All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
0
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96
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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(...
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(\...
CommunityBot
- 1
2
votes
0
answers
45
views
Additivity of squared Schatten $p$-norm with respect to spatial partition
Consider a Hilbert-Schmidt operator $A$ on $L^2(\mathbb R^d)$ with integral kernel $A(x,y)$. Let $\Omega\subset \mathbb R^d$ and $1_{\Omega}(x)$ denote its characteristic function as well as the ...
- 91
3
votes
0
answers
121
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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
1
answer
176
views
Classical fixed-point argument and invertible function
Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...
- 128k
1
vote
1
answer
3k
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Is point to set distance continuous?
Assume $\mathbf{d}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}_0^+$ is a metric such that the function $\psi(x)=\mathbf{d}(x,y)$ for any $y\in\mathbb{R}^n$ is continuous in the Euclidean ...
- 667
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:
$$
\|...
- 39.1k
8
votes
1
answer
734
views
Almost Arzela Ascoli
Definitions:
We say a sequence of continuous functions $f_n: [0, 1] \to \mathbb R$ is equicontinuous on average if for every $x \in [0, 1]$ and $\varepsilon > 0$ there exists some $\delta > 0$ ...
- 128k
2
votes
2
answers
495
views
Polynomial approximation (Weierstrass theorem) with bounds
Consider the closed interval $[0,1]$ and let $f \in C[0,1]$. Let $g$ be a real valued function on $[0,1]$ such that $g \leq f$.
Suppose $g = f$ at atmost finitely many points. Does there exist a ...
2
votes
1
answer
140
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Polynomial approximation of continuous function with constraints
Consider the closed convex subset $\mathcal{F} = \{f \in C[0,1] : 0 \leq f \leq 1, f(0)=0, f(1)=1\}$. Consider the polynomial class $\mathcal{P} = \{p \text{ is a polynomial} : p(0)=0, p(1)=1, 0 \leq ...
- 28k
3
votes
2
answers
324
views
An integral transform and the Stone-Weierstrass theorem
For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if
$$
\int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
- 17.2k
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
1
vote
2
answers
106
views
'Partial boundedness' of continuously parametrised power series
Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...
- 231
6
votes
2
answers
333
views
Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?
Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
- 437
3
votes
3
answers
226
views
Bounded $r$- variation function with a dense set of local maximum values
This is a sharpening of the following problem: $C^1$ function with a dense set of maximum values.
Problem set up:
Let $f \colon [0, 1] \to \mathbb R$ be a function on the unit interval. We say $y \in \...
- 17.2k
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 ...
- 128k
3
votes
1
answer
191
views
A convolution type singular integral operator with log
Define a convolution type operator $T_m$ by
$$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer.
Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
- 903
3
votes
2
answers
203
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 ...
- 266
2
votes
0
answers
65
views
Reference request for type of specific integral equation in two variable:
Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...
- 6,367
16
votes
2
answers
1k
views
How to generalize the various vector calculus theorems to distributions?
Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
- 760
2
votes
2
answers
1k
views
Decay estimate of Fourier transform of a compactly supported function
Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate
$$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$
for some $\...
- 903
3
votes
2
answers
417
views
Are equicontinuous function dominated by a continuous function?
Let $f_n: [0, 1] \to \mathbb R$ be an equicontinuous sequence of functions. Does there exist a continuous function $f$ that dominates $f_n $ in the following sense?
We say $f$ dominates the sequence $...
- 23.2k
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 ...
- 14.2k
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 ...
- 12.9k
2
votes
0
answers
132
views
Green's identity with a different norm
Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
- 66.3k
4
votes
1
answer
390
views
Existence of periodic solution to ODE
We shall consider the matrix-valued differential operator
$$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$
This is ...
- 192
5
votes
1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 2,155
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?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
- 64k
2
votes
0
answers
44
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
-3
votes
1
answer
315
views
Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
- 66.3k
6
votes
0
answers
107
views
Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 52.3k
2
votes
1
answer
174
views
Is the graph of a Sobolev function path connected?
Let $\Omega$ be a bounded, open, simply connected subset of $\mathbb R^n$ with Lipschitz boundary.
Question: Does every function in the Sobolev space $W^{1,1} (\Omega)$
admit a representative whose ...
- 6,213
3
votes
2
answers
2k
views
A condition under which an Lp function is L-infinity [closed]
I am looking for a condition under which a function in $L_p(\Omega)$ is also in $L_\infty(\Omega)$. The condition may be on the function itself, or on $\Omega$.
In other words, is there anything that ...
- 12.7k
1
vote
0
answers
76
views
Regarding an integrability property of Schwartz class function
Let $f\in\mathcal{S}(\mathbb{R^n})$, Schwartz class, satisfying
$$\int_{\mathbb{R}^n}|f(x)|e^{g(||x||)}dx<\infty, $$
where $g:[0,\infty)\to[0,\infty)$ be an increasing function satisfying $\int_0^\...
- 99
0
votes
0
answers
247
views
Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
3
votes
2
answers
296
views
Is radial part of a Schwartz class function also in Schwartz class?
Let $f\in\mathcal{S}(\mathbb{R}^n)$, Schwartz class. Consider the function $g$ defined on $[0,\infty)$ by $$g(r)=\int_{S^{n-1}}f(rw)d\mu(w),$$
where $d\mu$ is the normalised surface measure of $S^{n-1}...
- 6,367
1
vote
0
answers
56
views
Monotonically increasing and bounded function is in $BV_{loc}$?
For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$ be monotonically increasing and $\lim_{x\to -\infty} f_n(x)=0$ and $\lim_{x\to \infty} f_n(x)=1$. It follows $f_n$ is differentiable a.e..
I'm ...
- 173
2
votes
0
answers
71
views
Strict Riesz's rearrangement inequality when function is not nonnegative
The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions $f,g,h,$ are all nonnegative and $g$ is strictly symmetric ...
- 379
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
5
votes
2
answers
338
views
Approximation of analytic function by a fixed number of monomials
This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum_{n=0}^K \frac1{n!} x^n
$
...
- 109k
1
vote
1
answer
72
views
Condition for the maximum to be non-increasing
Let $u\in\mathcal{C}^1(\mathbb{R}_+\times[0,1],\mathbb{R})$ such that, for any $t\geq 0$, for all $x_0\in[0,1]$ satisfying $u(t,x_0)=\sup_{x\in[0,1]}u(t,x)$, we have
$$\partial_t u(t,x_0)\leq 0.$$
Is ...
- 128k
0
votes
0
answers
67
views
Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 617
6
votes
1
answer
256
views
Perron-Frobenius and Markov chains on countable state space
The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer:
Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
- 12.6k
0
votes
0
answers
300
views
Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 5,380
0
votes
1
answer
116
views
Fractional Laplacian and support
Let $u:\mathbb [-1,1] \to \mathbb R$ such that $\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on $u$ does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$
where $(-\...
- 17.2k
1
vote
1
answer
181
views
Unique solution for 2$\times$2 Fredholm integral equations system
Consider the following system of Fredholm integral equations with constant kernel matrix
$$
f(x)=K(x)\int_{0}^{1}f(s)ds
$$
where $K(x)\in C([0,1];M_{2\times 2}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\...
- 2,494
0
votes
0
answers
239
views
Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
0
votes
4
answers
1k
views
Does the Leibniz (product) rule hold for the spectral fractional Laplacian?
Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
- 3,065
5
votes
1
answer
155
views
Which averages of products of a function give a norm?
Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a_1, a_2, b_1, b_2$ with $a_1+b_1=a_2+b_2=1$ consider the quantity
$$N(f)=\int_{[0,1]} \int_{...
- 109k
1
vote
0
answers
258
views
Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99