All Questions
47 questions
27
votes
2
answers
1k
views
Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
- 28k
21
votes
1
answer
1k
views
Does summing divergent series using cutoff functions give consistent results?
One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...
- 313
21
votes
1
answer
564
views
Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 19.6k
13
votes
5
answers
3k
views
Reference request: Oldest calculus, real analysis books with exercises?
Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there.
Edit. Unsolved exercises ...
13
votes
2
answers
2k
views
An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
- 1,727
12
votes
1
answer
1k
views
A generalization of intermediate value theorem on R^k
Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...
- 207
12
votes
1
answer
1k
views
Proof of Green's formula for rectifiable Jordan curves
$\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
- 128k
12
votes
1
answer
239
views
Interval arithmetic with different definitions of intervals
Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}...
- 497
11
votes
2
answers
1k
views
Two divergent series conspiring?
Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus
$$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad
\sum_{n\geq0}\frac{a_n}{2n+...
- 43.2k
11
votes
2
answers
802
views
Functions that Calculate their $L_p$ Norm
are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\...
- 13.2k
10
votes
1
answer
571
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 299
9
votes
2
answers
1k
views
A tricky integral to evaluate
I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
- 43.2k
8
votes
2
answers
2k
views
Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
7
votes
1
answer
352
views
Tight upper bounds on trigonometric polynomials
According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
- 10.4k
6
votes
2
answers
513
views
Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
- 335
5
votes
2
answers
1k
views
real analyticity, Fourier coefficients [duplicate]
Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
- 43.2k
5
votes
2
answers
840
views
Decompostition of a Lipschitz domain
We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \...
- 291
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 ...
- 1,649
5
votes
2
answers
917
views
Is the inclusion of Lebesgue spaces compact?
[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-...
- 39k
5
votes
1
answer
136
views
Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'
Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
- 1,771
5
votes
0
answers
163
views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
- 20.2k
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
- 1,776
4
votes
2
answers
734
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
- 8,512
4
votes
0
answers
84
views
Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms
Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
- 128k
4
votes
0
answers
672
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
3
votes
1
answer
334
views
Does this function have any exponential growth?
Has anyone seen any function of the following type?
$$
g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant $c>...
- 1,649
3
votes
0
answers
55
views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
- 1,385
3
votes
0
answers
373
views
An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only
The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
- 128k
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
2
votes
1
answer
531
views
Radius of the ball where the inverse of Lipschitz maps exists
I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in On the inverse function ...
- 203
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that
$$
\left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
- 217
1
vote
2
answers
859
views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
- 128k
1
vote
1
answer
123
views
Where is the maximum of the product of two logistic curves?
I've got an asymmetric peak-like function $y(x) = y_1(x)y_2(x)$,
where $y_1(x) = 1 / (1 + f_1(x)) = 1 / ( 1 + e^{( -r_1(x-x_1))})$ is an increasing logistic function
and $y_2(x) = 1 / (1 + f_2(x)) ...
- 11
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
1
answer
262
views
Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)
Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
user89890
1
vote
1
answer
164
views
Two trigonometric integrals: looking for a transformation
I have two integrals of trigonometric functions and I would like to ask:
QUESTION. Is there a transformation rule (or general principle) to show this equality?
$$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
- 43.2k
1
vote
1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
0
answers
151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
1
vote
0
answers
218
views
Asymptotic inverses and de Bruijn conjugates (etc.) for complex-valued functions
I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT"), and it's been an absolute revelation for my research. The thing is, my current work centers ...
- 1,284
1
vote
0
answers
120
views
Interpolation functional for BV spaces?
Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
- 121
1
vote
0
answers
42
views
Error bounds for approximation with dyadic sums of polynomials
Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?
In the 2-dimensional case the ...
- 13.2k
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
0
votes
1
answer
191
views
Calculating derivatives of arbitrary-order at an operator's root
Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of ...
- 583
0
votes
1
answer
86
views
Kolmogorov entropy of a subset of $L^1$
How can we estimate the Kolmogorov $\epsilon$-entropy
$$H_\epsilon (A,L^1(\mathbb R))$$
where
$
A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\}
$?
- 839
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
0
votes
0
answers
99
views
Does $\sum_{m=0}^{\infty} \left|c_m g(x)^{2m+1}\right|$ converge absolutely to an integrable function?
Consider the integral
\begin{equation}
\int_{0}^{t}J_1(f(t)-f(s))\mathop{ds}=\int_{0}^{t}\sum_{m=0}^{\infty} c_m (f(t)-f(s))^{2m+1}\mathop{ds},
\end{equation}
such that $J_1$ is the Bessel function of ...
- 79
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665