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2 votes
4 answers
1k views

An inequality question

Let $M$ be a $3\times2$ matrix. Is it true that for any $x\in\mathbb{R}^{2}$ with $\left\Vert x\right\Vert _{3}=1$ there is some subspace $V$ with dimension $2$ of $\mathbb{R}^{3}$, such that $\left\...
2 votes
1 answer
412 views

General Sobolev Inequalities

In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated: Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
0 votes
2 answers
2k views

fundamental solution of radial wave equation

i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
7 votes
3 answers
1k views

A Question concerning the Fourier Transform of $\mathbb{R}$

Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$. Consider the subspace ...
2 votes
1 answer
1k views

Range of the Radon Transform

Let us consider the Radon transform in two dimensions: $$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$ where $r\in\mathbb{R}$ and $0\...
4 votes
2 answers
2k views

Inclusions of $C^{k,\alpha}$ spaces

When is $C^{k,\alpha}(\bar{\Omega})$ a subset of $C^{k',\alpha'}(\bar{\Omega})$? Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
2 votes
4 answers
358 views

When do functions near F have zeros near a zero of F?

Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...
8 votes
2 answers
8k views

Version of the Poincaré Inequality

Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality $$\|u\|_p \le C \|\nabla u\|_p$$ ($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)...
2 votes
1 answer
1k views

Green's function for wave equations in R² or R³

Hello, For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
1 vote
1 answer
706 views

Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??

Hi! The Plancerel-Polya inequality can be stated as follows: Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
3 votes
1 answer
2k views

fourier transform of radon measure

hi, assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e. $$ \left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
3 votes
1 answer
362 views

Cartesian product of test function spaces

Mini introduction Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \...
1 vote
0 answers
283 views

Density of Dolean exponentials in L2 and Wiener Measure

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
-1 votes
1 answer
311 views

A differential equation

let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function" Assume that $0<\gamma<1$, $v$ is a positive number, and $$\frac{dg}{ds}+(v\...
27 votes
1 answer
4k views

Criteria for boundedness of power series

Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$. Can one give necessary and sufficient criteria the ...
5 votes
2 answers
3k views

Uniform convergence of difference quotient

Let $\phi\in C^\infty_c(\mathbb R)$ be a smooth function with compact support. For $h>0$ define the difference quotient $\phi_h\in C^\infty_c(\mathbb R)$ by $\phi_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$...
6 votes
1 answer
581 views

A puzzling question on real interpolation

Suppose an operator $T$ is bounded on $L^2$ and also bounded from $L^{1}$ to $L^{1}$-weak. Then by Marcinkewicz interpolation one gets that $T$ is bounded on every $L^{p}$ for p between 1 and 2. ...
4 votes
0 answers
487 views

Convolutions and Toeplitz Operators

Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$. Let be $...
2 votes
2 answers
354 views

A bound on linear functionals over cotype 2 spaces

This is a modification of the somewhat naive question that I asked below. Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
5 votes
1 answer
403 views

Local form of a real-analytic function taking values in a Banach space

Let $B$ be an infinite-dimensional Banach space, and let $M\subset\mathbb{R}^n$ be a neighborhood of the origin in $\mathbb{R}^n$. Suppose that $I:M\to B$ is a real-analytic function with $I(0)=0$ ...
3 votes
3 answers
584 views

Polynomials and L^p(R)

As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
8 votes
0 answers
605 views

convergence rate in Wiener's approximation theorem

Wiener has the following fantastic results about approximations using translation families: Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
26 votes
3 answers
2k views

Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
7 votes
3 answers
2k views

What are some interesting sequences of functions for thinking about types of convergence?

I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...

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