All Questions
6,014 questions
7
votes
1
answer
834
views
Representing $\Gamma(a-x)$ in terms of $\Gamma(kx)$ and $\Gamma(a)$ and elementary functions
I asked this question on MSE here.
I wonder if it is possible to represent $\Gamma(a-x)$ in terms of powers of $\Gamma(a)$, powers of $\Gamma(kx)$, and elementary functions. I am not looking for any ...
- 541
1
vote
1
answer
79
views
PDF of the difference of two Beta Prime distribution
I am struggling to find the PDF of the difference of two Beta Prime distribution.
Definition
A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...
- 393
0
votes
0
answers
60
views
The size of super level sets and the symmetry on a sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define
$$
S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.
$$
Suppose ...
- 434
4
votes
1
answer
165
views
Dual spaces of Banach-valued $L^{p}$-spaces
Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
- 1,429
2
votes
1
answer
139
views
Domain of the infinitesimal generator of a composition $C_0$-semigroup
In the paper [1] the following $C_0$-group is presented,
$$
T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E
$$
where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
1
vote
0
answers
54
views
Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
-1
votes
1
answer
204
views
Cauchy reduction formula with measure (a variation)
The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
- 141
3
votes
2
answers
435
views
Closed form for $ \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \dotsb + x_n^p} \, \mathrm{d}x_1 \dotsm \mathrm{d}x_n $
I asked this question on MSE, but received no answer.
Recently, reading this problem, I found out that
$$ \lim_{n\to \infty} \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \...
- 387
3
votes
1
answer
346
views
Prove that $\lim\limits_{n\to\infty}\left(\sum\limits_{r=0}^{n-1}\sqrt{1-\frac{r^2}{n^2}}-\frac{\pi}{4}n\right)=\frac{1}{2}$
I came across the above question in a mathematical problem. It is not difficult to see that
$$
\lim\limits_{n\to\infty}\left(\frac{1}{n}\sum\limits_{r=0}^{n-1}\sqrt{1-\frac{r^2}{n^2}}\right)=\int\...
- 41
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
0
votes
0
answers
22
views
Approximation of Lipschitz functions by convex combinaison of Lipschitz functions depending on projections
Let $K\subset \mathbb R^2$ be compact. For any $c>0$, denote by ${\rm Lip}_c(K)$ the collection of Lipschitz functions $f:K\to\mathbb R$ whose Lipschitz constant is less than or equal to $c$. Set $...
- 1,334
2
votes
0
answers
75
views
Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
- 969
6
votes
1
answer
528
views
A functional equation
I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
- 63
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
0
votes
0
answers
53
views
Vectors of complex exponentials span $\mathbf{C}^N$
Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
- 171
1
vote
1
answer
127
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
- 171
9
votes
8
answers
1k
views
$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$
Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:
$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right).$$
I am struggling to find ...
- 393
1
vote
1
answer
125
views
Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
- 19
1
vote
0
answers
95
views
Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
1
vote
1
answer
168
views
About the sigma algebra generated by the Hausdorff measure on $\mathbb R^n$
Let $\mathcal{H}^k$ be the $k-$dimensional Hausdorff measure on $\mathbb R^n$, with $k \in \{1, \ldots n\}$. By Carathéodory's theorem we know that there exists a sigma algebra $\mu(\mathcal{H}^k)$ of ...
- 119
1
vote
0
answers
96
views
Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
- 4,359
7
votes
2
answers
324
views
For this continuous non differentiable function $f$ How to determine $\sup\{a\}$ s.t $\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h^\alpha}=0$ for all $x$?
I asked this question on MSE here.
Define $g(x)= |x|$ for $|x|\in [-1,1]$ , $g(x+2)=g(x)$
$$f(x)= \sum_{n \ge 1} \frac{3^n g\left(4^n x\right) }{4^n}$$
This function is a famous example of a ...
- 541
2
votes
1
answer
229
views
Does the existence of derivatives in the average sense imply absolute continuity?
Let $f: \mathbb R \to \mathbb R$ be a measurable function. Suppose there exists some integrable function $g$, and a measurable set $E$ of full measure such that
$$\lim_{r \to 0_+} \sup_{x \in E} \left ...
- 6,215
2
votes
1
answer
197
views
Prékopa-Leindler style inequality?
Does anyone know a simple proof of the following Prékopa-Leindler style inequality:
If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
- 125
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,145
0
votes
1
answer
53
views
Rate of convergence of the minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
0
votes
1
answer
77
views
Decay rate of minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
0
votes
0
answers
57
views
Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
- 75
1
vote
0
answers
92
views
Modulus of Continuity, Heat Flow, and Derivative Estimates
Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by
\begin{align}
(P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right],
\end{align}
where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
- 801
0
votes
2
answers
397
views
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 25
0
votes
1
answer
102
views
On weighted Fourier transforms
Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that
$$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$
...
- 4,145
2
votes
0
answers
116
views
For Polish $X,Y$, $L^p(X,Y)$ is separable
Let $X$ and $Y$ be Polish spaces. Equip $X$ with a Borel probability measure $\mu_X$ and $Y$ with a metric $d_Y$. We can define the $L^p$ space as follows:
Definition. Define
$\begin{align}L^p(X,Y) = \...
- 305
0
votes
0
answers
151
views
Help me find the antiderivative of $W(W(x))$ where $W$ denotes the Lambert W Function
Let $W$ denote the Lambert W Function. I must know the antiderivative of $W^2 = W(W(x))$.
I'm already convinced this function is not elementary. This does nothing to settle up my curiosity, as I ...
- 409
4
votes
2
answers
276
views
A function that maps every perfect set to $\mathbb{R}$
It's known that some real functions map every nonempty open subset onto $\mathbb{R}$.
Is there any function from $\mathbb{R}$ to $\mathbb{R}$ that maps every nonempty perfect set onto $\mathbb{R}$?
- 637
0
votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
80
views
Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
- 434
19
votes
2
answers
949
views
Etymology of “real numbers"
I would like to know why the real numbers are called “the real numbers.” I would also like to know the meaning of “real” in the phrase “real number.”
Further questions and clarifications:
I’d like to ...
- 319
1
vote
0
answers
95
views
Function whose derivatives eventually vanish almost everywhere
As a takeaway of this post we have the following result.
P. Let $f:[0,1]\to\mathbb{R}$ be infinitely differentiable such that for all $x\in[0,1]$ the sequence $\{f^{(n)}(x)\}$ is eventually $0$. Then ...
- 637
0
votes
0
answers
33
views
Reference request: injectivity of CWT, density of dilations and translations in $L^p$
Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
- 807
0
votes
0
answers
70
views
Multiplication with dilations of nonzero measurable function is injective
Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true:
Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
- 807
2
votes
0
answers
138
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
- 969
0
votes
0
answers
136
views
Antiderivatives via Taylor series and the FT of Calculus
If $f$ is a real function on an interval $[a,b]$ such that
$f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
- 827
-1
votes
1
answer
113
views
Lipschitz function which is surjective on subset implies that the subset is dense
Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a Lipschitz-function. Suppose $A \subseteq \mathbb{R}^n$ is an $(n-1)$-connected subset such that $f(A) = \mathbb{R}^n$. I would like to show that $A\subseteq ...
- 261
1
vote
0
answers
128
views
Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
- 1,043
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
- 4,074
0
votes
0
answers
44
views
Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?
We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
- 835
1
vote
1
answer
57
views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 835