All Questions
Tagged with real-analysis reference-request
419 questions
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 ...
- 12.9k
2
votes
0
answers
325
views
Examples of RKHS that are "classical"
Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs?
It is easy to construct example of RKHSs by applying the ...
- 21
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
5
votes
1
answer
505
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 188k
2
votes
1
answer
127
views
Partition of unity of simplex
Let $$\chi_S(x,y)=\begin{cases}1&\text{ if }0< x<y< 1\\0&\text{ else }\end{cases}$$
be the indicator function of the simplex $S=\{(x,y)\in (0,1)^2:x<y\}$. I am interested in an ...
- 128k
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 $(m-1)$-...
- 28k
6
votes
3
answers
536
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 10.5k
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
1
vote
1
answer
117
views
Product/quotient of factorials beget dyadic powers
I am writing up some notes and the following occurred to me and I would like to see if there are a variety of ways to prove it. Just for reference, the identity pops out of equality between constant ...
- 43.2k
1
vote
0
answers
59
views
Factoring a smooth map as a function to a linear map
I am searching for a reference to the following fact about smooth functions.
If $f \in C^k(\mathbb{R}^n, \mathbb{R}^m)$ such that $f(0) = 0$, then there exists $g \in C^{k - 1}(\mathbb{R}^n, \...
- 2,834
2
votes
0
answers
159
views
Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
- 1,375
14
votes
1
answer
416
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 24.2k
2
votes
0
answers
56
views
Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
- 1,101
3
votes
0
answers
176
views
A variant of the Laplace principle
$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
- 12.9k
1
vote
0
answers
165
views
Question about stationary phase with Hessian close to $0$
Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define
$$
I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
- 869
6
votes
2
answers
503
views
Computing a limit on the unit sphere: Riemann Lebesgue?
Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align*}
\lim_{|\xi|\to \infty}
\int_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma_{d-1}(w)
= \int_{\mathbb{S}^{d-1}}u(w)d \sigma_{d-1}(...
- 271
2
votes
2
answers
382
views
Asymptotics of an integral requested
Given an integer $n\geq2$, consider the following integral
$$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
QUESTION. Is this true? It appears to be so.
$$\lim_{n\...
- 1,440
2
votes
1
answer
152
views
Proof of Szegö asymptotic theorem
Consider the truncated exponential series
$$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$
The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...
- 91.8k
1
vote
1
answer
184
views
Quantitative version of Lebesgue points theorem
Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
- 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)) ...
- 128k
2
votes
1
answer
307
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 12.5k
4
votes
1
answer
95
views
Limiting values of particular functions
Let's define the functions
$$A_n(q)=\sum_{k=0}^n(-1)^k\cdot\frac{(1+q)q^k}{1+q^{2k+1}}\cdot\frac{2k+1}{n+k+1}\binom{2n}{n-k}.$$
I'm interested in the following:
QUESTION. Let $n\geq1$ be integers. ...
- 1,440
2
votes
2
answers
316
views
Properties of the topology of sequential convergence $\tau_\text{seq}$
Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_\text{seq}$ has the ...
- 12.9k
7
votes
3
answers
986
views
Mixtures of log-convex functions are log-convex: a reference
A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
- 571
4
votes
2
answers
391
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 6,155
7
votes
1
answer
1k
views
Signed variant of the Flint Hills series
I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one:
$$
\sum_{n = 1}^\infty \...
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 61.9k
1
vote
1
answer
295
views
Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...
- 5,038
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 ...
- 4,713
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
6
votes
1
answer
135
views
Small shifts of weakly converging sequences in $L^1$
$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
- 128k
20
votes
3
answers
2k
views
Convergence of convex functions
I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...
- 28k
6
votes
1
answer
404
views
Baire class $1$ functions and Baire's characterization theorem
Kechris in his Classical Descriptive Set Theory book gives the following definition (Definition 24.1) and characterization (Theorem 24.15) of Baire class $1$ functions:
Definition. Let $X,Y$ be ...
- 4,459
2
votes
0
answers
47
views
A function $f_r$ where $f_r (x)$ is defined as the ratio between $f(x)$ and the average value of $f$ over $B(x, r)$
Let $E := \mathbb R^d$. Let $f:E \to \mathbb R_{>0}$ be continuous and integrable. For $r>0$, we define
$$
f_r (x) := \frac{f(x)}{ \frac{1}{|B(x, r)|} \int_{B(x, r)} f(y) \, \mathrm{d} y} \quad \...
- 657
4
votes
2
answers
191
views
Reference request: "Tangent relation" in metric spaces
Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
- 27.7k
10
votes
1
answer
755
views
The $9$th tetration of $-\sqrt2$
Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
- 128k
4
votes
1
answer
134
views
On partial absolute continuity
$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
- 3,808
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 \...
- 2,821
2
votes
0
answers
198
views
Sets and their characteristic functions
There are some nice connections between properties of sets and properties of their characteristic functions.
For instance:
a set $C\subset \mathbb{R}$ is closed (resp. open) IFF the characteristic ...
- 2,196
6
votes
3
answers
1k
views
Discontinuous functions without removable discontinuities
A function $f:\mathbb{R}\rightarrow \mathbb{R}$ has a removable discontinuity at a given real $x$ in case the left and right limits are equal but not to the function value, i.e. $f(x+)=f(x-)$ but $f(x)...
6
votes
0
answers
309
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
2
votes
1
answer
127
views
Hausdorff dimension and non-empty intersections with lines
Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a ...
- 5,038
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
- 128k
16
votes
1
answer
687
views
Fourier's proof of reality of all roots of Bessel function $J_0(x)$
In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real.
I want to ask if there is a modern version of this proof exist in literature?
If someone ...
- 82.7k
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
5
votes
2
answers
321
views
If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too
Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$.
Let $\tilde u = u$ a.e. Is it true ...
- 12.9k
4
votes
2
answers
446
views
About Euclidean distances
$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$.
Let $d$ denote the Euclidean distance in $\R^n$.
Do then ...
- 128k
1
vote
0
answers
103
views
Real analytic map with connected fibers
Let $X,Y$ be compact real analytic varieties. Suppose $Y$ is connected and there is a surjective analytic map $f:X\to Y$ such that each fiber of $f$ is connected. How to prove that $X$ is connected as ...
- 14.3k
0
votes
1
answer
129
views
Seeking an integral formulation for an algebraic function
While working with a generating function for the Catalan numbers, I came across the integral representation
$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...
- 7,226
1
vote
0
answers
155
views
Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
- 307