Pointwise Product Research Articles

Pointwise Products of Uniformly Continuous Functions

The problem of characterizing the metric spaces on which the pointwise product of any two uniformly continuous real - valued functions is uniformly continuous is investigated. A sufficient condition is given; furthermore, the condition is shown to be necessary for certain types of metric spaces, which include those with no isolated point and all subspaces of Euclidean spaces. It is not known if the condition is always necessary. 2000 Mathematics Subject Classification. Primary: 54C10, 54C30; Secondary: 20M20

Noncompact uniform universal approximation

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space Rn. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions φ that are continuous, nonpolynomial, and asymptotically polynomial at ±∞. When φ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let Nφl(Rn)¯ denote the vector space of functions that are uniformly approximable by neural networks with l hidden layers and n inputs. For all n and all l≥2, Nφl(Rn)¯ turns out to be an algebra under the pointwise product. If the left limit of φ differs from its right limit (for instance, when φ is sigmoidal) the algebra Nφl(Rn)¯ (l≥2) is independent of φ and l, and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of φ equals its right limit, Nφl(Rn)¯ (l≥1) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of l≥1, whereas in the former case Nφ2(Rn)¯ is strictly bigger than Nφ1(Rn)¯.

A Particle Fusion Approach for Distributed Filtering and Smoothing

Distributed state estimation is an important tool for coordinated team decision-making and typically involves sharing information between robots in order to outperform individual state estimation. The shared information typically takes the form of relative measurements which allow team members to act as virtual sensors for other robots where the virtual sensor uncertainty is corrupted also by the team member’s state uncertainty. However, incorporating relative measurements commonly depends on a pointwise product operation in most distributed estimation techniques which is well defined for continuously-valued distributions but ill-defined for particle-based distributions. We propose a drop-in replacement for the pointwise product based on using the generalized Hölder’s inequality to upper-bound the product over a series of grid cell sets that discretize the state space. This upper-bound is well defined for particle-based distributions and allows for tighter approximations by decreasing the volume of the sets. We leverage the approach to realize two distributed estimation strategies that use a pointwise product, the Kullback–Leibler Average and Belief Propagation and use these methods in simulations and experiments with a pair of miniature autonomous blimps. We found that after distributed smoothing, we were able to achieve an average improvement of ∼25.1% and ∼35.2% in the position tracking error, indicating our distributed smoothing strategy is able to improve the tracking performance of our initial filtering estimates.

On Normed Algebras and the Generalized Maligranda–Orlicz Lemma

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications.

Dahlberg degeneracy for homogeneous Besov and Triebel–Lizorkin spaces

AbstractWe consider the composition operators acting on the real‐valued homogeneous Besov or Triebel–Lizorkin spaces, realized as dilation invariant subspaces of , denoted as . If and , then any function acting by composition on is necessarily linear. The above conditions are optimal: (i) in case , (Besov space), (Triebel–Lizorkin space), is a quasi‐Banach algebra for the pointwise product, (ii) in case , , , any function such that is a finite measure, and , acts by composition on .

A Local Douglas formula for Higher Order Weighted Dirichlet-Type Integrals

We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $${\mathcal {H}}_{\pmb \mu },$$ induced by an m-tuple $$\pmb \mu =(\mu _1,\ldots ,\mu _{m})$$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order m, for $$m\geqslant 3,$$ forms an algebra under pointwise product. We also prove that every non-zero closed $$M_z$$ -invariant subspace of $${\mathcal {H}}_{\pmb \mu },$$ has codimension 1 property if $$m\geqslant 3$$ or $$\mu _2$$ is finitely supported. As another application of this local Douglas formula obtained in this article, it is shown that for any $$m\geqslant 2,$$ weighted Dirichlet-type space of order m does not coincide with any de Branges–Rovnyak space $${\mathcal {H}}(b)$$ with equivalence of norms.

Bilinear wavelet representation of Calderón–Zygmund forms

We represent a bilinear Calder\'on-Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a sparse $T(1)$-type bound, which in turn yields directly new sharp weighted bilinear estimates on Lebesgue and Sobolev spaces. Moreover, we apply the representation theorem to study fractional differentiation of bilinear operators, establishing Leibniz-type rules in weighted Sobolev spaces which are new even in the simplest case of the pointwise product.

Orlicz Algebras Associated to a Banach Function Space

In this paper, we study the spaces ${\mathcal X}^\Phi$ as Banach algebras, where $\mathcal X$ is a quasi-Banach space and $\Phi$ is a Young function, and extend some well-known facts regarding Lebesgue and Orlicz spaces on this new structure. Also, for each $p\geq 1$, we give some necessary condition for the space $\mathcal{X}^p$ to be a Banach algebra under the pointwise product.

The product of two high-frequency Graph Laplacian eigenfunctions is smooth

In the continuous setting, we expect the product of two oscillating functions to oscillate even more (generically). On a graph G=(V,E), there are only |V| eigenvectors of the Laplacian L=D−A, so one oscillates ‘the most’. The purpose of this short note is to point out an interesting phenomenon: if ϕ1,ϕ2 are delocalized eigenvectors of L corresponding to large eigenvalues, then their (pointwise) product ϕ1⋅ϕ2 is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.

Star product approach for loop quantum cosmology

Guided by recent developments toward the implementation of the deformation quantization program within the loop quantum cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.

Symmetry reduction of states II: A non-commutative Positivstellensatz for [formula omitted

We give a non-commutative Positivstellensatz for CPn: The (commutative) ⁎-algebra of polynomials on the real algebraic set CPn with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C1+n, restricted to the real (2n+1)-sphere inside C1+n, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on C1+n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative ⁎-algebra of polynomials on C1+n, the Weyl algebra, and give an algebraic description of the real-valued U(1)-invariant polynomials that are positive in certain ⁎-representations on Hilbert spaces of holomorphic sections of line bundles over CPn. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive (semidefinite) elements. As an application, all ⁎-representations of the quantization of the polynomial ⁎-algebra on CPn, obtained e.g. through phase space reduction or Berezin–Toeplitz quantization, are determined.

Integrable Solutions for Gripenberg-Type Equations with m-Product of Fractional Operators and Applications to Initial Value Problems

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line R+=[0,∞). We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called L1N-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems.

Inner functions of matrix argument and conjugacy classes in unitary groups

Let $\mathrm{B}_n$ denote the set of complex square matrices of order $n$ whose Euclidean operator norms are less than one. Its Shilov boundary is the set $\operatorname{U}(n)$ of all unitary matrices. A holomorphic map $\mathrm{B}_m\to\mathrm{B}_n$ is inner if it sends $\operatorname{U}(m)$ to $\operatorname{U}(n)$. On the other hand we consider the group $\operatorname{U}(n+mj)$ and its subgroup $\operatorname{U}(j)$ that is embedded in $\operatorname{U}(n+mj)$ in the block-diagonal way ($m$ blocks $\operatorname{U}(j)$ and a unit block of size $n$). To any conjugacy class of $\operatorname{U}(n+mj)$ with respect to $\operatorname{U}(j)$ we assign a ‘characteristic function’, which is a rational inner map $\mathrm{B}_m\to\mathrm{B}_n$. We show that the class of inner functions that can be obtained as ‘characteristic functions’ is closed with respect to such natural operations as pointwise direct sums, pointwise products, compositions, substitutions into finite-dimensional representations of general linear groups and so on. We also describe explicitly the corresponding operations on conjugacy classes. Bibliography: 24 titles.

On recurrence in zero-dimensional locally compact flow with compactly generated phase group

We define recurrence for a compactly generated para-topological group G acting continuously on a locally compact Hausdorff space X with dimX=0, and then we show that if Gx‾ is compact for all x∈X, the conditions are pairwise equivalent: (i) this dynamic is pointwise recurrent, (ii) X is a union of G-minimal sets, (iii) the G-orbit closure relation is closed in X×X, and (iv) X∋x↦Gx‾∈2X is continuous. Consequently, if this dynamic is pointwise product recurrent, then it is pointwise regularly almost periodic and equicontinuous; moreover, a distal, compact, and nonconnected G-flow has a nontrivial equicontinuous pointwise regularly almost periodic factor.

On the homological and algebraical properties of some Feichtinger algebras

Abstract Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space B min in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of B min with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that B min is a symmetric Segal subalgebra of L 1(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of B min on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which A min $\mathcal{A}_{\min}$ with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product.

Twisted submanifolds of {mathbb {R}}^n

We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of {mathbb {R}}^n determined by a set of smooth equations f^a(x)=0. We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) star -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra Xi _t of vector fields that are tangent to all the submanifolds that are level sets of the f^a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted Xi _t. We can consistently project a connection from the twisted {mathbb {R}}^n to the twisted M if the twist is based on a suitable Lie subalgebra {mathfrak {e}}subset Xi _t. If we endow {mathbb {R}}^n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra {mathfrak {k}}subset {mathfrak {e}} of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and star -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean {mathbb {R}}^3 and twisted hyperboloids embedded in twisted Minkowski {mathbb {R}}^3 [these are twisted (anti-)de Sitter spaces dS_2,AdS_2].

Dual Convolution for the Affine Group of the Real Line

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on L^2({{mathbb {R}}}^times , dt/ |t|). In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on L^p({{mathbb {R}}}^times , dt/ |t|) for pin (1,2)cup (2,infty ).

Admissible Measurements and Robust Algorithms for Ptychography

We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say $$x_0\in \mathbb {C}^{d}$$ ) are illuminated one at a time, often with a physical mask between the sample and light source. The corresponding measurements are the noisy magnitudes of the Fourier transform coefficients resulting from the pointwise product of the mask and the sample. The goal is to recover the original signal from such measurements. The algorithmic framework we study herein relies on first inverting a linear system of equations to recover a fraction of the entries in $$x_0 x_0^*$$ and then using non-linear techniques to recover the magnitudes and phases of the entries of $$x_0$$ . Thus, this paper’s contributions are three-fold. First, focusing on the linear part, it expands the theory studying which measurement schemes (i.e., masks, shifts of the sample) yield invertible linear systems, including an analysis of the conditioning of the resulting systems. Second, it analyzes a class of improved magnitude recovery algorithms and, third, it proposes and analyzes algorithms for phase recovery in the ptychographic setting where large shifts—up to $$50\%$$ the size of the mask—are permitted.

Some endpoint estimates for bilinear Coifman-Meyer multipliers

In this paper we establish mapping properties of bilinear Coifman-Meyer multipliers acting on the product spaces H1(Rn)×bmo(Rn) and Lp(Rn)×bmo(Rn), with 1<p<∞. As application of these results, we obtain some related Kato-Ponce-type inequalities involving the endpoint space bmo(Rn), and we also study the pointwise product of a function in bmo(Rn) with functions in H1(Rn), h1(Rn) and Lp(Rn), with 1<p<∞.

On Pointwise Product Vector Measure Duality

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.