All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
2
votes
1
answer
102
views
Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 115
2
votes
1
answer
140
views
An inequality about embedding of cube into metric spaces
A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An edge of a cube is a pair of points $\{\psi(\epsilon_1),\psi(\epsilon_2)\}$ in $X$ such that $\epsilon_1$ and $\epsilon_2 $ differ in ...
- 1,245
2
votes
1
answer
223
views
Infinite sum of asymptotic expansions
I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^...
- 21
2
votes
1
answer
412
views
Convergence in norm of Sobolev spaces
I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\lbrace{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\rbrace}$ and I have to show that the function $$f(x)=\...
- 21
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
2
votes
1
answer
942
views
A singular value inequality
Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
$$\left|s_{1}\...
- 173
2
votes
1
answer
466
views
What is the regularity of the argument of a complex function?
Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
- 305
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 143
2
votes
1
answer
474
views
Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?
Let
$$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$
be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
2
votes
2
answers
151
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
- 825
2
votes
1
answer
189
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 75
2
votes
1
answer
60
views
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^...
- 3,477
2
votes
1
answer
276
views
Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm
Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$
It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...
- 97
2
votes
1
answer
455
views
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
$$
\frac{\partial^2\gamma}{\partial x \, \partial y} = 0\:\:\...
- 134
2
votes
1
answer
255
views
On the infimal convolution of two norms on $\mathbb R^n$
$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation*}
K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1),
\end{equation*}
\begin{equation*}
M:=M_{n,...
- 128k
2
votes
1
answer
245
views
Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?
For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by
$$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
2
votes
1
answer
181
views
Characterization of extendible distributions
I asked this question on Mathematics Stackexchange, but got no answer.
I found the following question which characterize the extension of a distribution in $\mathbb{R}$:
Let $f \in L_{\text{loc}}^{1}(...
- 509
2
votes
1
answer
174
views
Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex.
For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that
$$
(f'...
- 1,095
2
votes
2
answers
231
views
Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu_f, \mu_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (...
- 463
2
votes
1
answer
145
views
Estimate for an oscillatory integral of the first kind
I am confused in finding the right bound for the following oscillatory integral
$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth ...
- 159
2
votes
1
answer
74
views
Two variable polynomials that behave like Lagrange polynomials [closed]
Let us consider different points $z_i=(x_i,y_i)$ in the plane where $i=1,\cdots n$.
Q Do there exist two variable polynomials $P_i(x,y)$ with minimal degree such that $P_i(z_j)=\delta_{ij}$?
- 4,058
2
votes
1
answer
143
views
How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
- 21
2
votes
1
answer
239
views
Injectivity of an integral transform
For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that
$$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
- 303
2
votes
1
answer
168
views
A question about series involving a Sobolev functions
Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
- 339
2
votes
1
answer
218
views
If an estimate is false on $L^{1}$, then it is false for the $\delta$ distribution?
Let $u=\int e^{\dot{\imath}K(x,y)} f(y) dy$. My advisor told me that we can disprove an integrability estimate
$$\|u\|_{L^p}\lesssim \|f\|_{L^{1}}\label{1}\tag{1}$$
by disproving it when $f=\delta$, ...
- 852
2
votes
2
answers
155
views
Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients
Consider equation
$$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$
with initial condition $u(0, x) = g(x).$
Suppose that $c(t, x)$ and $...
- 1,399
2
votes
2
answers
251
views
inequality involving the fractional Sobolev space
Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...
- 407
2
votes
1
answer
134
views
Common eigenvalues for two Sturm-Liouville problem
Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form
$$y''(x)=\lambda^2 y(x)+\lambda a(x)y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
- 617
2
votes
2
answers
485
views
Dual space of the completion of the space of Lipschitz functions
This question is a continuation of this post : Metrization of a topological vector space
Let $C_{lip}(\mathbb R^d)$ be the space of Lipschitz functions on $\mathbb R^d$. We endow $C_{lip}(\mathbb R^...
user128095
2
votes
1
answer
846
views
Proof of Helmholtz-Hodge decomposition, poor man's version
Helmholtz (-Hodge) decomposition commonly used in physics includes decomposition of a (sufficiently smooth) vector field $F = -\mathrm{grad}(U) + \mathrm{curl}(W)$ on bounded simply connected domain $\...
- 259
2
votes
1
answer
307
views
Box counting dimension of a set and Lipschitz functions
If $f$ is Lipschitz, then the following holds for the Hausdorff dimension:
$$\dim_H f(A) \le \dim_H A.$$
Is the same true for the box counting dimension?
- 839
2
votes
1
answer
997
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
user124345
2
votes
1
answer
328
views
Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
Question 1.
How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$?
Question ...
- 839
2
votes
1
answer
563
views
Density in fractional Sobolev space
Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define
$$
H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right),
$$
$$
H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}.
$$
Q: Is $C^\...
- 515
2
votes
1
answer
300
views
Necessary and Sufficient conditions for integrable function [closed]
Suppose that $a, b$ and $c$ are constant.
Is there the necessary and sufficient conditions of $a ,b, c$ for the following integration is integrable? i.e.
$$\int_0^\infty \int_0^\infty \int_0^\infty ...
- 119
2
votes
1
answer
615
views
Convolution with Schwartz class function
Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution).
Consider distribution as follows:
$$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
- 305
2
votes
1
answer
162
views
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
- 1,209
2
votes
1
answer
141
views
small perturbation of BV function
consider an interesting real analysis question:
define average operator on $[0,1]$:
$A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $
( may clarify ...
- 553
2
votes
1
answer
263
views
Schwartz space on $\bigcup_{n=1}^CR^n$
I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
- 21
2
votes
1
answer
311
views
Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
2
votes
1
answer
223
views
Strongly continuous semigroup: continuous or continuous componentwise?
Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$
Then, one can consider the function
$f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space ...
- 536
2
votes
1
answer
963
views
Is the Delta distribution a continuous functional on $H^1(\mathbb{R})$? [closed]
While it is easy to see that $H^1(\mathbb{R})$ are Hölder $1/2$-continuous, I started wondering whether this implies that $\delta_x(\varphi)=\varphi(x)$ is continuous as a functional
$$\delta_x:H^1(\...
- 95
2
votes
1
answer
186
views
Equivalence of definitions of the space of test functions
Let $\Omega\subseteq\mathbb{R}^n$ be some nonempty open, and use the notation $U\Subset V$ to imply that $U$ is a compact subset of $V$. Then, for all $K\Subset\Omega$, we can define the space $$\...
- 1,247
2
votes
1
answer
301
views
Simplicity of eigenvalues
Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
- 369
2
votes
1
answer
268
views
Monotonicity of the Hellinger integral/distance
Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\...
- 128k
2
votes
1
answer
140
views
interpret of Picone inequality for non-regular functions
Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
- 371
2
votes
1
answer
186
views
Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?
I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that
$$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to \...
- 235
2
votes
2
answers
144
views
First order pde with characteristics [closed]
Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...
- 21
2
votes
2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 201
2
votes
1
answer
578
views
When is the bound in Riesz-Thorin Interpolation Theorem attained?
Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
- 717