All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
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Existence of a smooth extension
In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface
$$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$
Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...
- 4,135
1
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2
answers
588
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Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 11
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1
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154
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Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 128k
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1
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Which kind of convergence can we get from Laplace transform convergence?
This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that
\begin{equation}
\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
- 128k
5
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1
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340
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How to give a counterexample of this estimate related to Paley-Littlewood theorem?
I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^...
2
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1
answer
455
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A periodic integral inequality
(This problem comes in connection with a geometric problem exposed here.)
Let $\gamma(x,y)$ be a (real) function on the unit disk such that
$$
\frac{\partial^2\gamma}{\partial x \, \partial y} = 0\:\:\...
CommunityBot
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1
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0
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99
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Proving more stronger fomula for discrepancy of a sequence [closed]
I am reading famous book about uniform distribution of sequences by Kuipers and Niederreiter and have questions about solving below exercise from that book. Before going to main exercise I will write ...
- 111
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1
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301
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Vague convergence VS Laplace transform convergence?
If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
- 128k
0
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1
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327
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Deduce that a function is zero on interval $[0,M]$
I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
- 141
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245
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Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 141
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2
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382
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Singular support: equivalent definition
Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in ...
- 7,817
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175
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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}^{\...
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2
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708
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Approximation of Hölder continuous functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
6
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2
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326
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Looking for references to study $U^p$ and $V^p$ spaces
I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...
- 1,075
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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
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1
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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)$....
- 3,425
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2
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213
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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 \...
- 12.9k
8
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3
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429
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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\...
- 109k
3
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156
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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
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1
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281
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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
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1
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216
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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\|_{...
- 61.9k
4
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1
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522
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 ...
- 5,146
3
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1
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275
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Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?
$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
CommunityBot
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2
answers
281
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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 $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981
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1
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100
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Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?
Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
- 6,367
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2
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166
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Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions
Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by
$$
F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}.
$$
It is clear that $F$ is strictly ...
- 6,853
1
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0
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64
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Sequential Hölder-norm for functions in $H_{\alpha}([0,1]^{d})$?
I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even ...
- 149
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4
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Proofs of Young's inequality for convolution
For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
CommunityBot
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4
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How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 446
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0
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106
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Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 286
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2
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199
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What does "a universal tree" mean?
It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the ...
- 33
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142
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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
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2
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316
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Properties of the topology of sequential convergence $\tau_\text{seq}$
Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_\text{seq}$ has the ...
- 12.9k
2
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1
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60
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Specific estimation of the norm for a linearly transformed function in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$
According to the standard definition, $\mathcal{S}_0^{\beta}(\mathbb{R})$ is a subspace of smooth functions on $\mathbb{R}$ with the property that
\begin{equation}
\lvert x^k f^{(q)}(x) \rvert \leq CA^...
- 128k
1
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1
answer
102
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Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?
I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
- 39k
4
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2
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391
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Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 6,195
6
votes
2
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463
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Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
- 42.8k
4
votes
1
answer
128
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On the domain of functionals in measure with singular kernels
this post is concerned with functionals defined in measures. Consider the following functional
$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$
were we define $-\log\...
- 6,367
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0
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94
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A division of real analytic functions
Problem statement
Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$.
Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...
- 981
2
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1
answer
546
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Limit involving regularized gamma function and its inverse
Let
$$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$
where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function $\Gamma(...
- 4,713
3
votes
1
answer
394
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Approximations for real functions
Is there some set of real functions ($S$) that has precisely the cardinality of the continuum, but is dense in the space of all the real functions in a sense that every real function can be ...
- 14.2k
4
votes
1
answer
140
views
Whether a functional which preserves maximum for comonotone functions is monotone?
Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\...
- 41.8k
5
votes
1
answer
353
views
Family of functions with prescribed derivatives
Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
- 128k
6
votes
1
answer
135
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Small shifts of weakly converging sequences in $L^1$
$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
- 128k
0
votes
1
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161
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Verifying the proof of a bilinear estimate in $L^2$
$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
- 159
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0
answers
131
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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
4
votes
0
answers
140
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
4
votes
2
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158
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A ball with slit at the radius is not $W^{1,1}$-extension domain
Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
- 1,101
2
votes
2
answers
752
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Derivative of the absolute value
Let $f \in W^{1,p}(U)$, then how to prove that $|f| \in W^{1,p}(U)$, where $W$ means the sobolev space over some open subset $U \in \mathbb{R}^n$.
In Lieb's Analysis he prove that Let $f$ be in $W^{1,...
- 28k
4
votes
2
answers
191
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Reference request: "Tangent relation" in metric spaces
Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
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