twice differentiable function $p(a)$ such that $p(0) = 0$ and $\forall a$, $p'(a) > 0$ and $p''(a) < 0$ and $p$ is ...
- 60.5k
0
votes
0
answers
193
views
Boundedness of Riemann-like sums on unbounded interval
Hi
I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that:
\begin{equation}
\sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+...
3
votes
2
answers
466
views
Question on a Basel-like sum
Hello all,
I have happened upon the following sum:
$ 1^2 + \Big(1 \times \frac{1}{3} + \frac{1}{3} \times 1 \Big)^2 + \Big(1 \times \frac{1}{5} + \frac{1}{3} \times \frac{1}{3} + \frac{1}{5} \times ...
- 79.9k
5
votes
0
answers
270
views
Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
CommunityBot
- 1
19
votes
3
answers
1k
views
functions from Q to itself with derivative zero
Let $f: {\bf Q} \rightarrow {\bf Q}$ be a "${\bf Q}$-differentiable" function whose "${\bf Q}$-derivative" is constantly zero; that is, for all $x \in {\bf Q}$ and all $\epsilon > 0$ in ${\bf Q}$, ...
CommunityBot
- 1
0
votes
1
answer
721
views
Pointwise limit at Lebesgue's point
Dear MOs,
I am sorry if this problem is too elementary for someone. I just want to get confirmation.
Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
- 1,786