All Questions
6,015 questions
1
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limsup of sequence
Let $\mathbb{Z}_{\geq 0}[|t|]$ be the ring of power series with non-negative integer coefficients and consider the power series
$$P(t) = \sum_{i=0}^ \infty a_i t^i \in \mathbb{Z}_{\geq 0}[|t|]$$
$$P^2(...
- 81
1
vote
0
answers
50
views
Bound an integral with parameter
Let us define, for $x_0 > 0$ and $x_0 \ll 1$,
$$K(x) = \int_{-x_0}^{x - g(x)^2} \frac{f(y)}{x - y}dy, \quad \text{for } x \in [-x_0/4, x_0],$$
and
$$g(x) = \frac{(x_0 - x)|\log(x_0)|}{|\log(x_0 - x)...
- 452
5
votes
1
answer
258
views
On the continuity of a Set-Valued function (correspondence) [closed]
Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued function defined by
\begin{equation*}
f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left(
x\right) ^{T}y\...
- 61
2
votes
1
answer
164
views
Higher-order convexity
Let $f \in C^\infty(\mathbb R)$.
$f^{(1)}=f'\geq 0$ iff $\forall (a,b) \in\mathbb R^2,a\leq b$ then $f(a) \leq f(b)$
$f^{(2)}=f''\geq 0$ iff $\forall (a,b)\in\mathbb R^2,\forall t\in [0,1], f(ta+(1-t)...
- 4,074
17
votes
1
answer
580
views
Aperiodic monotile in $\mathbb{R}$
Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
- 48.9k
4
votes
2
answers
415
views
The set of all possible values of subseries of a convergent positive term series
Inspired by The set of all limits of sub-series of an absolute convergent series is the following true?:
Let $a_n$ be a strictly decreasing sequence and $\sum_1^\infty a_n=\ell<\infty$ is a ...
- 356
2
votes
1
answer
77
views
Total sets for $L^p$ for every $1\leq p < \infty$
Consider $L^p[ 0,1]$ for $1\leq p < \infty$ or, if you prefer, $L^p(\mu)$ where $\mu$ is a finite Borel measure with compact support. Let $(\phi)_{i\in I}$ be a subset of measurable functions that ...
- 355
2
votes
1
answer
141
views
Injectivity of two sided Laplace transform
Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
- 769
7
votes
0
answers
270
views
Between real analysis and mathematical logic
This question lies in the intersection of real analysis and logic, so I try to keep things rather basic.
First of all, logicians care about the following kind of formula:
Let $\varphi(n, x)$ be a ...
- 4,359
5
votes
0
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417
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
3
votes
1
answer
353
views
An integral on the interval depending on the integrand
Let $C_p\equiv C_p(\mathbb R_+,\mathbb R_+)$ be the space of right-continuous piecewise constant functions $f: \mathbb R_+\to \mathbb R_+$, i.e. $f\in C_p$ iff
$$f(t)=\sum_{k=1}^n {\mathbf 1}_{[t_{k-1}...
- 1,399
2
votes
1
answer
232
views
Existence of diffeomorphism interpolating affine map and identity
$\newcommand{\R}{\mathbb{R}}$Suppose $\Omega$ is a bounded, convex domain in $\R^{m}$. Fix $x_1, x_2\in\Omega$ and an invertible matrix $A\in\mathrm{GL}^{+}(m)$ with positive determinant.
Let $U\...
- 193
1
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1
answer
154
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Dense properties of weighted Sobolev space define on $\mathbb{R}^n$
Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....
- 561
5
votes
3
answers
630
views
If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?
Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
$$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
Do we have $\|f\...
- 517
3
votes
1
answer
242
views
Closed subset of unit ball with peculiar connected components
Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$...
- 722
1
vote
1
answer
184
views
Quantitative version of Lebesgue points theorem
Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
- 697
1
vote
2
answers
213
views
How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that
\begin{equation}\tag{1}\label{1}
\int_a^b \...
1
vote
1
answer
137
views
Integral inequality implies majorization by solution of ODE
Let $f:[0, \infty)\to [0, \infty)$ be non-increasing (and not necessarily differentiable nor continuous) and satisfy $$f(t)\leq f(0)-C\int_{0}^{t}f(s)^{1/2}ds,$$ where $C>0$. How can one show that ...
- 459
2
votes
0
answers
110
views
Real analytic periodic function whose critical points are fully denegerated
I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
- 21
5
votes
1
answer
328
views
Implicit function theorem with singularities of any order
Let $\mathcal{U}\subset \mathbb{R}\times \mathbb{C}$ a neighborhood of $(0,0)$, and $f:\mathcal{U}\to \mathbb{C}$ differentiable in the first variable and holomorphic in the second variable, with $f(0,...
- 211
1
vote
0
answers
135
views
Continuity of derivative
Consider a homeomorphism of the real line $F : \mathbb{R} \to \mathbb{R}$ such that it is differentiable everywhere and the derivative is bounded. Does it follow that the derivative is continuous, ...
0
votes
1
answer
225
views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
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
3
votes
0
answers
94
views
Question on an integral inequality
I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...
- 53
1
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0
answers
115
views
Higher dimensional Cauchy interlacing theorem
If $A$ is a Hermitian matrix and $A_j$ the principal minor with the $j$ row and column deleted and $\phi_A(x)$ the characteristic polynomial. The Cauchy interlacing iheorem states that the roots of $\...
- 1,362
3
votes
2
answers
248
views
Exceptional set for Marstrand's projection theorem
If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $...
- 265
3
votes
0
answers
157
views
Growth of the constants from the Stone-Weierstrass Theorem
The Stone Weierstrass theorem for $C([0,1])$ claims that for any continuous function $f:[0,1]\to\mathbb{R}$ and each $n\in\mathbb{N}$, there is a polynomial $p_{n,f}(x)=\sum_ia_{f,n,i}x^i$ such that $\...
- 10.6k
8
votes
3
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429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,153
0
votes
1
answer
139
views
What kind of functions can be represented as infinite linear combinations of exponential functions?
Let $f(x)$ be a real-valued function defined in $(0, \infty)$. I am curious what kind of $f(x)$ has the following representations:
$$
f(x) = \sum_{j=0}^\infty a_j e^{-jx}, \quad \forall x \in (0, \...
- 81
6
votes
1
answer
345
views
Characterization of sums of periodic functions over the real line
Is there any known characterization of the functions $\mathbb{R \to R}$ that can be written as a sum of (a finite family of) periodic functions? Not assuming any regularity condition (not even ...
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,153
0
votes
1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
4
votes
1
answer
523
views
Is there any strengthened version of Rademacher's Theorem or any counterexample?
The following theorem is well-known in the ordinary analysis textbook:
Theorem: Assume the function $f:U\to\Bbb R^n$ is Lipschitz continuous on an open set $U\subset\Bbb R^m$, then $f$ is almost ...
- 300
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1
answer
213
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Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 581
3
votes
2
answers
206
views
Getting Wasserstein closeness from a derivative estimate
In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate:
$$
|\mathbb{E}_{\mu}(f)-\...
- 662
0
votes
1
answer
281
views
Roots of linear combination of $x \sin x$
Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$.
Under which condition on $\theta$, such that the following function ...
- 405
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
5
votes
1
answer
234
views
Can a continuous bounded variation function be $C^0$-reparametrized to be continuously differentiable?
Let $f: [0, 1] \to \mathbb R$ be a function of bounded variation. We say that $g$ is a $C^0$ reparametrization if $g = f \circ s$ for $s$ a continuous increasing bijection from a finite interval $I$ ...
- 6,223
4
votes
1
answer
642
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 697
2
votes
0
answers
231
views
Where does this trig. identity hold?
Fix an integer $n\geq1$.
QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that
$$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...
- 43.2k
1
vote
2
answers
120
views
Integral inequality implies $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-increasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 459
2
votes
1
answer
299
views
Eigenvalues of a specific matrix
I have a block matrix
$$M=\begin{bmatrix}
I_0& I_1& \cdots& I_1\\
I_2& I_0& \ddots& \vdots\\
\vdots& \ddots& \...
- 43
0
votes
0
answers
107
views
The eigenstructure of the symmetric tridiagonal matrix whose entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$
Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and
$$a_{1,2}=\cdots=a_{n-1,n}=1$$
Any approach to ...
- 4,058
0
votes
0
answers
124
views
Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
- 4,058
2
votes
1
answer
116
views
Can continuous correspondence be represented via continuous functions?
Let $\Theta \subset \mathbb{R}^n, \mathcal{X} \subset \mathbb{R^m}$, and suppose that $C: \Theta \rightrightarrows \mathcal{X}$ is a correspondence defined by $f: \Theta \times \mathcal{X}\to \mathbb{...
- 53
4
votes
2
answers
283
views
Can every symmetric function be factorized through symmetric polynomials?
A symmetric function is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x_1,\ldots,x_n)=f(\sigma(x_1,\ldots,x_n))$ for every permutation $\sigma\in S_n.$
The most commonly encountered symmetric ...
- 361
3
votes
1
answer
202
views
Local properties of Baire 1 functions
A Baire 1 function $f:[0,1]\rightarrow \mathbb{R}$ need not be bounded. However, thanks to the Baire category theorem, we know there is $N\in \mathbb{N}$ and a sub-interval $(c, d) \subset [0,1]$ ...
- 4,359
0
votes
0
answers
124
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Calculation of first correction to Selberg type integral
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix.
$\Tr U$ will denote the character ...
6
votes
3
answers
267
views
Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let
$$
F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)
$$
be the ...
- 127
2
votes
2
answers
281
views
Most general reverse Hölder inequality for polynomials
Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981