All Questions
1,485 questions with no upvoted or accepted answers
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136
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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
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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-...
- 825
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votes
0
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112
views
Characterization for the multipliers of Schwartz space
Is the following true?
A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following:
For every $\alpha$ ...
- 938
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0
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80
views
Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
- 85
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71
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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) $f$ ...
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32
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Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions
For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
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279
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Are there "gaps" between Lipschitz functions and $C^1$ functions?
I was wondering about this fact. By Rademacher theorem, it is well known that a Lipschitz function in $\mathbb{R}^d$ is $\mathcal{L}^d$-a.e. differentiable. Of course, a $C^1$ function is everywhere ...
- 697
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0
answers
73
views
Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
- 19
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votes
0
answers
28
views
Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
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1
answer
135
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On polynomial equation of fourth order depending on two parameters and bound on a maximal root
I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:
\begin{eqnarray}
F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...
- 371
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votes
0
answers
63
views
Computing the eta invariant of a rather contrived operator on the circle
For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...
- 111
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103
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Who first gave a result stronger-or-equal to this one on ODEs
After some thinking I've come to the following conclusion.
Consider the initial value problem $$\text{(P)}\begin{cases}x'(t)=f(t,x(t)),\quad t\geq t_0\\x(t_0)=x_0 \end{cases}$$ where $f:D\subset\...
- 637
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22
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An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
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618
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The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
- 305
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126
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A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
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122
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Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
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53
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Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 393
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votes
0
answers
36
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Convergence of numerical scheme for HJB equation
Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...
- 393
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180
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Proof that the zeroes of certain polynomials are increasing with respect to degree
Choose $k+1$ positive integers $d_j\in\{0,1,2,3,\ldots\}$ and let $d=(d_1,\ldots,d_k)$.
Consider the following polynomial equation over the positive reals:
$$
\sum_{j=1}^{k}\frac1{x^{d_j}} = x^{d_{k+1}...
- 21
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0
answers
52
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
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0
answers
76
views
Implicit function theorem for non $C^1$ mappings
I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
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272
views
How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
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0
answers
97
views
Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
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34
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It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
- 677
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119
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About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
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0
answers
120
views
Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
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0
answers
66
views
Lower bound of the derivative $(f*g_\sigma)'$ at the zero-crossing point
I am stuck with the following problem. Let consider $f$ a smooth real function such that:
$f$ is negative before 0,
$f$ is positive after 0,
we have $|f'(0)|>0$.
Let $\sigma>0$ and $g_\sigma$ ...
- 393
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0
answers
145
views
Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
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149
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Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477
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votes
0
answers
101
views
Does the tensor product of mollifiers work for $L^{p,q}$ spaces?
Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively.
For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \...
- 3,477
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votes
0
answers
94
views
Oscillation of a polynomial
Recently I came across a statement in a paper that I am unable to verify. Namely, it roughly says that the oscillation of a polynomial on a cube can be controlled by the oscillation of the polynomial ...
- 1,045
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0
answers
138
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
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0
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102
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Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \Delta u=0, $$
the quantity $|\nabla u|^2$ is subharmonic, i.e.
$$\Delta (|\nabla u|^2) \geq 0. $$
Reason:
$$\Delta (|\nabla u|^2)= 2 \nabla (...
- 371
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votes
0
answers
122
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A bound for the Bessel function of the first kind J_0
I have proved the following bound for the Bessel function of the first kind:
$$
J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2}
$$
which is
$$
|J_0(x)|\le \frac1{\sqrt[4]{1+x^2}}
$$
but I ...
- 258
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48
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On $[0,1]$ with the Lesbegue measure, is it possible to have $\lVert \cdot \rVert^{1/n}_p \leq \lVert \cdot \rVert_q$ for $p>q$ and $n$ large?
The question is as in the above.
In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$.
(I deleted the last question ...
- 3,477
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votes
0
answers
75
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Morse functions on subset $\bar \Omega$ of $\mathbb {R}^d$ and its level sets
Let $f$ be a $C^{2}(\bar{\Omega})$ Morse function, where $\Omega$ is a bounded open set of $\mathbb{R}^d$: this means that
$$
\begin{cases}
f(x) = 0 \\
\nabla f (x) \neq 0
\end{cases}\text{ on }\...
- 61
0
votes
0
answers
111
views
Sum power series not continuous unit circle
This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
- 7,300
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0
answers
101
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Sobolev estimates on domain with boundary
Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...
- 61
0
votes
0
answers
107
views
$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 48.9k
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votes
0
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108
views
linear span of linear in C_0(R)
We consider the set $C_0(\mathbb R)$ of real continous functions $f:\mathbb R\rightarrow \mathbb R$ with $\lim_{|x|\rightarrow \infty}f(x)=0$ endowed with the supremum norm.
Is there $f\in C_0(\mathbb ...
- 1
0
votes
0
answers
317
views
What is the "best" good kernel?
A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied:
$\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$,
$...
- 3,062
0
votes
0
answers
120
views
The sufficient condition for a function equals to zero when it vanishes in an open set
For the class od analytic functions in one variable, there is an identity theorem: "If two holomorphic functions $f$ and $g$ on a domain $D$ agree on a set $S$ which has an accumulation point $c$ ...
- 101
0
votes
0
answers
175
views
Does l2 projection of sequences preserve l1 norm convergence?
Let $\ell^2$ denote the set of square summable sequences with inner product $\langle x,y\rangle=\sum_{i=1}^{n}x(i)y(i)$ and $\ell^2$ norm $\|x\|_2=\sqrt{\langle x,x\rangle}$. Let $\|x\|_1=\sum_{i=1}^{\...
- 45
0
votes
0
answers
85
views
Lagrange's interpolating polynomial
Let $f:[a,b]\rightarrow R$ be a function that is not $C^{(n+1)}$ on $[a,b]$ but its $n$-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does ...
- 1
0
votes
0
answers
124
views
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 ...
0
votes
0
answers
342
views
Main ideas behind the proof of the Carleson theorem
I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a ...
- 587
0
votes
0
answers
142
views
Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
0
votes
0
answers
155
views
Implicit function theorem on curves
I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
0
votes
0
answers
131
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
- 307
0
votes
0
answers
168
views
How does one make sense of singular solutions to constant mean curvature equation?
Background:
Consider the following ODE:
$$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$
where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
- 537