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Hilbert Space Techniques
The goal of this chapter is to collect the main features of the Hilbert spaces and to introduce the Lax–Milgram theorem which is a key tool for...
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Translation-invariant Operators in Reproducing Kernel Hilbert Spaces
Let G be a locally compact abelian group with a Haar measure, and Y be a measure space. Suppose that H is a reproducing kernel Hilbert space of...
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On the Bergman Projection and Kernel in Periodic Planar Domains
We study Bergman kernels \(K_\Pi \) and projections...
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Quantum space, quantum time, and relativistic quantum mechanics
We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity,...
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From Kernel Methods to Neural Networks: A Unifying Variational Formulation
The minimization of a data-fidelity term and an additive regularization functional gives rise to a powerful framework for supervised learning. In...
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Optimal Recovery from Inaccurate Data in Hilbert Spaces: Regularize, But What of the Parameter?
In Optimal Recovery, the task of learning a function from observational data is tackled deterministically by adopting a worst-case perspective tied...
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A universal formula for derivation operators and applications
We give a universal formula describing derivation operators on a Hilbert space for a large class of interpolation methods. It is based on a simple...
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Gaussian RBF kernels via Fock spaces: quaternionic and several complex variables settings
In this paper, we study two extensions of the complex-valued Gaussian radial basis function (RBF) kernel and discuss their connections with Fock...
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Octonionic Kerzman–Stein Operators
In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field...
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A Note on the Completeness of Generalized Eigenfunctions of the Hamiltonian of Reggeon Field Theory in Bargmann Space
The aim of the present paper is to review some old results of Intissar (Commun Math Phys 113:263–297, 1987) and to study some new spectral properties...
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Moment Estimates of the Cloud of a Planar Measure
With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of...
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Singular integral operators
The map** properties of T will of course heavily depend on the assumptions made on the kernel K that we will discuss in more detail in this chapter.
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Szegő and Widom Theorems for Finite Codimensional Subalgebras of a Class of Uniform Algebras
We establish versions of Szegő’s distance formula and Widom’s theorem on invertibility of (a family of) Toeplitz operators in a class of finite...
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Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results
We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and...
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Stochastic saddle-point optimization for the Wasserstein barycenter problem
We consider the population Wasserstein barycenter problem for random probability measures supported on a finite set of points and generated by an...
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Positive matrices in the Hardy space with prescribed boundary representations via the Kaczmarz algorithm
For a singular probability measure μ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation...
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Stable interpolation with exponential-polynomial splines and node selection via greedy algorithms
In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial...
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Impulsive Dirac System on Time Scales
We consider an impulsive Dirac system on Sturmian time scales and present an existence theorem for this system. Maximal, minimal, and self-adjoint...