All Questions
Tagged with real-analysis reference-request
419 questions
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
11
votes
4
answers
4k
views
When is the infimum of an arbitrary family of measurable functions also measurable?
Let $(X,\Sigma,\mu)$ be a measure space and consider a family of $\mu$-measurable functions $f_i:X \to \mathbb{R}$ for $i$ lying in some index set $I$. Define $$f(x) = \inf_{i \in I} f_i(x)$$
I think ...
- 15.5k
4
votes
1
answer
471
views
Ask for theory about the weighted L^2(R^d) space.
Dear MOs,
I am now considering the following norm:
$$
||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.
$$
where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
- 1,649
0
votes
1
answer
939
views
Asymptotic equivalence for functions with zeros
I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.
...
- 2,480
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 ...
4
votes
2
answers
977
views
Articles with examples of Darboux functions without fixed points
A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \...
- 2,222
5
votes
1
answer
540
views
Cosets of groups of functions
Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$.
The set $\mathcal ...
- 4,312
6
votes
2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 1,649
12
votes
5
answers
2k
views
analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 19.7k
6
votes
2
answers
812
views
A dual theory to the theory of currents?
The k-currents are defined as dual space to the spaces of all smooth k-forms.
(These monsters are used to work with the minimal k-surfaces.)
Assume I want to look at the generalized k-forms;
they can ...
- 1,785
11
votes
1
answer
1k
views
Proof of the "Neo-classical Inequality", a fractional extension of the binomial theorem
I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$:
$$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\...
- 4,952
9
votes
1
answer
958
views
Quantitative bounds for multivariate central limit theorem
For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:
https://...
- 93
5
votes
2
answers
917
views
Is the inclusion of Lebesgue spaces compact?
[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-...
- 39k
4
votes
2
answers
3k
views
Power series with non-integer exponents
Motivation:
For the sake of concreteness, I'll state a very particular context, but my question is a little more general. I'm trying to find a function $\gamma\colon [0,\delta) \to [0,\delta')$ that ...
- 8,903
4
votes
1
answer
346
views
approximately linear functions -- more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
- 123
7
votes
1
answer
2k
views
approximately linear functions
i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies
$f(x-y)=f(x)-f(y)+const$
then it is necessarily linear.
are there any general ...
- 123
6
votes
1
answer
778
views
Inverse function theorem for DC-functions
I would like to have an inverse (or/and) implicite function theorem for DC-functions.
It seems that I have right definitions, but I fail to prove it...
Definitions:
Let $h:\mathbb R^n\to\mathbb R$ ...
- 45k
4
votes
2
answers
734
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
- 8,512
7
votes
2
answers
505
views
The set of non-smooth points of a convex function is (m - 1)-rectifiable
I am looking for a reference to the following result.
Let $f:\mathbb R^m\to\mathbb R$ be a convex function.
Then $f$ is differentiable at all points of outside of a countable union of $(m-1)$-...
- 45k