Numerical Algorithms

, Volume 78, Issue 2, pp 661–672 | Cite as

Remarkable Haar spaces of multivariate piecewise polynomials

Original Paper


Some families of Haar spaces in \(\mathbb {R}^{d},~ d\ge 1,\) whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis.


Scattered data interpolation Lobachevsky splines Gaussians Positive definite functions 

Mathematics subject classification 2010

65D05 65D07 42A82 


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The author is very grateful to the anonymous referee for many accurate and helpful comments on a draft of this note.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics “G. Peano”University of TurinTurinItaly

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