All Questions
Tagged with fa.functional-analysis real-analysis
1,448 questions
4
votes
0
answers
115
views
Delta distributions that are smooth on strata of a singular manifold
This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
- 7,962
1
vote
1
answer
48
views
Inverse transform of $\int K(s,t) \rho(s) dS $
Assume that $V$ is a n-dimensional surface, or the disjoint union such surfaces.
Let $K(s,t)$ be a function $V^2\to \mathbb R$. For the sake of simplicity, it can be assumed that $K$ is continuous, ...
- 687
5
votes
2
answers
700
views
Ground state for non-linear Schrödinger
When studying the blow-up for focusing non-linear Schrödinger equation (NLS) one often compares the initial-state to a stationary solution.
In the energy-critical case, this stationary solution is ...
- 536
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)?
user139845
1
vote
0
answers
98
views
Two definitions of Sobolev spaces and the trace theorem
Let $M=[0,\infty) \times S^2$. We have the regular regular Sobolev space $H^1(M)$.
We also have the space $H^1\bigg([0,\infty); H^1(S^2)\bigg)$. Are those two spaces the same? Does one contain the ...
- 969
3
votes
1
answer
374
views
Positive part of Cauchy sequence of Sobolev functions is again Cauchy
Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
- 49
0
votes
0
answers
166
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
2
votes
3
answers
217
views
Equivalence of operators
let $T$ and $S$ be positive definite (thus self-adjoint) operators on a Hilbert space.
I am wondering whether we have equivalence of operators
$$ c(T+S) \le \sqrt{T^2+S^2} \le C(T+S)$$
for some ...
- 23
4
votes
1
answer
221
views
Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
- 4,115
4
votes
0
answers
194
views
What are the "local degrees of freedom" in the space of smooth functions?
Let $C^k$ be the set of $k$th-order smooth real functions $f:\mathbb{R}\to\mathbb{R}$, and $C^\infty$ the set of smooth real functions. One can specify an $f\in C^k$ by specifying all its derivatives ...
- 49
2
votes
1
answer
721
views
Injectivity of an integral operator
Consider the operator
$$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$
with $k\in L^2((0,1)\times(0,1)).$
I want to know under what assumption the kernel is reduced to zero. i....
- 617
11
votes
4
answers
668
views
Is every non-negative test function the limit of a sequence of sums of squares of test functions?
Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-...
- 7,817
11
votes
2
answers
1k
views
Harmonic oscillator in spherical coordinates
It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...
- 113
3
votes
1
answer
203
views
Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$
Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
user139844
1
vote
1
answer
176
views
Classical fixed-point argument and invertible function
Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...
- 237
0
votes
1
answer
971
views
Is the pointwise supremum of a continuous function continuous?
Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...
- 3
0
votes
1
answer
241
views
Dense sub-algebra of $C_{b}((0,1))$ which is not smooth
I am looking for a dense sub-algebra $B$ in $C_{b}((0,1))$ in uniform topology such that it satisfy following requirements:
$B\cap C^{\infty}_{b}((0,1))=\mathbb{R}$ (No polynomial, no bump function).
...
- 523
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
6
votes
2
answers
1k
views
Properties of heat equation
** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...
user121558
8
votes
1
answer
380
views
Lavrentiev phenomenon between $C^1$ and Lipschitz
Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf_{y\in Lip([a,b])}F(y)<\inf_{...
- 1,309
5
votes
1
answer
260
views
Approximate Sobolev embedding
It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define ...
- 124
4
votes
1
answer
213
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
- 536
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
$$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
- 303
3
votes
0
answers
96
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
- 969
2
votes
0
answers
55
views
An integral average condition and its relationship with BMO, VMO, and Sobolev spaces
Let $V: \mathbb R^n \to \mathbb R^n$ be a vector field which satisfies
$$
\lim_{l \to \infty} \sup_{x \in \mathbb R^n} \left|\frac{1}{l^n} \int_{[0,l]^n}V(x+y) dy \right| = 0
$$
What is the ...
- 839
3
votes
2
answers
218
views
Analogue of decay of Fourier coefficients of a smooth function on $\mathbb{S}^1$
Let $\nu$ be the uniform measure on the unit circle $\mathbb{S}^1 \subset \mathbb{R}^2$, normalised so that $\nu(\mathbb{S}^1) = 1$. Suppose $\mu$ is a Borel probability measure on $\mathbb{S}^1$ ...
- 399
3
votes
2
answers
676
views
Compact-open limit of continuous functions is continuous?
Let $X$ be a topological space and $Y$ a metric space.
A classical result states that compact-open topology on the space $C(X,Y)$ of continuous functions is the same as the topology of uniform ...
- 829
2
votes
0
answers
252
views
A sequence of compact operators, weak convergence versus strong convergence
Let $(X,\mu)$ be a finite measure space.
We consider two sequences of bounded linear operators $\{T_n\}_{n=1}^\infty$ and $\{S_n\}_{n=1}^\infty$ on $L^2(X,\mu)$. We denote by $\mathcal{L}$ the space ...
- 721
9
votes
1
answer
511
views
Do these surfaces intersect?
For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
- 91
2
votes
1
answer
163
views
Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem)
In the paper "The Kadison-Singer Problem" by Marcin Bownik (https://arxiv.org/pdf/1702.04578.pdf), the following Lemma (3.8) is proven:
Lemma:
Let $p, q\in \mathbb{R}[x]$ be stable monic ...
- 203
0
votes
1
answer
603
views
A concave function as supremum of upper semi continuous is upper semi continuous
We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
- 1,043
11
votes
2
answers
552
views
Smoothness of finite-dimensional functional calculus
Assume that $f:\mathbb R\to\mathbb R$ is continuous.
Given a real symmetric matrix $A\in\text{Sym}(n)$, we can define $f(A)$ by applying $f$ to its spectrum. More explicitly,
$$ f(A):=\sum f(\lambda)...
- 3,146
-1
votes
1
answer
114
views
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
- 217
7
votes
1
answer
1k
views
A variational problem - some guidance
This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.
Let
$\...
user125029
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of 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 $$\{f_{n}\psi_m(...
- 161
10
votes
1
answer
595
views
Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
- 13.8k
2
votes
3
answers
308
views
What are the necessary conditions on $f$ if $f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$?
I am aware that the statement:
$$f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$$
is not true for all $f$. For example, $f$ can be $x$ to any constant power or any constant to the $x$th power but it cannot be ...
0
votes
1
answer
345
views
Embedding of fractional Sobolev space into BMO
Is it true that $$\Vert u \Vert_{BMO(\mathbb R^2)} \lesssim_{s} \Vert u \Vert_{\dot H^s(\mathbb R^2)},$$
for $s \in (0,1)$, where $\dot H^s(\mathbb R)$ is the homogeneous fractional Sobolev space?
- 99
7
votes
1
answer
856
views
Compactness of set of indicator functions
Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...
- 395
3
votes
1
answer
404
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
- 178
3
votes
2
answers
498
views
if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?
This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
user147204
2
votes
1
answer
174
views
Is the graph of a Sobolev function path connected?
Let $\Omega$ be a bounded, open, simply connected subset of $\mathbb R^n$ with Lipschitz boundary.
Question: Does every function in the Sobolev space $W^{1,1} (\Omega)$
admit a representative whose ...
- 6,313
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
1
vote
1
answer
125
views
Functions such that $ \Vert\tfrac{d^4}{dx^4}f\Vert_{L^2(0,1)} < \sqrt{2} \Vert f \Vert_{L^1(0,1)}$
Is there a (non-constant) function $f \in C^4((0,1))$ that is zero in an interval $(a,b) \subset (0,1)$ and such that the inequality
$$\Vert\tfrac{d^4}{dx^4}f\Vert_{L^2(0,1)} < \sqrt{2}\Vert f \...
- 131
1
vote
1
answer
2k
views
Sobolev embedding in the space of continuous functions [duplicate]
Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be ...
- 11
-1
votes
2
answers
129
views
Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]
Under which condition can it form a Hilbert space? Or what space can it form?
You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
- 11
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 ...
- 820
4
votes
1
answer
121
views
Condition for existence of a continuous function realizing a partition
Let $\{U_i\}_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup_{i =1 }^{ I} \overline{U_i}$. ...
6
votes
1
answer
2k
views
Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold
How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
- 839
5
votes
2
answers
597
views
Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
- 42k