All Questions
Tagged with fa.functional-analysis real-analysis
1,448 questions
0
votes
1
answer
150
views
Total variation of composition of BV function and diffeomorphism [closed]
Let $f:\mathbb R \to \mathbb R$ be a $BV$ function and $g:\mathbb R \to \mathbb R$ be a diffeomorphism. What is the total variation of $f \circ g$?
My guess is
$$
TV(f\circ g) \le TV(f) \Vert (g^{-1})'...
- 303
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 131
2
votes
1
answer
143
views
How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
- 21
0
votes
0
answers
106
views
Extension of a Hilbert basis
The picture below is taken from this paper: http://real.mtak.hu/22877/.
The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
- 617
5
votes
0
answers
100
views
What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?
Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence.
Question: What is a minimal topological condition on $X$ ...
- 315
0
votes
1
answer
191
views
$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}$
Let $l$ be a continuous bounded function ($l$ is not differentiable). I want to prove for $x$ large enough that
$$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\...
0
votes
1
answer
236
views
Estimate on total variation of composition of functions
Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is
$$
\int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ?
$$
...
- 303
1
vote
1
answer
115
views
$K(x,y)\in L^{\infty}(R^n\times R^n, m\times m)$, $K(x,y)=K(y,x)$, so $K(x,y)=\sum_{k=1}^{\infty}\lambda_k \phi_k(x)\phi_k(y)$, are $\phi_k$ bounded?
Consider a symmetric function
$$
K(x,y):R^n \times R^n \to R
$$
satisfying $K(x,y)=K(y,x)$ and
$$
\int_{R^n\times R^n} K^2(x,y)dm(x) dm(y) <\infty.
$$
Let $m$ be a probability measure on $R^n$.
...
4
votes
1
answer
226
views
Approximate constant function
Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert_{L^2[0,1]}=1.$
Does there exist a constant $c>0$ such that any such function ...
- 192
3
votes
2
answers
164
views
A question on convergence in $\operatorname{Lip}_0(\mathbb R^n)$
$\DeclareMathOperator\Lip{Lip}$This question arose when I read Godefroy and Lerner - Some natural subspaces and quotient spaces of $L^1$.
Let $\Lip_0(\mathbb R^n)$ be the space of Lipschitz functions ...
user128095
1
vote
0
answers
121
views
Does this integral have a closed form solution? [closed]
Let $x,y\in \mathbb{R}^2$, $B_r(0)=\{x||x|\leq r\}$. Does the following function (denoted as $f(r)$) have a closed form expression?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
- 379
24
votes
3
answers
6k
views
Density of smooth functions under "Hölder metric"
This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
- 515
3
votes
0
answers
468
views
Opposite of the curl operator and Biot-Savart kernel
Note: I just realized that using $\omega$ and $w$ might not have been the smartest choice of notation -- Sorry about that.
Let $\renewcommand{\div}{\operatorname{\div}}Q_0, Q_1$ be two real numbers, $...
- 1,326
13
votes
0
answers
395
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
- 854
25
votes
2
answers
2k
views
$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$
Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a ...
- 295
0
votes
1
answer
110
views
Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 131
0
votes
1
answer
329
views
Domain of the fractional Laplacian operator
If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$
but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-...
- 109
22
votes
2
answers
652
views
Does every positive continuous function have a non-negative interpolating polynomial of every degree?
Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
- 1,209
3
votes
0
answers
121
views
Schatten norm estimate of spatially truncated resolvent of Laplacian
Consider the operator $-\Delta$ on $L^2(\mathbb R^d)$. In my studies I stumbled upon operators of the form
$$1_{\Gamma_n} (-\Delta -z)^{-1} 1_{\Gamma_m},$$
where $1_{\Gamma_m}$ denotes multiplication ...
- 91
1
vote
0
answers
144
views
Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99
0
votes
1
answer
120
views
Density property fractional heat kernel
Let us consider $$p_t^{(n+2)}(\tilde x) , $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\...
- 109
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
21
votes
1
answer
3k
views
Density of polynomials in $C^k(\overline\Omega)$
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
- 4,034
1
vote
1
answer
388
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
3
votes
1
answer
465
views
Existence of function satisfying $f(f'(x))=x$ almost everywhere
My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...
- 1,503
2
votes
0
answers
132
views
When does a nonnegative $C^1$ function on $[a,b]$ have finitely many zeros?
It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.
It is also known that you can have a ...
- 133
1
vote
1
answer
135
views
Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$
Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$
and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
- 123
3
votes
1
answer
237
views
Measure theory on abstract Boolean ring
Since a σ-algebra in measure theory is indeed an algebra over $\mathbb{Z}_2$ with addition given by symmetric difference and multiplication given by intersection, does it mean we can put measure on ...
- 43
5
votes
2
answers
582
views
Efficiently reversing the triangle inequality with additional information
Suppose $f$ and $g$ are bounded functions, having whatever niceness properties you want, on some space of finite measure. Assume they are normalized so that $\int |f|^2=\int|g|^2=1$. I am looking for ...
- 61
1
vote
1
answer
72
views
Condition for the maximum to be non-increasing
Let $u\in\mathcal{C}^1(\mathbb{R}_+\times[0,1],\mathbb{R})$ such that, for any $t\geq 0$, for all $x_0\in[0,1]$ satisfying $u(t,x_0)=\sup_{x\in[0,1]}u(t,x)$, we have
$$\partial_t u(t,x_0)\leq 0.$$
Is ...
- 449
6
votes
0
answers
107
views
Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
1
vote
1
answer
131
views
Fractional Laplacian problem on half-line
Is it possible to obtain an explicit solution for the following fractional problem on the half-line?
$$(-\Delta)^\alpha u(x) + M u'(x) + K u(x) + C = 0 \quad \text{ in } (0,\infty)$$
$$u(x) = a, \quad ...
- 99
4
votes
1
answer
378
views
Every convex set is of locally finite perimeter
I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter.
$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...
- 73
5
votes
1
answer
151
views
Existence of operator with certain properties
I am curious to know the answer to the following question:
Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated ...
user69109
3
votes
2
answers
410
views
Is a bounded sequence of $H^1(\Omega)$ tight?
Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$.
Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
- 1,101
13
votes
2
answers
653
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 536
1
vote
1
answer
181
views
Unique solution for 2$\times$2 Fredholm integral equations system
Consider the following system of Fredholm integral equations with constant kernel matrix
$$
f(x)=K(x)\int_{0}^{1}f(s)ds
$$
where $K(x)\in C([0,1];M_{2\times 2}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\...
- 617
3
votes
1
answer
299
views
Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)
Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align*}
|f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n.
\end{align*}
For $n \ge 2$, can we ...
- 1,741
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
2
votes
1
answer
217
views
Given functions $A(x), B(x)$ find $f(x)$ s.t. $A\big(f(x)\big)=f\big(B(x)\big)$
Currently, I am facing this problem:
Given two real functions $A( \vec x )$ and $B( \vec x ):\Bbb R^N\to \Bbb R$, I want to find a third real, monotonic function $f(x):\Bbb R\to\Bbb R$ such that:
$$...
0
votes
1
answer
324
views
Injectivity of analytic functions
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:
Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
- 431
1
vote
1
answer
385
views
Approximating $1/x$ by a polynomial on $[0,1]$
For every $\varepsilon > 0$, is there a polynomial of $x^4$ without constant term, i.e., $p(x^4) = a_1 x^4 + a_2 x^8 + \cdots +a_n x^{4n}$, such that
$$\|p(x^4)x^2 - x\| < \varepsilon $$
for ...
- 481
1
vote
0
answers
34
views
$L^p$-continuity for discrete linear causal systems
Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...
- 41
0
votes
1
answer
142
views
Some multivariate Taylor series and corresponding smoothness balls
Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form
$$f(\mathbf x) = \sum\limits_{\mathbf k \in \mathbb N^d} a_{\mathbf k} \...
- 686
1
vote
1
answer
372
views
Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?
The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions :
Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
- 128k
0
votes
1
answer
106
views
Existence of uniform approximator that also approximates derivative
Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to ...
- 119
2
votes
1
answer
218
views
If an estimate is false on $L^{1}$, then it is false for the $\delta$ distribution?
Let $u=\int e^{\dot{\imath}K(x,y)} f(y) dy$. My advisor told me that we can disprove an integrability estimate
$$\|u\|_{L^p}\lesssim \|f\|_{L^{1}}\label{1}\tag{1}$$
by disproving it when $f=\delta$, ...
- 852
6
votes
1
answer
128
views
Equivalence of antiderivative in L1 sense and in the usual sense
We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that:
$$
\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0
$$
where $\Vert \cdot \Vert_1$ is the $L_1$ ...
- 165
2
votes
0
answers
164
views
What are (the different aspects of) harmonic analysis good for?
Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
- 2,081
2
votes
0
answers
136
views
Banach limit with added properties
Let $c=\{a:\mathbb{N}\rightarrow \mathbb{C}: \exists \alpha\in \mathbb{C} \textrm{ so that }
\lim\limits_{n\rightarrow \infty} a(n)=\alpha=:L_c(a)\}\subset \ell^\infty$, where $\ell^\infty$ is the ...
- 21