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Remarkable Haar spaces of multivariate piecewise polynomials. (English) Zbl 1391.65030

Summary: Some families of Haar spaces in \(\mathbb {R}^{d}\), \(d\geq 1\), 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.

MSC:

65D05 Numerical interpolation
65D07 Numerical computation using splines
42A82 Positive definite functions in one variable harmonic analysis

Software:

Matlab; GaussQR
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Full Text: DOI

References:

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