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V. A. Menegatto

Publications and source records attributed to V. A. Menegatto.

6 recordsLinked to original sources
arXiv2021

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$.\ The method is consistent with a generalized version of Aitken's integral formula for Gaussians.\ In the case where $X$ is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation.\ In addition, it can be interpreted as an abstract multivariate generalization of the well-established Gneiting's model for constructing space-time covariances commonly cited in the literature.\ Many parametric models discussed in statistics can be interpreted as particular cases of the method.

math.FA↗
arXiv2016

Strict positive definiteness on a product of compact two-point homogeneous spaces

We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of dimension greater than 1 and the other is not a circle. The result complements similar characterizations previously obtained for products of high dimensional spheres.

math.FA↗
arXiv2015

Strictly positive definite kernels on two-point compact homogeneous spaces

We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously obtained by D. Chen at all (2003) in the case in which the space is a sphere of dimension at least 2 and Menegatto at all (2006) in the case in which the space is the unit circle. As an application, we use the characterization to improve upon a recent result on the differentiability of positive definite kernels on the spaces.

math.FA↗
arXiv2015

Differentiable positive definite functions on two-point homogeneous spaces

In this paper we study continuous kernels on compact two point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the $d$-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order $\lfloor (d-1)/2 \rfloor$ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two point homogeneous spaces.

math.CA↗
arXiv2015

An extension of a theorem of Schoenberg to products of spheres

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.

math.CA↗