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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\...
Doofenshmert's user avatar
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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\...
Alexander's user avatar
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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 ...
Alma Arjuna's user avatar
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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 ...
Zhang Yuhan's user avatar
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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 ...
Zhang Yuhan's user avatar
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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 $n$ ...
Joe Shipman's user avatar
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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-...
Akira's user avatar
  • 835
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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 $\alpha$ ...
Liding Yao's user avatar
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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^...
PPB's user avatar
  • 85
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0 answers
71 views

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) $f$ ...
William Kong's user avatar
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32 views

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 ...
Vladimir B.'s user avatar
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279 views

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 ...
tommy1996q's user avatar
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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+...
Ben's user avatar
  • 19
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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{...
chrisv's user avatar
  • 21
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1 answer
135 views

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^...
Vladimir's user avatar
  • 371
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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 ...
Blind Miner's user avatar
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103 views

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\...
aleph2's user avatar
  • 637
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22 views

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 $\mathfrak{S}$. ...
Alexander's user avatar
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616 views

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 ...
Kaira's user avatar
  • 305
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0 answers
126 views

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 $M$ ...
WaoaoaoTTTT's user avatar
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122 views

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 ...
Zachary Hoelscher's user avatar
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53 views

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) =...
NancyBoy's user avatar
  • 393
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0 answers
36 views

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 ...
NancyBoy's user avatar
  • 393
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0 answers
180 views

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}...
chrisv's user avatar
  • 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 ...
tommy1996q's user avatar
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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....
Lorenzo Vanni's user avatar
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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 ...
Clara Torres-Latorre's user avatar
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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 ...
jsb's user avatar
  • 403
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0 answers
34 views

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]...
Ilovemath's user avatar
  • 677
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0 answers
119 views

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 ...
Elio Li's user avatar
  • 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}...
Josh's user avatar
  • 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 $g_\sigma$ ...
NancyBoy's user avatar
  • 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 $\...
Bobo's user avatar
  • 101
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0 answers
149 views

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 ...
Isaac's user avatar
  • 3,477
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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 \...
Isaac's user avatar
  • 3,477
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0 answers
93 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 ...
Severin Schraven's user avatar
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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\...
Sayan Dutta's user avatar
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0 answers
102 views

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 (...
Hheepp's user avatar
  • 371
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0 answers
122 views

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 ...
van der Wolf's user avatar
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0 answers
48 views

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 ...
Isaac's user avatar
  • 3,477
0 votes
0 answers
75 views

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 }\...
L19's user avatar
  • 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}{...
Libli's user avatar
  • 7,300
0 votes
0 answers
101 views

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 ...
L19's user avatar
  • 61
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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+...
Dominic van der Zypen's user avatar
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0 answers
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 ...
david's user avatar
  • 1
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0 answers
316 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: $\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$, $...
Dr. Pi's user avatar
  • 3,062
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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 $c$ ...
fractal's user avatar
  • 101
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0 answers
174 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}^{\...
Stephen Berg's user avatar
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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 ...
Adi's user avatar
  • 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 ...
Sergii Voloshyn's user avatar

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