Name
Abstract | ||
|---|---|---|
The purpose of this paper is to construct universal, auto-adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper) sphere $\mathbb{S}^q$ ($q\ge2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well-known formulas as Driscoll-Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178. |
Year | DOI | Venue |
|---|---|---|
2008 | 10.1137/060678555 | SIAM J. Numerical Analysis |
Keywords | Field | DocType |
high degree,essential ingredient,learning theory on the sphere. ams classification: 65d32,spherical polynomial,driscoll-healy formula,quadrature formulas,numerical experiment,41a10,polynomial quasi-interpolation,41a25,particular manner,scattered data,scattered site,localized kernels,discrete fourier projection polynomial,localization property,localized linear polynomial operators,learning theory,numerical analysis,least square | Least squares,Mathematical optimization,Polynomial,Polynomial interpolation,Mathematical analysis,Vieta's formulas,Linear map,Operator (computer programming),Quadrature (mathematics),Numerical analysis,Mathematics | Journal |
Volume | Issue | ISSN |
47 | 1 | 0036-1429 |
Citations | PageRank | References |
20 | 1.07 | 13 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Q. T. Le Gia | 1 | 93 | 12.64 |
| Hrushikesh Narhar Mhaskar | 2 | 257 | 61.07 |