Small Support Spline Riesz Wavelets in Low Dimensions

HAN Bin; MO Qun; SHEN Zuowei

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Small Support Spline Riesz Wavelets in Low Dimensions

HAN Bin 1,MO Qun 2 * #,SHEN Zuowei 3

1.DepartmentofMathematicalandStatisticalSciences,UniversityofAlberta

2.DepartmentofMathematics,ZhejiangUniversity

3.DepartmentofMathematics,NationalUniversityofSingapore

*Correspondence author

#Submitted by

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Funding: 国家自然科学基金（No.10971189）, 新加坡国立大学科研基金（No.11071）, 博士点新教师基金（No.20070335176）, 加拿大NSERC基金（No.228051）, 中央高校基本科研业务费专项基金（No.2010QNA3018）

Opened online:18 February 2011

Accepted by: none

Citation: HAN Bin,MO Qun,SHEN Zuowei.Small Support Spline Riesz Wavelets in Low Dimensions[OL]. [18 February 2011] http://en.paper.edu.cn/en_releasepaper/content/4408648

B.~Han and Z.~Shen constructed afamily of univariate short support Riesz waveletsfrom uniform B-splines in 2006. A bivariate spline Riesz wavelet basis fromthe Loop scheme was derived by B.~Han and Z.~Shen in 2005.Motivated bythese two papers, we develop in this article ageneral theory and a construction method to derive small supportRiesz wavelets in low dimensions from refinable functions. Inparticular, we obtain small support spline Riesz wavelets frombivariate and trivariate box splines. Small support Riesz waveletsare desirable for developing efficient algorithms in variousapplications. For example, the short support Riesz wavelets constructed byB.~Han and Z.~Shen were used in a surface fitting algorithm byM.J.~Johnson, Z.~Shen and Y.H.~Xu in 2009, and the Riesz wavelet basis from the Loop scheme wasused in a very efficient geometric mesh compression algorithm byA. Khodakovsky, P. Schr"oder and W. Sweldens in 2000.

Keywords:Function theory, Riesz wavelet bases, box splines, linear independence, Sobolev spaces

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