Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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112 views

Building representation of an arbitrary umbral calculus

Consider a set of integrable functions on the interval $(0,1)$. Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function). In such system the ...
0 votes
1 answer
250 views

Ellipsoid in $L^p([0,1],\lambda)$ spaces?

Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $ 1\leq p < q \leq +\infty$ we ...
1 vote
1 answer
132 views

Complemented C*-algebras

Let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
-4 votes
2 answers
467 views

Inverse square-law as a positive definite kernel?

Newtons law for gravity states that: $$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$ The function : $$k(x,y):=\exp(-| x-y|^2)$$ is known to be a positive definite function, called the RBF-kernel. It ...
3 votes
2 answers
148 views

Dimension of spectral projection subspaces under strong convergence of operators

I have a possibly simple question regarding estimating bounds on spectral projection subspace. Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...
5 votes
0 answers
166 views

Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions

Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
1 vote
0 answers
54 views

Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
2 votes
1 answer
113 views

Koopman operators on $L^p(X)$

On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
2 votes
0 answers
96 views

Poincare inequality on the hemisphere

Background: Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
9 votes
1 answer
637 views

How do people prove $\Gamma$-convergence in more complicated settings?

This is a soft question, I guess. $\Gamma$-convergence is a notion of convergence of functionals so that if $F_n$ $\Gamma$-converges to $F$, then cluster points of $\arg\inf F_n$ are minimizers of $F$....
2 votes
0 answers
141 views

Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
0 votes
1 answer
85 views

Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
1 vote
0 answers
102 views

Infinite tensor product of Hilbert spaces [duplicate]

Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
3 votes
1 answer
99 views

Fréchet-valued symbols

Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
1 vote
0 answers
81 views

Definition of second quantization

The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write: Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
4 votes
0 answers
107 views

Flatness of $C_0(S)$-module $L_\infty(S,\mu)$

Let $S$ be a locally compact Hausdorff space. By $C_0(S)$ we denote the space of continuous functions vanishing at infinity. Let $\mu$ be a finite Borel regular measure om $S$, then consider $L_\infty(...
6 votes
1 answer
298 views

How are coordinate charts constructed in noncommutative geometry?

In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...
3 votes
2 answers
384 views

Functional equations based on composition

I have asked this question here (*), but there are no answer. Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
1 vote
1 answer
149 views

Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?

It seems too good to be possible, but: Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space? Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
0 votes
1 answer
170 views

A continuous injection from the Hilbert cube to the real line?

Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question: Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
1 vote
1 answer
102 views

Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex

Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
1 vote
1 answer
635 views

Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
3 votes
0 answers
170 views

Generalized family of Hölder inequalities

Is the "only if" direction of the following fact known? For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^...
3 votes
1 answer
139 views

Uniformly closed ideals of smooth/real analytic functions

Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
3 votes
0 answers
124 views

Takesaki's duality in representation theory of $C^*$-algebras

In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting. ...
16 votes
1 answer
684 views

Unbalancing lights in higher dimensions

In ''The Probabilistic Method'' by Alon and Spencer, the following unbalancing lights problem is discussed. Given an $n \times n$ matrix $A = (a_{ij})$, where $a_{ij} = \pm 1$, we want to maximise the ...
0 votes
0 answers
69 views

Intersection of Sobolev Spaces

Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\...
5 votes
1 answer
2k views

Finite element method inverse estimate

$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma: Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{...
1 vote
0 answers
46 views

Best approximation rates of various classes of functions by truncated Fourier series

Let $f\in C([-1,1]^d)$ have periodic boundary, $N$ be a positive integer, and let $S_N(f)$ be the best approximation of $f$ by its truncated Fourier expansion truncated approximation $$ S_N(f):=\sum_{...
0 votes
0 answers
55 views

Existence of a measurable maximizer

Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
0 votes
0 answers
82 views

A question about associated operator on continuous functions space equiped with L2 norm

$$\text{For M a connected compact manifold, T is in }C^{1+\nu}(M,M) \text{ with } \nu\in(0,1),\\ \text{i.e. DT is some Hölder continuous function with Hölder exponent }\\ \text{, Denote m as the ...
3 votes
2 answers
217 views

Question about the Bessel operator

For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by \begin{equation*} L_\...
0 votes
0 answers
132 views

Dependence of functional integral on the function space

In physics, the following functional integral is considered \begin{gather} Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf )) \end{gather} It is usually said that the integration is performed ...
3 votes
1 answer
282 views

Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$

Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be ...
1 vote
0 answers
107 views

Looking for examples of kernels with scalar Pick property but not the complete Pick property

I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy. A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
0 votes
0 answers
57 views

Series representation of functions

Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let $$ V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\} $$ ...
15 votes
4 answers
3k views

Universal $C^*$-algebra with generators and relations

We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations $R_1,...,...
2 votes
1 answer
241 views

Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?

Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
2 votes
2 answers
139 views

Schauder bases in Banach spaces with a symmetric $k$-FDD

The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into ...
3 votes
3 answers
453 views

Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously, It should be at least twice differentiable. It should have a ...
2 votes
1 answer
141 views

Topology of ${\mathcal D}(\Omega)$ (space of test functions)

I have seen two approaches to the topology of ${\mathcal D}(\Omega)$: (i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...
4 votes
0 answers
139 views

Isomorphic copies of $c_0$ in the projective tensor products

There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\...
2 votes
0 answers
75 views

Array-determined operator ideals

For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology). An array in the Banach space $X$ is a sequence of sequences, $(...
3 votes
1 answer
475 views

A strange condition of convexity?

During my research, I come across this question. Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$. Is it true that $\forall x \in \mathbb R, f''(x) \...
2 votes
1 answer
153 views

Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\operatorname{div}g(x)\...
2 votes
1 answer
137 views

Prove if the fractional Laplacian of a function is bounded

Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$. Here $(-\Delta)^s$ is the ...
2 votes
2 answers
227 views

Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
2 votes
0 answers
46 views

Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
2 votes
0 answers
188 views

If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality

Related: On a deceptively tricky calculus problem. The way that Leonard Gross proves the log Sobolev inequality is in the following stages: He proves that for any operator $B$ that satisfies the log ...
2 votes
0 answers
171 views

What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows: Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...