All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
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112
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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
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answers
112
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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
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votes
0
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148
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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
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votes
0
answers
84
views
Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
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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
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165
views
Compact embedding of Lipschitz continuous functions
Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
- 3,388
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48
views
Surjectivity of the limiting operator
Consider the operator
\begin{eqnarray*}
K_{n} &:&L^{2}(0,1)\longrightarrow L^{2}(0,1)^{n}, \\
u(x) &\mapsto &A_{n}U_{n}(x)=A_{n}(u(\frac{x}{n}),u(\frac{x+1}{n}),...,u(%
\frac{x+n-1}{n})...
- 617
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95
views
A property of the Hilbert transform involving the cotangent function
A lemma of a paper by T. Elgindi and I.-J. Jeong (Arch. Rational Mech. Anal. 235 (2020) 1763–1817, Lemma 2.2) states the following:
Let $g(z)=\operatorname{sgn}(z)k(|z|^\alpha)$ with $k$ smooth and $k(...
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106
views
Extension of a Hilbert basis
The picture below is taken from this paper: http://real.mtak.hu/22877/.
The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
- 617
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247
views
Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
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0
answers
67
views
Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 617
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239
views
Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
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53
views
Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
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81
views
What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
- 6,853
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150
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Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
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299
views
Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 205
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answers
92
views
Can $\ell_1(E)$ be embedd into the dual of continuous function space?
Let $E$ be a compact seperable space, for example $E=[0,1]\subset\mathbb{R}$. Denote by
$$\ell_1(E):=\{u:=(u_x\in\mathbb{C}:x\in F):F\text{ is a countable subset of }E \text{ and } \|u\|_{\ell_1(E)}:=\...
- 131
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1
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115
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Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?
The following inequality is trivially true
$$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
- 161
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168
views
Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 29
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0
answers
152
views
Predual of $BMO(\mathbb{T}^d) $
In 1971, Fefferman characterized the predual of $BMO(\mathbb{R}^d)$ as the Hardy space $H^1(\mathbb{R}^d)$.
Is there a characterization of the predual of $BMO(\mathbb{T}^d$)?
- 11
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1
answer
603
views
A concave function as supremum of upper semi continuous is upper semi continuous
We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
- 1,043
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95
views
When does a potential function with given partial derivatives exist
I am looking for the answer to the following question:
Consider an integrable function $f:X\rightarrow X$ with $X$ being a compact subset of $\mathbb{R}^n$. What are the conditions on $f$ so that a ...
- 37
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49
views
Non-square multiplication operator matrix
Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...
- 617
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76
views
Constructing a small Radon-Nikodym derivative
Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that:
$0<h(x)$.
$\int_{x \in \mathbb{R}^n} |h(x)|<\infty$,
$\sup_{x \in ...
- 5,405
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answers
147
views
Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$
The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas.
Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
- 307
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255
views
Span of a nonlinear function
Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...
- 159
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85
views
Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?
$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that.
For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
user128095
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0
answers
91
views
Does $L^1$ convergence preserve the regularity of this sequence of functions?
Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges
$...
- 143
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0
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135
views
Help showing F is weakly lower semicontinuous
Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...
- 383
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55
views
Smooth compactly supported function with good scaling with respect to the fractional Laplacian
Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user139845
0
votes
0
answers
479
views
What are the sets on which norm-closedness implies weakly closedness?
Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
- 369
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63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
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0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
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0
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115
views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
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113
views
Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user123457
0
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0
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171
views
What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?
I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...
- 723
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75
views
Dense Egoroff theorem
Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...
- 5,405
0
votes
1
answer
113
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Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
- 1,903
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votes
0
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60
views
Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
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votes
0
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324
views
Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
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0
answers
299
views
When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
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0
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81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
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votes
0
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308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 617
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votes
0
answers
58
views
in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
- 43.2k
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votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
0
votes
0
answers
271
views
Convolution Integral involving an unknown function
I've got the following problem I'm working on which is related to some of my research.
I am trying to solve the following equation for the function $f$.
$$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \...
- 281
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0
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85
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Some problems about symmetric convolution semigroup on the unit circle
These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes".
Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
- 369
0
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0
answers
64
views
Approx the jump point of a $BV$ function from both hand side
Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as
$$
u(x)=
\begin{cases}
0,&\text{ if }x\in(-1,0)\\
1,&\text{ if }x\in(0,1)
\end{cases}
$$
Clearly, we have $u\in ...
- 679
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votes
0
answers
470
views
Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
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0
answers
63
views
The union of weighted compact supported continuous function
Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
- 679