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Zonal function network frames on the sphere.

H N Mhaskar1, F J Narcowich, J D Ward

  • 1Department of Mathematics, California State University, Los Angeles, CA 90032, USA. hmhaska@calstatela.edu

Neural Networks : the Official Journal of the International Neural Network Society
|March 12, 2003
PubMed
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We developed new zonal function network frames for analyzing scattered spherical data. These frames offer improved localization and increasing polynomial moments for higher resolution analysis.

Area of Science:

  • Mathematical analysis
  • Spherical data analysis

Background:

  • Analyzing data on the unit sphere presents challenges due to scattered sampling.
  • Existing methods may lack optimal localization or adaptability for varying scales.

Purpose of the Study:

  • Introduce a novel class of zonal function network frames.
  • Enhance the analysis of data collected at scattered sites on the unit sphere.
  • Investigate frame properties related to scale and localization.

Main Methods:

  • Construction of frames using zonal function networks.
  • Analysis of frame localization properties.
  • Exploration of vanishing polynomial moments in wavelet spaces.
  • Application within separable Hilbert spaces.

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Main Results:

  • Developed well-localized frames suitable for spherical data.
  • Demonstrated that frames in higher scale wavelet spaces possess more vanishing polynomial moments.
  • Established a general technique applicable in separable Hilbert spaces.

Conclusions:

  • The proposed zonal function network frames provide an effective tool for spherical data analysis.
  • The frames offer tunable properties for improved resolution and localization.
  • The methodology extends to general separable Hilbert spaces.