A Characterization of Sobolev Spaces on the Sphere and an Extension of Stolarsky’s Invariance Principle to Arbitrary Smoothness

In this paper, we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^{d} \subset\mathbb{R}^{d+1}$ . The reproducing kernel is given by an integral representation using the truncated power function $(\mathbf{x} \cdot\mathbf{z} - t)_{+}^{\beta-1}$ supported on spherical caps centered at z of height t, which reduces to an integral over indicator functions of open spherical caps if β=1, as studied in Brauchart and Dick (Proc. Am. Math. Soc. 141(6):2085–2096, 2013). This is analogous to a generalization of the reproducing kernel to arbitrary smoothness on the unit cube by Temlyakov (J. Complex. 19(3):352–391, 2003). We show that the reproducing kernel is a sum of the Euclidean distance ∥x?y∥ of the arguments of the kernel raised to the power of 2β?1 and an adjustment in the form of a Kampé de Fériet function that ensures positivity of the kernel if 2β?1 is not an even integer; otherwise, a limit process introduces logarithmic terms in the distance. For $\beta\in\mathbb{N}$ , the Kampé de Fériet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel. Stolarsky’s invariance principle states that the sum of all mutual distances among N points plus a certain multiple of the spherical cap $\mathbb{L}_{2}$ -discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap $\mathbb{L}_{2}$ -discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over $\mathbb{S}^{d}$ of smoothness (d+1)/2 provided with the reproducing kernel 1?C _d ∥x?y∥ for some constant C _d . Using the new function spaces, we establish an invariance principle for a generalized discrepancy extending the spherical cap $\mathbb{L}_{2}$ -discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over $\mathbb{S}^{d}$ of arbitrary smoothness s=β?1/2+d/2. Previously, Warnock’s formula, which is the analog to Stolarsky’s invariance principle for the unit cube [0,1] ^s , has been generalized using similar techniques in Dick (Ann. Mat. Pura Appl. (4) 187(3):385–403, 2008).     

The Root Operator on Finite Zero Based Invariant Subspaces of the Weighted Bergman Space

For a nonnegative integer α, we study and compute the root functions ${R_{\alpha}^{I}(z, w) = (1-\overline{w}z)^{2+\alpha}K_{\alpha}^{I}(z, w)}$ of finite zero based invariant subspaces I of the weighted Bergman space ${A_{\alpha}^{2}}$ , where ${K_{\alpha}^{I}}$ is the reproducing kernel of I. Furthermore, we estimate ranks of the corresponding root operators.     

Kuelbs–Li Inequalities and Metric Entropy of Convex Hulls

Let T be a precompact subset of a Hilbert space. We make use of a precise link between the absolutely convex hull $\operatorname{aco}(T)$ and the reproducing kernel Hilbert space of a Gaussian random variable constructed from T. Firstly, we avail ourselves of it for optimality considerations concerning the well-known Kuelbs–Li inequalities. Secondly, this enables us to apply small deviation results to the problem of estimating the metric entropy of $\operatorname{aco}(T)$ in dependence of the metric entropy of T.     

The weighted Weiss conjecture and reproducing kernel theses for generalized Hankel operators

The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterized by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space ${H^2(\mathbb{D})}$ (discrete time) or the right-shift semigroup on ${L^2(\mathbb{R}_+)}$ (continuous time). To contrast and complement these counterexamples, in this paper, positive results are presented characterizing weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis.     

Invertible Factorization over Multiplier Algebras

Let ${\mathcal{A}}$ denote the multiplier algebra of an E-valued reproducing kernel Hilbert space, ${H_E^2(k)}$ . Then when H ²(k) is nice, we give necessary and sufficient conditions that T > 0 factors as A*A, where A and ${A^{-1} \in \mathcal{A}}$ . Such nice spaces include the Bergman and Hardy spaces on the unit polydisk and unit ball in ${\mathbb{C}^d}$ .     

A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space

(A) The celebrated Gaussian quadrature formula on finite intervals tells us that the Gauss nodes are the zeros of the unique solution of an extremal problem. We announce recent results of Damelin, Grabner, Levesley, Ragozin and Sun which derive quadrature estimates on compact, homogenous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets. (B) Given $\mathcal{X}$ , some measurable subset of Euclidean space, one sometimes wants to construct, a design, a finite set of points, $\mathcal{P} \subset \mathcal{X}$ , with a small energy or discrepancy. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that these two measures of quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^2(=\mathcal{X}\times\mathcal{X}) \to \mathbb{R}$ . The error of approximating the integral $\int_{\mathcal{X}} f(\mathbf{\mathit{x}}) \, {\rm d} \mu(\mathbf{\mathit{x}})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms the energy or discrepancy of $\mathcal{P}$ . The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. (C) Let $\mathcal{X}$ be the orbit of a compact, possibly non Abelian group, $\mathcal{G}$ , acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$ . This allows us to calculate explicit representations of equilibrium measures. There is an extensive literature on the topics (A–C). We emphasize that this paper surveys recent work of Damelin, Grabner, Levesley, Hickernell, Ragozin, Sun and Zeng and does not mean to serve as a comprehensive survey of all recent work covered by the topics (A–C).     

A Family of Measures of Noncompactness in the Space {L^1_{\text{loc}}(\mathbb{R}_+)} and its Application to Some Nonlinear Volterra Integral Equation

The aim of this paper is to study a new family of measures of noncompactness in the space ${L^1_{\text{loc}}(\mathbb{R}_+)}$ consisting of all real functions locally integrable on ${\mathbb{R}_+}$ , equipped with a suitable topology. As an example of applications of the technique associated with that family of measures of noncompactness, we study the existence of solutions of a nonlinear Volterra integral equation in the space ${L^1_{\text{loc}}(\mathbb{R}_+)}$ . The obtained result generalizes several ones obtained earlier with help of other methods.     

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\) , we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\) -matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szegö kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.     

Inequalities and exponential stability and instability in finite delay Volterra integro-differential equations

We use Liapunov functionals to obtain sufficient conditions that ensure exponential stability of the nonlinear Volterra integro-differential equation $$x^{\prime }(t)=p(t)x(t)-\int \limits _{t-\tau }^{t}q(t,s)x(s)ds,$$ where the constant $\tau $ is positive, the function $p$ does not need to obey any sign condition and the kernel $q$ is continuous. Our results improve the results obtained in literature even in the autonomous case. In addition, we give a new criteria for instability.