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 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.     

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).     

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.     

Convergence to steady states of solutions to nonlinear integral evolution equations

We prove that any global bounded solution of the nonlinear evolutionary integral equation $$\dot{u}(t) + \int\limits_0^t a(t-s)\mathcal{E}'(u(s))ds =f(t), \quad t >0 $$ tends to a single equilibrium state for long time (i.e., ${\mathcal{E}'(\vartheta)=0}$ where ${\vartheta= \lim_{t \rightarrow \infty} u(t)}$ on a real Hilbert space), where ${\mathcal{E}'}$ is the Fréchet derivative of a functional ${\mathcal{E}}$ , which satisfies the ?ojasiewicz?CSimon inequality near ${\vartheta}$ . The vector-valued function f and the scalar kernel a satisfy suitable conditions.     

Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ and Solutions of the Helmholtz Equation

The purpose of this article is to extend to \(\mathbb {R}^{n}\) known results in dimension 2 concerning the structure of a Hilbert space with reproducing kernel of the space of Herglotz wave functions. These functions are the solutions of Helmholtz equation in \(\mathbb {R} ^{n}\) that are the Fourier transform of measures supported in the unit sphere with density in \(L^{2}(\mathbb {S}^{n-1})\). As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in \(\mathbb {R}^{n}\) belonging to weighted Sobolev type spaces \(\mathcal {H}^{p}\) having in a non local norm that involves radial derivatives and spherical gradients. We calculate the reproducing kernel of the Herglotz wave functions and study in \(\mathcal {H}^{p}\) and in mixed norm spaces, the continuity of the orthogonal projection \(\mathcal {P}\) of \(\mathcal {H}^{2}\) onto the Herglotz wave functions.     

The subelliptic heat kernel on the CR sphere

We study the heat kernel of the sub-Laplacian $L$ on the CR sphere $\mathbb{S }^{2n+1}$ . An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian $-L+n^2$ that was obtained by Geller (J Differ Geom 15:417–435, 1980), and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration $\mathbb{S }^{2n+1} \rightarrow \mathbb{CP }^n$ .     

Lattice rules for nonperiodic smooth integrands

The aim of this paper is to show that one can achieve convergence rates of $N^{-\alpha + \delta }$ for $\alpha > 1/2$ (and for $\delta > 0$ arbitrarily small) for nonperiodic $\alpha $ -smooth cosine series using lattice rules without random shifting. The smoothness of the functions can be measured by the decay rate of the cosine coefficients. For a specific choice of the parameters the cosine series space coincides with the unanchored Sobolev space of smoothness 1. We study the embeddings of various reproducing kernel Hilbert spaces and numerical integration in the cosine series function space and show that by applying the so-called tent transformation to a lattice rule one can achieve the (almost) optimal rate of convergence of the integration error. The same holds true for symmetrized lattice rules for the tensor product of the direct sum of the Korobov space and cosine series space, but with a stronger dependence on the dimension in this case.     

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form \(k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}\), and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H _d ² in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H _d ² . Thus, a family of multiplier algebras of Dirichlet series is exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H _d ² and when its multiplier algebra is isometrically isomorphic to Mult(H _d ² ).     

Basic Relations Valid for the Bernstein Space B^{p}_{\sigma} and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from B^{p}_{\sigma}

There are various basic relations (equations and inequalities) that hold in Bernstein spaces $B_{\sigma}^{p}$ but are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, one may expect that the corresponding relation is not violated drastically. It should hold with a remainder that involves the distance of f from $B_{\sigma}^{p}$ . First we establish a hierarchy of spaces that generalize the Bernstein spaces and are suitable for our studies. It includes Hardy spaces, Sobolev spaces, Lipschitz classes and Fourier inversion classes. Next we introduce an appropriate metric for describing the distance of a function belonging to such a space from a Bernstein space. It will be used for estimating remainders and studying rates of convergence. In the main part, we present the desired extensions. Our considerations include the classical sampling formula by Whittaker-Kotel’nikov-Shannon, the sampling formula of Valiron-Tschakaloff, the differentiation formula of Boas, the reproducing kernel formula, the general Parseval formula, Bernstein’s inequality for the derivative and Nikol’ski?’s inequality estimating the $l^{p}(\mathbb{Z})$ norm in terms of the $L^{p}(\mathbb{R})$ norm. All the remainders are represented in terms of the Fourier transform of f and estimated from above by the new metric. Finally we show that the remainders can be continued to spaces where a Fourier transform need not exist and can be estimated in terms of the regularity of f.     

A class of \ell ^p saturated Banach spaces

We present a new class of reflexive \(\ell ^p\) saturated Banach spaces \(\mathfrak{X }_p\) for \(1<p<\infty \) with rather tight structure. The norms of these spaces are defined with the use of a modification of the standard method yielding hereditarily indecomposable Banach spaces. The space \(\mathfrak{X }_p\) does not embed into a space with an unconditional basis and for any analytic decomposition into two subspaces, it is proved that one of them embeds isomorphically into the \(\ell ^p\) -sum of a sequence of finite dimensional normed spaces. We also study the space of operators of \(\mathfrak{X }_p\) .     

Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term

In this paper we propose primal-dual interior-point algorithms for semidefinite optimization problems based on a new kernel function with a trigonometric barrier term. We show that the iteration bounds are $O(\sqrt{n}\log(\frac{n}{\epsilon}))$ for small-update methods and $O(n^{\frac{3}{4}}\log(\frac{n}{\epsilon}))$ for large-update, respectively. The resulting bound is better than the classical kernel function. For small-update, the iteration complexity is the best known bound for such methods.