All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
2
votes
0
answers
90
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 93
2
votes
0
answers
191
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
2
votes
0
answers
120
views
On mollifiers acting between $L^2$ and Sobolev spaces
(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.)
Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by
$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
- 505
2
votes
0
answers
57
views
Mappings that preserve local or global minimum
In the most general form, I'm interested in any non-trivial results of the following question.
Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
- 275
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 434
2
votes
0
answers
75
views
Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
- 969
2
votes
0
answers
138
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
- 969
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 71
2
votes
0
answers
103
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Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
2
votes
0
answers
170
views
finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
- 4,143
2
votes
0
answers
120
views
Closure of Laplacian
Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...
- 1,171
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
2
votes
1
answer
165
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
- 1,043
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
2
votes
0
answers
325
views
Examples of RKHS that are "classical"
Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs?
It is easy to construct example of RKHSs by applying the ...
- 21
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
2
votes
0
answers
103
views
Schwartz kernel theorem for restricted operators
Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
- 1,171
2
votes
0
answers
175
views
Banach space of vector measures
Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...
- 552
2
votes
0
answers
203
views
Schrödinger representation of the Heisenberg group
Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
- 521
2
votes
0
answers
94
views
A division of real analytic functions
Problem statement
Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$.
Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...
- 981
2
votes
0
answers
201
views
Green function of a 2D exterior domain
Consider solutions of the laplace equation
\begin{equation}
\begin{split}
-\Delta u=f, \ \ u|_{\partial D}=0,
\end{split}
\end{equation}
where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
- 379
2
votes
0
answers
193
views
Commutative Banach algebras with zero-dimensional maximal ideal space and disjoint supports of Gelfand transforms
Let $A$ be a commutative semi-simple unital Banach algebra and let $\Delta$ be the maximal ideal space of $A$. Denote by $\widehat{\cdot}\colon A\to C(\Delta)$ the Gelfand transform.
If $\Delta$ is ...
- 21
2
votes
0
answers
65
views
Is it possible to extend Borel's lemma to the case of functional derivatives?
Let us think of a collection of tempered distributions $\{ T(x_1, \cdots, x_n)\}_{n=0}^\infty$. Here I will specifically set $x_i \in \mathbb{R}^4$ since I am considering quantum field theory and ...
- 3,477
2
votes
0
answers
56
views
Existence of a suitable smooth kernel
Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
- 4,143
2
votes
0
answers
172
views
Fourier transform harmonic oscillator eigenstates
The normalized eigenfunctions of the quantum harmonic oscillator are
$$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$
where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...
- 124
2
votes
0
answers
72
views
Semilinear elliptic equations in complex plane
Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
- 4,143
2
votes
0
answers
170
views
Equivalence of implicit function theorem and Peano existence theorem in ODEs?
I was recently reading a book about the implicit function theorem (IFT): The implicit function theorem: history, theory, and applications, and before that I learned that Peano's existence theorem can ...
- 181
2
votes
0
answers
122
views
Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc
Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$:
$$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
- 321
2
votes
0
answers
76
views
Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
- 343
2
votes
0
answers
216
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
155
views
Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 319
2
votes
0
answers
161
views
The Laplace transform and the Lagrange compositional inversion formula
I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
- 10.5k
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
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
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 ...
2
votes
0
answers
68
views
Core for Neumann Laplacians
Let $d$ be a positive integer. We write $\mathbb{H}^d$ for the closed $d$-dimensional upper-half space: $\mathbb{H}^d=\{(x_1,\ldots,x_d) \in \mathbb{R}^d,\,x_d \ge 0\}$. We consider the Neumann ...
- 721
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
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
2
votes
0
answers
229
views
Weighted Sobolev norm in terms of Spherical harmonics coefficients
Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
- 969
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
2
votes
0
answers
164
views
(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis
It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
- 1,245
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
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
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 ...
- 41
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