caa a22 4500 605496862 CHVBK 20210128100538.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s10543-014-0522-0 doi (NATIONALLICENCE)springer-10.1007/s10543-014-0522-0 Levin David School of Mathematical Sciences, Tel-Aviv University, 69978, Tel Aviv, Israel aut Between moving least-squares and moving least- $$\ell _1$$ ℓ 1 [Elektronische Daten] [David Levin] Given function values at scattered points in $${\mathbb {R}}^d$$ R d , possibly with noise, one of the ways of generating approximation to the function is by the method of moving least-squares (MLS). The method consists of computing local polynomials which approximate the data in a locally weighted least-squares sense. The resulting approximation is smooth, and is well approximating if the underlying function is smooth. Yet, as any least-squares based method it is quite sensitive to outliers in the data. It is well known that least- $$\ell _1$$ ℓ 1 approximations are not sensitive to outliers. However, due to the nature of the $$\ell _1$$ ℓ 1 norm, using it in the framework of a "moving” approximation will not give a smooth, or even a continuous approximation. This paper suggests an error measure which is between the $$\ell _1$$ ℓ 1 and the $$\ell _2$$ ℓ 2 norms, with the advantages of both. Namely, yielding smooth approximations which are not too sensitive to outliers. A fast iterative method for computing the approximation is demonstrated and analyzed. It is shown that for a scattered data taken from a smooth function, with few outliers, the new approximation gives an $$O(h)$$ O ( h ) approximation error to the function. Springer Science+Business Media Dordrecht, 2014 Moving least-squares nationallicence Outliers nationallicence Multivariate approximation nationallicence BIT Numerical Mathematics Springer Netherlands 55/3(2015-09-01), 781-796 0006-3835 55:3<781 2015 55 10543 https://doi.org/10.1007/s10543-014-0522-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10543-014-0522-0 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Levin David School of Mathematical Sciences, Tel-Aviv University, 69978, Tel Aviv, Israel aut NATIONALLICENCE 773 0- BIT Numerical Mathematics Springer Netherlands 55/3(2015-09-01), 781-796 0006-3835 55:3<781 2015 55 10543