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A class of blending functions with \(C^{\infty }\) smoothness

Abstract

In this work, by combining a class of local support and infinitely differentiable functions together with the sinc function, we construct a new class of univariate blending functions with three local shape parameters αi, βi, and λi. The new blending functions have the properties of \(C^{\infty }\) smoothness, compact support, and partition of unity. The shape parameter αi has tension property, and βi can adjust the support of the blending functions. With λi, the given blending functions can be used to interpolate sets of points partly or entirely without solving a linear system of equations. Some simple conditions for the blending functions possessing nonnegativity and/or linear independence are developed. Based on the new univariate blending functions, tensor product blending functions and local tensor product blending functions are also developed.

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Acknowledgments

We wish to express our gratitude to the referees for their valuable remarks for improvements.

Funding

The research is supported by the National Natural Science Foundation of China (No. 61802129), and the Natural Science Foundation Guangdong Province, China (No. 2018A030310381).

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Correspondence to Yuanpeng Zhu.

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Zhu, Y. A class of blending functions with \(C^{\infty }\) smoothness. Numer Algor 88, 555–582 (2021). https://doi.org/10.1007/s11075-020-01049-7

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Keywords

  • Blending function
  • Smoothness
  • Shape parameter
  • Interpolation
  • Approximation