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<p>The continuity and differentiability of positive semidefinite kernels depending on a given norm has been studied to a limited extent, mainly in the case of radially symmetric scalar-valued kernels in Euclidean spaces. This paper considers matrix-valued measurable kernels defined over a variety of domains and depending on different norms. The measurability assumption relaxes that of continuity, which has been customarily used in earlier literature from the 80s. We extend the standing literature and show that, for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d&gt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, a radial measurable kernel in a <i>d</i>-dimensional Euclidean space, open ball, or sphere, is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lfloor \frac{d-1}{2} \rfloor \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mfrac> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation>-times continuously differentiable in the interior of its domain of definition. We further generalize this result to componentwise isotropic kernels defined in products of Euclidean spaces and/or spheres.</p>

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On the continuity and differentiability of matrix-valued covariance kernels depending on Euclidean or spherical distances

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Xavier Emery,
  • Emilio Porcu

摘要

The continuity and differentiability of positive semidefinite kernels depending on a given norm has been studied to a limited extent, mainly in the case of radially symmetric scalar-valued kernels in Euclidean spaces. This paper considers matrix-valued measurable kernels defined over a variety of domains and depending on different norms. The measurability assumption relaxes that of continuity, which has been customarily used in earlier literature from the 80s. We extend the standing literature and show that, for \(d>1\) d > 1 , a radial measurable kernel in a d-dimensional Euclidean space, open ball, or sphere, is \(\lfloor \frac{d-1}{2} \rfloor \) ⌊ d - 1 2 ⌋ -times continuously differentiable in the interior of its domain of definition. We further generalize this result to componentwise isotropic kernels defined in products of Euclidean spaces and/or spheres.