Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer’s theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.     

Kernels of L-functions of cusp forms

We give a new expression for the inner product of two kernel functions associated to a cusp form. Among other applications, it yields an extension of a formula of Kohnen and Zagier, and another proof of Manin’s Periods Theorem. Cohen’s representation of these kernels as series is also generalized.     

Potential inequality

We prove a general inequality for kernels satisfying the maximum principle. This is then used to derive a sufficient condition for the kernel to define a continuous map of Lebesgue spaces. Exactly this condition happens to be necessary and sufficient for the validity of Hardy’s inequality with weights in one dimension. Some applications indicating the unifying nature of the potential inequality are given. A part of this paper was presented at the 865th AMS meeting, March 1991, Tampa, Florida.     

Quasi-compactness and absolutely continuous kernels

We show how the essential spectral radius r _e (Q) of a bounded positive kernel Q, acting on bounded functions, is linked to the lower approximation of Q by certain absolutely continuous kernels. The standart Doeblin’s condition can be interpreted in this context, and, when suitably reformulated, it leads to a formula for r _e (Q). This results may be used to characterize the Markov kernels having a quasi-compact action on a space of measurable functions bounded with respect to some test function, when no irreducibilty and aperiodicity are assumed.      

The probabilistic representation formulae for cosine operator functions

In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach, which include Webb’s [1] and Shaw’s [2] formulae and some new one as special cases. We also give the quantitative estimations for the general formulae.     

Locally stationary stochastic processes and Weyl symbols of positive operators

The paper treats locally stationary stochastic processes. A connection with the Weyl symbols of positive operators is observed and explored. We derive necessary conditions on the two functions that constitute the covariance function of a locally stationary stochastic process, some of which use this connection to time-frequency analysis and pseudodifferential operators. Finally, we discuss briefly the subclass of Cohen’s class of time–frequency representations having separable kernels, which is related to locally stationary stochastic processes.     

Mode testing via higher-order density estimation

We investigate the use of fourth-order (reduced bias) variants on kernel density estimation in Silverman’s (J Roy Stat Soc B 43:97–99, 1981) critical bandwidth test of multimodality. Simulations suggest that variable-bandwidth estimates in the style of Abramson (Ann Stat 10:1217–1223, 1982) appear to be especially favorable to mode testing. Conversely, the use of fourth-order kernels leads to extremely poor properties and should be strongly discouraged.     

Interior Capacities of Condensers in Locally Compact Spaces

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a finite collection of arbitrary sets with sign + 1 or − 1 prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in \mathbb Rⁿ\mathbb R^n and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying Fuglede’s condition of consistency between the strong and vague (= weak*) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We describe those solutions and the corresponding equilibrium constants, analyze their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in \mathbb Rⁿ\mathbb R^n, which is important in applications.     

Initiation of Cracks in Griffith’s Theory: An Argument of Continuity in Favor of Global Minimization

The initiation of a crack in a sound body is a real issue in the setting of Griffith’s theory of brittle fracture. If one uses the concept of critical energy release rate (Griffith’s criterion), it is in general impossible to initiate a crack. On the other hand, if we replace it by a least energy principle (Francfort–Marigo’s criterion), it becomes possible to predict the onset of cracking in any circumstance. However this latter criterion can appear too strong. We propose here to reinforce its interest by an argument of continuity. Specifically, we consider the issue of the initiation of a crack at a notch whose angle ω is considered as a parameter. The result predicted by the Griffith criterion is not continuous with respect to ω, since no initiation occurs when ω>0 while a crack initiates when ω=0. In contrast, the Francfort–Marigo’s criterion delivers a response which is continuous with respect to ω, even though the onset of cracking is necessarily brutal when ω>0. The theoretical analysis is illustrated by numerical computations.     

Continuous tree-like scales

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵ _ω is consistent with Martin’s Maximum.     

Embedding problems with cyclic kernel of order 4

We consider Galois theoretical embedding problems with kernelC ₄, and prove that such an embedding problem can be ‘constructively’ reduced to two embedding problems, where the kernels are groups of roots of unity.     

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

We show how ‘test’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in terms of the inradius of a plane domain. Received: 13 June 2005     

A remark on absolutely continuous functions in ℝ n

We introduce the notion ofα, λ-absolute continuity for functions of several variables and we compare it with the Hencl’s definition. We obtain that eachα, λ-absolutely continuous function isn, λ-absolutely continuous in the sense of Hencl and hence is continuous, differentiable almost everywhere and satisfies change of variables results based on a coarea formula and an area formula.     

Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind

Integral equations of first kind with periodic kernels arising in solving partial differential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential.     

Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain \mathbbRⁿ{\mathbb{R}^n} with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun’s extension theorem for Lipschitz functions and, therefore, establishes a link between Kirszbraun’s theorem and Whitney’s theorem. In fact this result holds not only in Euclidean \mathbbRⁿ{\mathbb{R}^n} but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable extension problem in Euclidean spaces and we show that no Brudnyi–Shvartsman-type theorem holds for this last problem. We conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was originally studied by Aronsson in the continuous case.     

Spectral representation for generalized operator-valued Toeplitz kernels

We prove an integral representation for operator-valued Toeplitz kernels. The proof is based on the spectral theory of the corresponding differential operator constructed from this kernel and acting in a Hilbert space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1698–1710, December, 2005.     

Fast computation of spectral centroids

The spectral centroid of a signal is the curve whose value at any given time is the centroid of the corresponding constant-time cross section of the signal’s spectrogram. A spectral centroid provides a noise-robust estimate of how the dominant frequency of a signal changes over time. As such, spectral centroids are an increasingly popular tool in several signal processing applications, such as speech processing. We provide a new, fast and accurate algorithm for the real-time computation of the spectral centroid of a discrete-time signal. In particular, by exploiting discrete Fourier transforms, we show how one can compute the spectral centroid of a signal without ever needing to explicitly compute the signal’s spectrogram. We then apply spectral centroids to an emerging biometrics problem: to determine a person’s heart and breath rates by measuring the Doppler shifts their body movements induce in a continuous wave radar signal. We apply our algorithm to real-world radar data, obtaining heart- and breath-rate estimates that compare well against ground truth.     

Factorisation of positive definite operators

In this paper we prove Reade’s result for the positive definite C ¹ kernels by using the factorisation method used by Kühn. Received: 10 April 2007, Revised: 3 December 2007     

A strong form of almost differentiability

We present a uniformization of Reeken’s macroscopic differentiability (see [5]), discuss its relations to uniform differentiability (see [6]) and classical continuous differentiability, prove the corresponding chain rule, Taylor’s theorem, mean value theorem, and inverse mapping theorem. An attempt to compare it with the observability (see [1, 4]) is made too.