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To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert–Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.

The standard Sobolev space $W_2^s({ℝ}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel

We revise the use of eight-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the eight-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures $\mathrm{\Xi }$ on a space S with measure $\lambda$, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, $H_{\ell },\ell =1,2$, which are assumed to be realized as $L^2$-spaces, $L^2(S_{\ell },{\lambda }_{\ell })$, $\ell =1,2$, and introduce a bounded linear operator ${𝒲}:H_1\to H_2$ and its adjoint ${{𝒲}}^{\ast }:H_2\to H_1$. We show that if ${𝒲}$ is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP $({\mathrm{\Xi }}_1,K_1,{\lambda }_1)$ associated with ${{𝒲}}^{\ast }{𝒲}$. In addition, if ${{𝒲}}^{\ast }$ is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, $({\mathrm{\Xi }}_{\ell },K_{\ell },{\lambda }_{\ell })$, $\ell =1,2$. We also give a practical framework which makes ${𝒲}$ and ${{𝒲}}^{\ast }$ satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ($d\in \mathbb{N}$) series of infinite DPPs on $S={ℝ}^d$ and ${ℂ}^d$ are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

Since the first works laying its foundations as a subfield of Complex Analysis, the theory of reproducing kernels has proved to be a powerful tool in many fields of Pure and Applied Mathematics. The aim of this paper is to give some idea of how and why this theory interacts with Probability and Statistics.

We present a survey of recent results by the authors which show that reproducing kernels enjoy a closely knit interplay of algebraic, differential, integral and spectral properties (whenever all are defined). The matrix characterization of reproducing kernels implies a family of differential diagonal dominance inequalities, which in turn acts as the appropriate integrability condition when these are kernels of L² integral operators. This fact allows complete and optimal determination of eigenvalue asymptotics.

The constructive recovery of functions in multi-window spline-type spaces from irregular samples are treated, using the concepts of frames and reproducing kernels. It is well known that one can describe iterative methods, which guarantee stable reconstruction, convergent in a variety of functions spaces, at least if the sampling set is dense enough. The corresponding theoretical statements either work in the context of continuous variables or discrete signals, but rarely consider the effect of errors that unavoidably occur if the continuous problem is implemented in a discrete (finite) setting. The analysis of the situation, using function spaces methods (in particular Wiener amalgams) and the description of implementable variants of these iterative algorithms with guaranteed rates of convergence are treated in the necessary detail here. It relies on the results of an earlier paper by the authors, demonstrating the fact that one can guarantee a good approximation of the biorthogonal family for the set of generators of the space. It is important to note, that the imperfections due to discretization do not spoil the chance of perfect reconstruction (in the limit), but only lead to a mild degradation of the rate of convergence, compared to the ideal case.