All Questions
Tagged with fa.functional-analysis real-analysis
1,448 questions
3
votes
2
answers
477
views
Vanishing convolution between density and compactly supported function
Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:
$f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
$g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
- 75
0
votes
1
answer
55
views
Kernels of sequences of operators
Let $\left( S_{n}^{1}\right) $ and $\left( S_{n}^{2}\right) $ two sequences
of operators in $\mathcal{L}(E_{1},F_{1})$ and $\mathcal{L}(E_{2},F_{2})$
where $E_{i},F_{i},i=1,2$ are Hilbert spaces such ...
- 617
2
votes
1
answer
148
views
The regularity theorem, a non-regular minimizer problem
During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R ...
- 159
3
votes
2
answers
204
views
What is the distribution of the following limit?
Assume $x \in \mathbb{R}$. We already know that
$$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$
Here $\delta_x$ denotes the Dirac distribution. If we ...
- 903
0
votes
0
answers
83
views
Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 159
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 536
3
votes
1
answer
116
views
Ekeland's standardness-property inheritable?
Ekeland's inverse function theorem gives weak conditions under which a function $f:E\rightarrow F$ between two graded Fréchet-spaces is locally surjective. The theorem requires
the codomain $F$ to be ...
- 779
1
vote
1
answer
123
views
Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$
Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$.
To prove that, as $\epsilon \to 0$, the weak solution ...
user139844
2
votes
2
answers
316
views
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 ...
- 1,245
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
8
votes
2
answers
2k
views
Arzela-Ascoli for L_p-norm
Since I am from a different mathematical field and couldn't find it: Is there something which would be best called an Arzela-Ascoli version for the $L_p$-norm, namely:
Let $X,Y$ be two nice ...
- 255
0
votes
0
answers
71
views
Sufficient condition to be increasing, following a vector field
Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
- 449
4
votes
3
answers
251
views
A functional integral inequality
Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies
$$ \int_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$
Does it follow that $f\geq 0$ on $I$?
- 4,115
3
votes
0
answers
120
views
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2)$ then there is a subsequence such that $u_n(t)\rightharpoonup u(t)$ almost everywhere?
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?
I'm not sure if this question is trivial or not,...
- 153
11
votes
2
answers
759
views
Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x))^{2}+\sin ^{2} f(x)}dx)^{2}\ge 4\pi^{2}$
This problem has been posted on Math.SE but didn't receive any correct answer after a long time.
Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. ...
- 302
0
votes
0
answers
112
views
How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
7
votes
0
answers
619
views
Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 1,309
4
votes
1
answer
591
views
Derivative of trace
Consider two positive-semi definite matrices $T_1, T_2$ of unit trace.
Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two.
We then study $f(\lambda) := \operatorname{tr}(T(\...
- 536
7
votes
1
answer
755
views
Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
- 71
0
votes
0
answers
149
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
0
votes
1
answer
139
views
Build an explicit "small perturbation" of the identity satisfying some properties
How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant ...
user139844
4
votes
0
answers
81
views
Does this sequence of functions converge in a distributional sense?
Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let
$$\tilde{K}^{\epsilon} := K^{\...
- 469
3
votes
0
answers
187
views
Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 20.7k
1
vote
1
answer
357
views
Conditions that make a dual space strictly convex
We know that a normed space $X$ is said to be strictly convex if
$\Vert tx+(1-t)y\Vert<1$, $\forall t\in(0,1)$, $\forall x,y\in X$ with $\Vert x\Vert=\Vert y\Vert=1$ and $x\neq y$.
For example if $...
- 883
2
votes
0
answers
298
views
A question on convergence rates of Fourier series and strict convergence
Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$....
- 698
1
vote
1
answer
225
views
How to prove the reverse Hölder inequality for Laplace equations?
Let $ u\in H^1(2B) $ be a weak solution of $ \Delta u=0 $ in $ 2B $, where $ B=B(0,1) $ is a ball with center $ 0 $ and radius $ 1 $. Then there exists some $ p>2 $ such that
\begin{eqnarray}
\left(...
- 1,219
0
votes
1
answer
736
views
Proof: If a reproducing kernel exists for a Hilbert space, then it is unique
I really want to prove the statement in the title but I'm struggling with it. Here my current state:
Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ ...
- 13
3
votes
1
answer
216
views
Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
- 124
0
votes
0
answers
94
views
Is the space of affine continuous functions a Baire space
Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
- 675
1
vote
1
answer
1k
views
Young's convolution inequality for weighted norms
Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f * g \rVert_r \leq \lVert f\rVert_p \lVert g\rVert_q.$$
It is implicit here that the ...
- 75
2
votes
0
answers
166
views
Green's function for elliptic PDE with potential
$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
1
vote
1
answer
321
views
Is $(f \ast K)'' \in L^1(\mathbb R)$ for $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$?
Is it possible to deduce that $$(f \ast K)'' \in L^1(\mathbb R)$$ if $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$? What I can prove is that $(f \ast K)' \in L^1 \cap L^\infty$. Is ...
- 131
15
votes
3
answers
2k
views
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I've been trying to find an asymptotic expansion of the following series
$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$
and
$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
- 153
1
vote
0
answers
99
views
Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
3
votes
1
answer
261
views
The continuous dependence of the Green's function on a domain
Let $\Omega\in\mathbb{R}^2$ be a smooth bounded domain and $G(x,y)$ be the Green's function of $-\Delta$ in $\Omega$ with zero Dirichlet condition. Clearly $G(x,y)=-\frac{1}{2\pi}\ln|x-y|-h(x,y)$, ...
- 379
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
1
vote
1
answer
519
views
Asymptotic cone
Let $S$ be a subset in a real vector space $\mathbb{R}^n$. Define the asymptotic cone $S\infty:=\{y\in\mathbb{R}^n\mid\textrm{there exists a sequence }(y_k,\varepsilon_k)\in S\times\mathbb{R}^+\textrm{...
- 951
6
votes
1
answer
182
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,115
0
votes
0
answers
112
views
Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
- 1,334
2
votes
1
answer
177
views
Determine the sign (positive or negative) of an integral with the fractional Laplacian
Let $u,v:\mathbb R \to \mathbb R$ and $\phi: \mathbb R \to \mathbb R_+$ be smooth bounded functions. Assume also $\phi' \ge 0$. Assume that $u(0) - v(0) = 0$ and that $0$ is a strict global minimum of ...
- 839
1
vote
1
answer
180
views
Proof that Clarke generalized directional derivative is upper continuous
Let $X$ be a reflexive Banach space, $z, x, v \in X$. If $f: X \times X \rightarrow \mathbb{R}$ is continuous regarding its first argument and locally Lipschitz regarding its second argument. $\{z_i\},...
- 15
1
vote
1
answer
287
views
Examples of $C^{k,1}$ functions which are not $C^{k+1}$?
I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|...
- 5,405
6
votes
1
answer
257
views
Perron-Frobenius and Markov chains on countable state space
The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer:
Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
- 173
2
votes
1
answer
220
views
Diagonalise self-adjoint operator explicitly?
Consider the linear constant coefficient differential operator
$P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$
$$P= \begin{pmatrix} D_{z}+c & a \\ b & D_{z}+c \end{pmatrix}$$
where $D_z=-...
- 536
5
votes
1
answer
155
views
Which averages of products of a function give a norm?
Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a_1, a_2, b_1, b_2$ with $a_1+b_1=a_2+b_2=1$ consider the quantity
$$N(f)=\int_{[0,1]} \int_{...
- 2,288
1
vote
1
answer
277
views
Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$
Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...
- 119
1
vote
1
answer
518
views
On level sets of smooth functions in a bounded domain
Let $\Omega$ be a bounded domain in $\mathbb R^n$, $n\geq 2$, with a smooth boundary and let $f$ be a smooth function on $\bar\Omega$. Is there a natural condition that one can impose on $f$ ( say in ...
- 11
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
6
votes
0
answers
208
views
Can every weakly converging sequence be made to converge strongly after taking a subsequence and rearranging?
Let $f_i: [0, 1] \to \mathbb R$ be functions in $L^1 \cap L^\infty$ with $\sup_i \|f_i\|_{L^\infty} < M$ for some $M > 0$.
Suppose $f_i$ converge weakly in $L^1$ to some $L^1$ function $f$ - ...
- 6,313
2
votes
0
answers
66
views
Existence of saddle points under a $C^0$-perturbation of a continuous function
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and has a strict maximum point $a$ and strict minimum point $b$. Define $g(x,y)=f(x)+f(y)$ and $h_\varepsilon(x,y)$ be a family of continuous ...
- 379