Article
Mathematics
Guan-Tie Deng, Yun Huang, Tao Qian
Summary: In this paper, the theory of Bergman kernel is extended to the weighted case, obtaining a general form of weighted Bergman reproducing kernel which can be used to calculate concrete Bergman kernel functions for specific weights and domains.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Ernesto De Vito, Nicole Muecke, Lorenzo Rosasco
Summary: This study focuses on reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold, exploring the conditions under which Sobolev spaces are RKHS and characterizing their reproducing kernels. The introduction of smoother diffusion spaces is also discussed, with detailed examples illustrating the general results. The paper presents a self-contained study of connections between Sobolev spaces, differential operators, and RKHS on Riemannian manifolds, aiming to provide a useful resource for researchers interested in the topic.
ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Interdisciplinary Applications
Dah-Chin Luor, Liang-Yu Hsieh
Summary: This paper investigates the connections between fractal interpolation functions (FIFs) and reproducing kernel Hilbert spaces (RKHSs). By establishing a fractal-type positive semi-definite kernel, it is shown that the span of linearly independent smooth FIFs is the corresponding RKHS. Furthermore, the nth derivatives of these FIFs, properties of related positive semi-definite kernels, and the importance of subspaces in curve-fitting applications are studied.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Wei Qu, Tao Qian, Haichou Li, Kehe Zhu
Summary: This study explores the best kernel approximation problem for analytic functions on the unit disk D in the reproducing kernel Hilbert space H, proving the existence of the best kernel approximation for weighted Bergman spaces with standard weights.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics
Alexandru Aleman, Micheal Hartz, John E. Mccarthy, Stefan Richter
Summary: This paper investigates the inner-outer factorization of Hardy space functions and shows that under certain conditions, the factors of this factorization are essentially unique. Several applications of this factorization are also provided.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Isao Ishikawa
Summary: In this paper, the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold is investigated. It is proved that the boundedness of a composition operator strongly limits the behavior of the original map, and the theory of dynamical systems provides an effective method to investigate the properties of composition operators. Consequently, it is proven that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. It is also proven that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.
ADVANCES IN MATHEMATICS
(2023)
Article
Automation & Control Systems
Peter Koepernik, Florian Pfaff
Summary: This paper investigates the formal consistency of Gaussian process (GP) regression on general metric spaces, demonstrating that the variance of the posterior GP converges to zero almost surely and the posterior mean converges pointwise in L-2 to the unknown function.
JOURNAL OF MACHINE LEARNING RESEARCH
(2021)
Article
Mathematics, Applied
Jabar S. Hassan, David Grow
Summary: In this paper, we introduce new reproducing kernel Hilbert spaces on a trapezoidal semi-infinite domain and establish uniform approximation results for solutions to nonhomogeneous hyperbolic partial differential equations. We also demonstrate the stability of these solutions with respect to the driver and provide an example to illustrate the efficiency and accuracy of our results.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Masahiro Ikeda, Isao Ishikawa, Yoshihiro Sawano
Summary: In this paper, we investigate the functions that induce bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function. Our results show that only affine transforms can achieve this in a certain class of RKHSs, which includes more general RKHSs. We establish a connection between the behavior of composition operators and the asymptotic properties of the greatest zeros of orthogonal polynomials. Additionally, we examine the compactness of composition operators and prove that bounded composition operators cannot be compact in our situation.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
Fahimeh Arabyani-Neyshaburi, Ali Akbar Arefijamaal
Summary: This paper surveys the topic of weaving Hilbert space frames from the perspective of the duality principle, obtaining new properties and approaches for manufacturing pairs of woven frames. The study provides sufficient conditions under which a frame with its canonical dual, alternate duals, or approximate duals constitute concrete pairs of woven frames, and presents methods for constructing weaving frames using small perturbations. The findings demonstrate that the canonical duals of two woven frames are also woven.
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
(2021)
Article
Optics
Franco Gori, Rosario Martinez-Herrero
Summary: This paper introduces Hilbert spaces with reproducing kernels using mathematical tools such as Fourier optics and coherence theory. It covers basic definitions of such spaces, examples illustrating different forms of inner product, rules for building RKHS, properties of those spaces, eigenfunctions and eigenvalues of the reproducing kernel, integral representation, pseudomodal expansions, generalized forms of sampling, and applications of RKHS in wave optics.
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A-OPTICS IMAGE SCIENCE AND VISION
(2021)
Article
Mathematics, Applied
Danny Ofek, Satish K. Pandey, Orr Moshe Shalit
Summary: This paper explores the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set in which they exist. New variants of distance measures are introduced for quantifying the proximity between these spaces and algebras. The study shows that in finite dimensional quotients of the Drury-Arveson space, the closeness of spaces is reflected in the proximity of their multiplier algebras, and this closeness is linked to the proximity of their underlying point sets through biholomorphic automorphisms of the unit ball.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
K. P. Isaev, R. S. Yulmukhametov
Summary: We describe some radial Fock type spaces that have Riesz bases of normalized reproducing kernels for certain entire functions. The spaces are characterized by radial subharmonic functions, and we prove the existence of Riesz bases under certain conditions.
ANALYSIS AND MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
K. P. Isaev, R. S. Yulmukhametov
Summary: This paper considers the spaces F-phi of entire functions f such that fe(-phi) is an element of L-2(C), where phi(z) = phi(|z|) is a radial subharmonic function with some regularity property. It is proved that F-phi has a Riesz basis of normalized reproducing kernels if and only if (phi(e(r)))'' is bounded above.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Automation & Control Systems
Paul Scharnhorst, Emilio T. Maddalena, Yuning Jiang, Colin N. Jones
Summary: This study considers the problem of estimating out-of-sample bounds for an unknown ground-truth function. The main framework used is kernels and their associated Hilbert spaces, along with an observational model that includes bounded measurement noise. The noise can come from any compactly supported distribution, and no independent assumptions are made about the available data. The study shows how solving parametric quadratically constrained linear programs can compute tight, finite-sample uncertainty bounds. The properties of the approach are established and its relationship with another method is studied through numerical experiments.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2023)
