Abstract
In this chapter, we present a systematic digital-discrete method for obtaining continuous functions with smoothness of a certain order (C n) from randomly arranged data points. The new method is based on gradually varied functions and the classical finite difference method. This method is independent from existing popular methods such as the cubic spline method and the finite element method. The new digital-discrete method has considerable advantages for a large number of real data applications. This digital method also differs from other classical discrete methods that usually use triangulation.
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Acknowledgements
This research has been partially supported by the USGS Seed Grants through the UDC Water Resources Research Institute (WRRI) and Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University. Professor Feng Luo suggested the direction of the relationship between harmonic functions and gradually varied functions. Dr. Yong Liu provided many helps in PDE. UDC undergraduate Travis Branham extracted the application data from the USGS database. Professor Thomas Funkhouser provided helps on the 3D data sets and OpenGL display programs. The author would also like to thank Professor C. Fefferman and Professor N. Zobin for their invitation to the Workshop on the Whitney’s Problem in 2009.
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Chen, L.M. (2013). Digital-Discrete Approaches for Smooth Functions. In: Digital Functions and Data Reconstruction. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5638-4_7
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DOI: https://doi.org/10.1007/978-1-4614-5638-4_7
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