All Questions
5,629 questions
2
votes
0
answers
341
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Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 305
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
- 63
0
votes
1
answer
150
views
Solutions to Schrödinger equation parameter dependence
This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) =...
- 305
15
votes
4
answers
3k
views
No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
3
votes
1
answer
93
views
Does $p$ integrability in n-1 dimensions give higher integrability in $n$ dimensions?
Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that
$$
\int\limits_t\left(\int\limits_x|...
- 391
2
votes
1
answer
112
views
Enforcing an inequality on series
Let $f:[0,1]\to\mathbb{R}_+$ be a convex, strictly increasing function such that $f(0)=0$ (typically, $f$ is very flat at $0$, i.e. increases very slowly). I would like to prove or disprove the ...
- 14.4k
28
votes
0
answers
1k
views
Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
- 6,486
1
vote
1
answer
823
views
What is the growth of sum of absolute values of Fourier coefficients
For a periodic BV function $f$ which has jump discontinuties, is there any theorem in Fourier analysis which gives like $$\sum_{k=0}^n\left|c_k\right|\sim C\log\left(n\right)$$ where $C$ is a constant ...
- 698
15
votes
2
answers
2k
views
Where does the Lebesgue differentiation theorem fail?
The Lebesgue differentiation theorem says that for certain metric spaces $X$ (see below), any Borel measure $\mu$ that is finite on bounded sets and any $f: X \rightarrow \mathbb{R}$ locally $\mu$-...
- 1,368
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
1
vote
0
answers
91
views
Modulus of continuity of the Dirichlet Laplacian problem
I remember the following statement is correct but I cannot find a reference for that, can anybody help me to give one?
Let $\Omega\subseteq\mathbb{R}^{n}$ be an open, bounded, smooth domain,
$\varphi\...
- 31
4
votes
2
answers
334
views
what is this sum of squares of algebraic functions?
This question is inspired by the MO query here, although it has no direct implications.
Define the family of polynomial functions
$$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$
and ...
- 43.2k
1
vote
3
answers
653
views
When does $f^{-1}=\frac{1}{f}$ with $f$ a function mapping $\mathbb{R}^{*}$ to $\mathbb{R}$?
In mathematics, an inverse function is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ ...
9
votes
2
answers
244
views
If normal with respect to prime base then normal for all bases
I tried to find it on internet but couldn't so m asking this here. I want to ask if a number is normal with respect to all prime number base then do we know that it is normal with respect to any base. ...
- 335
5
votes
1
answer
101
views
Does minimum of an analytic map restricted to analytic curves implies minimum?
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that its restriction to any arbitrary analytic curve $\gamma$ passing through the origin $0\in \mathbb{R}^n$ attains a local ...
- 53
4
votes
0
answers
298
views
Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
- 41
8
votes
2
answers
1k
views
does this sum have a limit?
Define the sequence given by the finite sum
$$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$
Questions.
(1) Is $0<a_n<1$?
(2) Does the limit $\lim_{n\rightarrow\infty}...
- 43.2k
13
votes
2
answers
1k
views
Is the exponential function the sole solution to these equations?
Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
user78249
4
votes
1
answer
357
views
Lipschitz function admits Whitney stratification
I've been reading Topological Aspects of Nonsmooth Optimization by Vladimir Shikhman.
There I have found the following observation that:
Lipschitz functions $f : \mathbb{R}^n \to \mathbb{R}$ admit
...
- 99
2
votes
1
answer
71
views
Distances between probability distributions by the variance of the test functions
Let $P$ and $Q$ be two probability distributions on $\mathbb{R}$. The goal is to obtain a notion of ``distance'' between $P$ and $Q$, e.g., total variation distance, K-L divergence.
Let $f\colon \...
- 1,127
2
votes
0
answers
571
views
Integrating a product of integrals involving Bessel functions
I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable.
I ...
- 161
6
votes
4
answers
780
views
roots of higher derivatives of exponential
Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$.
Question. Is it true that $D^nf(z)=0$ has only real roots ...
- 43.2k
4
votes
0
answers
673
views
Proofs of the second fundamental theorem of calculus
I am referring to the following version of the theorem, in the setting of the Lebesgue integral.
Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
- 18.7k
0
votes
1
answer
139
views
Moment problem with wrong solution
I will write a problem with an answer that apparently is wrong. My question would be what is wrong with this solution.
Define $$B(s)=\sum_{i=0}^s{{2\,s-i-1\choose s-1}\frac {i}{s}{5}^{i}{2}^{s-i}}$$ ...
- 333
3
votes
1
answer
411
views
Continuation of a smooth function, whose every derivative is strictly monotonic
Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
3
votes
1
answer
236
views
computational complexity: do we gain acceleration?
There is a technique we developed for series acceleration using the Wilf-Zeilberger method. Here is a simple Maple code in this regard, you need to download this too.
The idea is you start with a WZ-...
- 43.2k
8
votes
2
answers
5k
views
Proof of Karlin-Rubin's theorem
I asked this question on Math Exchange, but as I did not receive a successful answer, maybe you could help me.
Karlin-Rubin's theorem states conditions under which we can find a uniformly most ...
- 141
21
votes
3
answers
1k
views
What is the set of all "pseudo-rational" numbers (see details)?
Define a “pseudo-rational” number to be a real number $q$ that can be written as
$q=\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$
Where $P(x)$ and $Q(x)$ are fixed integer polynomials (independent of n). ...
- 311
3
votes
0
answers
235
views
Is this "differentiation map" uniquely determined by these properties?
Let $A$ be the set of all real-valued functions having their domain a subset of $\Bbb R$ which are at least differentiable on an open set, and for $f \in A$, let $U_f$ be the largest open set on which ...
user101261
2
votes
1
answer
341
views
Lebesgue measure of the set $\frac{1+x}{1+y}$ with $x,y$ in a fat Cantor
Let $I_\alpha\subset[0,1]$ be an $\alpha$-Cantor set of Lebesgue measure $\alpha$ and let $I=I_\alpha+\{1\}=\{1+x:x\in I_\alpha\}$.
Q1. What is the Lebesgue measure of the set $\{\frac{t}{s}:t,s\in I\...
- 1,583
3
votes
1
answer
167
views
Recovering residue using local real information
Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$.
Compute the residue of $f(z)$ at z = 0 using just the ...
- 31
15
votes
0
answers
511
views
Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
- 6,541
3
votes
1
answer
277
views
Nowhere dense set with high multiplicity
For a subset $S\subset[0,1]$ with $0<|S|<1$ ($|S|$ is the Lebesgue measure of $S$) we define the multiplicity function of order $n$ $m_{n,S}:[0,1] \rightarrow \{0,1,\ldots,n\}$ in the following ...
- 549
2
votes
0
answers
881
views
Why does a convex function have to have a convex domain? [closed]
Other than convenience in convex optimization, is there a reason that the definition of a convex function includes the requirement for the domain to be a convex set?
- 121
44
votes
7
answers
4k
views
The missing link: an inequality
I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:
Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then
$$F_n(x)=\...
- 43.2k
2
votes
1
answer
210
views
asymptotic estimate for log-tan sum
I am finding the following first order estimate.
Question. As $y\rightarrow\infty$,
$$\sum_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\,
\sim\,\,\frac{\pi}4\log^2y.$$
Is it true?
- 43.2k
9
votes
1
answer
2k
views
Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 263
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
1
vote
1
answer
211
views
Representation of Hilbert transform by a singular integral
Hilbert transform defines as follow:
$$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$
$$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$
Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...
- 371
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
2
votes
0
answers
139
views
Existence of solution of a variational inequality
Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \...
- 21
3
votes
0
answers
97
views
Dimension of a graph
Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?
If not, what extra condition do we ...
- 2,135
4
votes
1
answer
393
views
Locally doubling measures
Let us say that a measure $\mu$ on $\mathbb{R}^d$ is locally doubling if for each
$x\in\mathbb{R}^d$ there is a constant $C(x)$ such that for all $r>0$,
$\mu(B(x,2r)) \le C(x) \mu(B(x,r))$,
where $...
- 6,541
2
votes
1
answer
800
views
Interpolation in Sobolev spaces
Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...
- 843
1
vote
1
answer
186
views
Almost binomial sum limit
I got the following sum with which I want to prove one limit fact:
$$
f_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} (a^t)^{n-t}
$$
I want to prove that $f_n(a) \to 1$ while $n \to \infty$ for $a\...
- 342
4
votes
1
answer
264
views
Density of the max set of a non-differentiable function
For $f : [0;1] \to \mathbb{R}$, let $M_f := \{x \in [0;1] \mid f(x)$ is a local strict maximum of $f\}$. It is easy to see that for any $f$, $M_f$ is at most countable. It is also easy to see that ...
- 15.9k
30
votes
4
answers
3k
views
A counterexample for Sard's theorem in $C^1$ regularity
I can't seem to find an example of a function $f \colon \mathbb{R}^2\to \mathbb{R}$ which is $C^1$ and such that the set of its critical values is not of zero measure.
What examples are there?
$...
- 478
7
votes
2
answers
470
views
Continuous functions and infinity
Suppose $f(x)$ is continuous on $\mathbb{R}$, for $\forall \delta>0, \forall x\in\mathbb{R}, \lim_{n\rightarrow\infty}f(x+n\delta)=+\infty$. Is it correct that $\lim_{x\rightarrow+\infty}f(x)=+\...
- 183
3
votes
1
answer
331
views
Solving recurrent relation
I have the following recurrent relation and I want to find a close form of it if it exists at all.
$$
P_n = (1-p)^{n-1}P_{n-1} + \sum\limits_{k=2}^{n} \binom{n-1}{k-1} p^{\binom{k}{2}} (1-p)^{k(n-k)} ...
- 342
3
votes
2
answers
1k
views
Properties of matrix exponential without using Jordan normal forms
There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:
All eigenvalues of $A$ have negative ...
- 791